An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

AN ALLEN–CAHN EQUA TION WITH JUMP-DIFFUSION NOISE F OR BIOLOGICAL D AMA GE AND REP AIR PR OCESSES ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MAR GHERIT A ZANELLA Abstra ct. This paper analyzes a sto c hastic Allen–Cahn equation for the dynamics of biomolecular dam- age and repair. The system is driv en by tw o distinct noise processes: a multiplicativ e cylindrical Wiener pro cess, mo deling con tinuous background stochastic ﬂuctuations, and a jump-t ype noise, modeling the abrupt, lo calized damage induced by external shocks. The drift of the equation is singular and cov ers the t ypical logarithmic Flory-Huggins p oten tial required in phase-separation dynamics. W e prov e w ell- p osedness of the mo del in a strong probabilistic sense, and analyze its long-time b ehavior in terms of existence and uniqueness of inv ariant measures, ergo dicit y , and mixing prop erties. Even tually , w e present an Euler–Maruyama sc heme to simulate the mo del and illustrate how it captures fundamental biological phenomena, suc h as damage clustering, stress-induced top ology perturbations, and damage dynamics. 1. Intr oduction The in tegrit y of DNA and cellular tissues is constan tly challenged b y in ternal and external stressors. Understanding ho w damage arises and ho w repair pro ceeds is a cen tral problem in bioph ysics and medicine. Man y relev ant incidents are neither smooth nor con tin uous: double-strand breaks, replication errors, c hemically induced lesions, or other lo calized failures can o ccur abruptly and unpredictably , and ma y cluster in space. A t the same time, bac kground v ariability (metab olic noise, reactive oxygen sp ecies, heterogeneous repair activity) acts persistently and on smaller scales. A realistic mesoscopic mo del should therefore com bine (i) bistable relaxation b et w een “healthy” and “damaged” states, (ii) diﬀusion-like repair co operation, and (iii) the coexistence of con tin uous ﬂuctuations with discrete, catastrophic damage hits. The Allen–Cahn equation, originally dev eloped to mo del phase separation (see [ 1 ]), provides a natural framew ork for describing such bistable systems. Its deterministic dynamics captures the ev olution to w ards one of t w o stable states (namely , “healthy” or “damaged”) with the diﬀusion term represen ting spatial repair in teractions. Sto chastic v arian ts of the Allen–Cahn equation t ypically take into accoun t only contin uous p erturbations by Gaussian noise. How ev er, accurately mo deling biological damage requires incorp orating also discrete, abrupt sho c ks. In this work, w e introduce and analyze a sto c hastic Allen–Cahn mo del with b oth multiplicativ e Wiener and jump noise terms. The mo del reads        d u − ε ∆ u d t + Φ ′ ( u ) d t = G ( u ) d W + Z Z J ( u − , z ) µ (d t, d z ) in (0 , T ) × O , α d u + α n ∂ n u = 0 on (0 , T ) × ∂ O , u ( · , 0) = u 0 in O , where u ( t, x ) denotes the local damage level at p osition x ∈ O and time t ≥ 0 , and the co eﬃcien ts α d , α n ∈ { 0 , 1 } with α d + α n = 1 allo w either Dirichlet or Neumann homogeneous b oundary conditions. The classical diﬀuse-in terface persp ectiv e of the Allen-Cahn mo deling requires u to b e conﬁned in the ph ysically-relev ant interv al [0 , 1] , with 0 and 1 represen ting the so-called pure phases, with the con v ention u = 0 (health y) , u = 1 (fully damaged) . Instead, in termediate v alues of u describ e sublethal, p oten tially repairable states: this is consistent with evidence that damage accumulates gradually rather than in an all-or-nothing manner [ 15 ]. Mathematically , 2020 Mathematics Subje ct Classiﬁc ation. 35Q35, 35R60, 60H15, 65C30, 76T06. Key words and phr ases. Sto c hastic Allen–Cahn equations; jump-diﬀusion noise; well-posedness; inv ariant measures; nu- merical sim ulations. 1 2 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA this is ensured by a suitable choice of the p oten tial Φ , which enforces barriers at the endp oin ts. The t ypical c hoice for Φ required in thermo dynamics is the well-kno w double-well Flory-Huggins p oten tial [ 21 , 29 ] Φ( u ) = θ  u ln u + (1 − u ) ln(1 − u )  − θ 0  u − 1 2  2 , u ∈ [0 , 1] . (1.1) The Allen-Cahn dynamics results then as a gradient ﬂow with resp ect to the L 2 metric of the free-energy u 7→ ε 2 Z O |∇ u ( x ) | 2 d x + Z O Φ( u ( x )) d x. where ε > 0 controls the thickness of interfacial lay ers separating healthy and damaged regions. F rom the mo deling point of view, the diﬀuse-interface dynamics capture reco v ery and cell-fate transitions, diﬀusion- driv en repair co op eration, and constrained bistable dynamics. In our analysis, the mo del will b e allow ed to b e ev en more general, in the form        d u − ε ∆ u d t + ∂ Ψ( u ) d t + F ( u ) d t ∋ G ( u ) d W + Z Z J ( u − , z ) µ (d t, d z ) in (0 , T ) × O , α d u + α n ∂ n u = 0 on (0 , T ) × ∂ O , u ( · , 0) = u 0 in O . (1.2) The deriv ative of the double-well p oten tial is replaced b y the m ulti-v alued sub-diﬀerential op erator ∂ Ψ + F , where F is a Lipschitz-con tin uous function and Ψ is a prop er, con vex, low er semicontin uous function on R with In t D (Ψ) = (0 , L ) , and L ∈ (0 , ∞ ] . The case L < + ∞ mo del double-barrier potentials while L = + ∞ corresp onds to a single-barrier in 0 only . In particular, this generality allo ws to p ossibly include non-smo oth energies and to model additional bio chemical regulation, suc h as time-scale separation in repair pathw a ys or environmen tal mo dulation. The sto chastic components enco de tw o biologically distinct p erturbations. The Wiener noise G ( u ) d W represen ts contin uous bac kground ﬂuctuations, while the jump term R Z J ( u − , z ) µ (d t, d z ) , driv en by a comp ensated Poisson random measure ¯ µ , mo dels abrupt lo calized damage hits o ccurring at random times and lo cations. In particular, we allow the Wiener noise intensit y G and the jump amplitude J to b e state dep enden t through a pre-factor that v anishes at the potential barriers, reﬂecting the idea that fully healthy and fully damaged conﬁgurations are comparatively insensitive to small additional perturbations, whereas in termediate conﬁgurations are the most susceptible. By selecting the ev ent statistics and spatial kernels appropriately , this framework can also b e sp ecialized to mo del ionizing radiation tracks as a concrete application, see [ 14 , 24 , 27 , 31 ]. F rom a mathematical persp ectiv e, the main c hallenge is to deal with the p ossible singularit y of the p oten tial Φ at the barriers 0 and L , and to conﬁne the pro cess u in to the ph ysically-relev ant in terv al [0 , L ] . The main idea is to suitable tune the noise intensities in a multiplicativ e fashion, in such a w a y that the singular energy proliferation at the p oten tial barriers is kept under control throughout the ev olution. As far as the Wiener noise is concerned, this has b een prop osed for the sto c hastic Allen-Cahn equation in [ 7 , 44 ], and also for coupled Allen-Cahn-Navier-Stok es systems in [ 18 , 20 ]. F or the jump contribution, the presence of a singular p oten tial is a nov el challenge, as the main literature on sto c hastic ev olutions equations with jump-t yp e noise mainly deal with regular nonlinearities [ 2 , 40 ]. The main idea to ov ercome suc h problem is to prescrib e a tuning of the jump intensit y as for the Wiener case: roughly sp eaking, this allo ws to control the total energy ov er time and not to “jump” outside the relev ant domain (0 , L ) . The ﬁrst main result of the pap er consists of w ell-p osedness for ( 1.2 ) in the strong probabilistic sense. The intuitiv e idea is that the jump in tensity close to the p oten tial barriers comp ensate the blow-up of the p otential. In general, such energy-noise balance may not b e preserved under standard approximation tec hniques. F rom the tec hnical p oint of view, the strategy of the pro of combines a Y osida regularization for the double-well potential with a ad-ho c counter-correction on the jump noise, in order to preserv e such energy-noise balance also at the approximate level. This allows to extend the classical frameworks for SPDEs with lo cally monotone nonlinearities and jump noise [ 12 ]. AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 3 The second main result of the pap er concerns existence and uniqueness of in v arian t measures for the asso ciated transition semigroup, as w ell as ergo dic and mixing properties. These are obtained b y deducing reﬁned energy estimates on the solution in order to cast our framework in the t ypical long-time b eha vior theory for sto c hastic ev olution equations. Ev en tually , w e demonstrate the model’s computational relev ance through an implicit–explicit Eu- ler–Maruy ama ﬁnite element scheme. This sim ulates damage-repair dynamics and visualizes the roles of sto c hastic ﬂuctuations and radiation trac ks in the most interesting case of the logarithmic p oten tial ( 1.1 ). By tuning the parameters gov erning noise and jumps, the simulations repro duce c haracteristic radiobi- ological resp onses such as lo calized damage clustering, partial recov ery , and cumulativ e loss of integrit y under sustained irradiation. F urthermore, we are also able to visualize an abstract random separation prop ert y , that was prov ed for the mo del without jumps in [ 8 , 38 ]. This reads, in its general form, P sup ( t,x ) ∈ R + ×O | u ( t, x ) | < 1 ! = 1 and conﬁrms also from a mathematical viewp oin t the fact that equilibrium dynamics for ( 1.1 ) are not attained at the pure phases. In conclusion, the Allen-Cahn equation has b een studied in the last decades under numerous p oin t of views. F or example, without aiming for generality , w e refer to [ 4 , 6 , 19 , 25 , 36 , 39 ] for con tributions on the Allen-Cahn equation with Wiener-type noise, [ 42 , 45 ] on its sharp interface limit, [ 17 , 37 ] for coupled systems, and [ 10 , 11 , 26 ] for sto chastic ev olution equations with jump-type noise, as well as the references therein. Up to the authors’ knowledge, the study of sto c hastic mo dels with jump-noise and singular p oten tials has b een an op en issue in the last years. This pap er provides a ﬁrst rigorous analytical treatmen t in a biophysically-motiv ated setting. Plan of the pap er. The con ten ts of the work are organized as follo ws. Section 2 is dev oted to illustrating the preliminary material, notation, and the necessary notions to state the main results. Section 3 contains the pro of of existence of probabilistically-strong solutions to the system. Section 4 contains the inv estigation of the longtime b ehavior of the system, in terms of existence and uniqueness of in v ariant measures. Section 5 delv es in to a n umerical in v estigation of the system. Finally , a collection of useful tec hnical estimates are presented in App endix A . 2. Preliminaries and main resul ts 2.1. Mathematical preliminaries and notation. In the following, w e illustrate the notation emplo y ed throughout the whole work. 2.1.1. Sto chastic fr amework. Let T > 0 b e an arbitrary ﬁnal time, and let us set (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) to b e a ﬁltered probabilit y space satisfying the usual conditions, i.e., the ﬁltration ( F t ) t ∈ [0 ,T ] is saturated and righ t-con tinuous. Let P denote the progressive σ -algebra on Ω × [0 , T ] , accordingly . F or every measurable space ( E , E ) , we denote by M ( E ) the set of p ositiv e measures on ( E , E ) . Often times, the space E is also equipp ed with the structure of Banach space: in these cases, we imply that the underlying σ -algebra E is the Borel σ -algebra B ( E ) . Moreo v er, for any p ositiv e real quantit y p ≥ 1 the symbol L p (Ω; E ) denotes the set of strongly measurable E -v alued random v ariables on Ω with ﬁnite moments up to order p . When the space E is allow ed to dep end on time, as in classical Bo c hner spaces, we ma y denote the resulting space with L p P (Ω; E ) to stress that measurability is in tended with resp ect to the progressiv e σ -algebra P . Let us no w p oin t out the precise in terpretation of the sto c hastic p erturbations app earing in ( 1.2 ). Con- cerning Itô noise, we denote by W a cylindrical Wiener pro cess on some ﬁxed separable Hilb ert space U . A ccordingly , we set { e k } k ∈ N ⊂ U to b e a ﬁxed arbitrary orthonormal system for U . Let us recall that by the prop erties of cylindrical Wiener pro cesses, the pro cess W admits the representation W = + ∞ X k =0 β k e k , (2.1) 4 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA where { β k } k ∈ N is a family of real and indep enden t Bro wnian motions. Ho w ev er, the series ( 2.1 ) can not b e exp ected to alw a ys con v erge in U . A w ell known result establishes that, in general, con v ergence can b e ac hiev ed in some larger Hilb ert space U 0 suc h that U  → U 0 with Hilbert–Schmidt embedding ι . Moreov er, it is p ossible to identify W as a Q 0 -Wiener pro cess on U 0 , for the trace-class op erator Q 0 := ι ◦ ι ∗ on U 0 (see [ 33 , Subsection 2.5.1]). In the following, w e implicitly assume this extension by simply saying that W is a cylindrical pro cess on U . This holds also for sto c hastic integration à la Itô with respect to W . Indeed, the symbol Z · 0 B ( s ) d W ( s ) := Z · 0 B ( s ) ◦ ι − 1 d W ( s ) , for ev ery pro cess B ∈ L 2 P (Ω; L 2 (0 , T ; L 2 ( U, K ))) , where K is an y real Hilb ert space and L 2 ( U, K ) is the space of Hilb ert–Sc hmidt op erators from U to K (a precise deﬁnition is giv en later on). It is well known that suc h a deﬁnition is well p osed and do es not dep end on the choic e of U 0 or ι (see [ 33 , Subsection 2.5.2]). Let us no w mov e to the jump noise term: we refer here to the classical theory [ 30 ]. Let ( Z , Z ) b e a Blac kw ell space (e.g. a Polish space with its Borel σ -algebra). A random measure is a map µ : Ω → M ([0 , + ∞ ) × Z , B ([0 , + ∞ )) ⊗ Z ) suc h that µ ( ω , { 0 } × Z ) = 0 for all ω ∈ Ω . A random measure µ is said to b e in teger-v alued if: (1) µ ( ω , { t } × Z ) ≤ 1 for ev ery ω ∈ Ω and t ≥ 0 ; (2) µ ( ω , A ) ∈ N ∪ { + ∞} for ev ery ω ∈ Ω and A ∈ B ([0 , + ∞ )) ⊗ Z ; (3) µ is optional and B ([0 , + ∞ )) ⊗ Z - σ -ﬁnite. F urthermore, given a σ -ﬁnite p ositiv e measure ν on ( Z, Z ) , we say that µ is a time-homogeneous P oisson random measure with intensit y ν if µ is an integer-v alued random measure suc h that: (1) µ ( · , A ) is a Poisson random v ariable with rate ( λ ⊗ ν )( A ) for ev ery A ∈ B ([0 , + ∞ )) ⊗ Z , where λ is the Leb esgue measure on [0 , + ∞ ) ; (2) { µ ( · , A i ) } n i =1 are indep endent for every disjoint A 1 , . . . , A n ∈ B ([0 , + ∞ ) ⊗ Z . The measure λ ⊗ ν on ([0 , + ∞ ) × Z , B ([0 , + ∞ ) ⊗ Z ) is called comp ensator of µ , and ¯ µ := µ − λ ⊗ ν is called comp ensated P oisson measure. The qualiﬁer “time-homogeneous” underlines the fact that the comp ensator is precisely the pro duct measure λ ⊗ ν . In general, this may not b e the case. F or every Hilb ert space K , for every T > 0 , and for ev ery f : Ω × [0 , T ] × Z → K that is P ⊗ Z / B ( K ) -measurable and with f ∈ L 1 (Ω; L 1 (0 , T ; L 1 ( Z, ν ; K ))) , the sto chastic pro cess ( f · ¯ µ )( t ) := Z (0 ,t ] Z Z f ( s, z ) ¯ µ (d s, d z ) , t ∈ (0 , T ] , is a w ell-deﬁned K -v alued martingale. Moreo ver, if also f ∈ L 2 (Ω; L 2 (0 , T ; L 2 ( Z, ν ; K ))) , then f · ¯ µ is a square-in tegrable martingale with predictable quadratic v ariation given b y ⟨ f · ¯ µ ⟩ ( t ) = Z t 0 Z Z ∥ f ( s, z ) ∥ 2 K ν (d z ) d s, t ∈ (0 , T ] . The concept of P oisson random measure arises somewhat naturally in the theory of P oisson p oin t pro cesses. F or conv enience, w e brieﬂy recall the main ideas. Giv en a coun table and lo cally ﬁnite subset S ⊂ (0 , + ∞ ) , a p oin t function on ( Z , Z ) is an y mapping p : S → Z . The coun ting measure asso ciated to the p oin t function p is the measure µ p on ((0 , + ∞ ) × Z, B ((0 , + ∞ )) ⊗ Z ) deﬁned as µ p ((0 , t ] × A ) := # { s ∈ S ∩ (0 , t ] : p ( s ) ∈ A } , t > 0 , A ∈ Z . This measure coun ts ho w many ev ents within A hav e happ ened at most at time t . A p oin t pro cess is then a random v ariable with v alues in the set of p oint functions on Z , equipp ed with a suitable σ -algebra in suc h a w a y that the maps p 7→ µ p ((0 , t ] × A ) are measurable for all t > 0 and A ∈ Z . Then, giv en a p oin t pro cess Π , it may happ en that the family of counting measures asso ciated to Π is a time-homogeneous P oisson random measure on ((0 , + ∞ ) × Z, B ((0 , + ∞ ) × Z ) . In this case, w e sa y that Π is a time-homogeneous P oisson p oin t pro cess. AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 5 2.1.2. A nalytic al fr amework. Given an y Banac h space E , we denote with the sym b ol E the pro duct space E d (or even E d × d , if no am biguities arise), and by E ∗ its top ological dual. The corresp onding dualit y pairing is denoted b y ⟨· , ·⟩ E ∗ ,E . Whenev er E is also a Hilb ert space, then the scalar pro duct of E is denoted b y ( · , · ) E . F or an y pair of Banach spaces E 1 and E 2 and any extended real num ber s ∈ [1 , + ∞ ] , the symbol L s ( E 1 ; E 2 ) indicates the usual spaces of strongly measurable, Bo c hner-in tegrable functions deﬁned on the Banac h space E 1 and with v alues in the Banac h spaces E 2 . If E 2 is omitted, it is understo o d that E 2 = R . If E 1 and E 2 are also Hilb ert spaces, then we are able to deﬁne the space of Hilb ert-Schmidt op erators from E 1 to E 2 as L 2 ( E 1 , E 2 ) , equipp ed with its usual norm structure giv en by ∥ · ∥ L 2 ( E 1 , E 2 ) . Within a smo oth and b ounded domain O ⊂ R d , with d ∈ { 1 , 2 , 3 } , we denote b y W s,p ( O ) the classical Sob olev spaces of order s ∈ R and p ∈ [1 , + ∞ ] , and we denote b y ∥ · ∥ W s,p ( O )) their classical norms (with the usual understanding that W 0 ,p ( O ) = L p ( O ) ). A ccordingly , w e deﬁne the Hilb ert space H s ( O ) := W s, 2 ( O ) for all s ∈ R , endo wed with its canonical norm ∥ · ∥ H s ( O ) . In the following, it will b e useful to introduce a sp eciﬁc notation for a v ariational structure, i.e., w e set H := L 2 ( O ) , V := ( H 1 0 ( O ) if ( α d , α n ) = (1 , 0) , H 1 ( O ) if ( α d , α n ) = (0 , 1) , endo w ed with their standard norms ∥·∥ H and ∥·∥ V , respectively . As usual, w e iden tify the Hilb ert space H with its dual through the corresp onding Riesz isomorphism, so that we hav e the v ariational structure V  → H  → V ∗ , with dense and compact em b eddings (b oth in the cases d = 2 and d = 3 ). In the Diric hlet case (i.e. ( α d , α n ) = (1 , 0) ), the zero-trace space H 1 0 ( O ) = { u ∈ H 1 ( O ) : u = 0 almost everywhere on ∂ Ω } is intended to b e endow ed with the H 1 -seminorm structure, owing to the Poincaré inequality , namely ( u, v ) H 1 0 ( O ) := ( ∇ u, ∇ v ) H , ∥ u ∥ H 1 0 ( O ) := ∥∇ u ∥ H , u, v ∈ V . The weak realization of the negative Dirichlet-Laplace op erator, i.e., the op erator A : H 1 0 ( O ) → H − 1 ( O ) , ⟨A u, v ⟩ H − 1 ( O ) ,H 1 0 ( O ) := ( ∇ u, ∇ v ) H for all v ∈ H 1 0 ( O ) is a linear isomorphism by the Riesz represen tation theorem. The sp ectral prop erties of A will b e later useful. W e denote by { ρ n } n ∈ N the coun table sequence of eigen v alues of A . In particular, it is an increasing sequence such that lim n → + ∞ ρ n = + ∞ . Finally , we recall that ρ − 1 2 1 is the optimal constant in Poincaré’s inequality ∥ u ∥ H ≤ ρ − 1 2 1 ∥ u ∥ H 1 0 ( O ) , (2.2) that holds for all v ∈ H 1 0 ( O ) . In all the presen t work, we reserve the symbols C and K (ev entually indexed) for constants dep ending on some or all the structural parameters of the problem. If the dep endencies of suc h constan ts are relev ant, they will b e explicitly p ointed out. The v alues of these constan ts may c hange within the same argument without relab eling. That b eing said, throughout this work, w e shall make use of several diﬀerent em b edding theorems. As sometimes it shall b e relev an t to explicitly trac k down the v alue of some constan ts, for the sake of clarity , w e precise some further notation. (i) The embedding V  → L q ( O ) for all ﬁnite q ≥ 1 in tw o dimensions and for q ∈ [1 , 6] in three dimensions, implies that ∥ u ∥ L q ( O ) ≤ K q ∥ u ∥ V for all u ∈ V . 6 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA (ii) The embedding L ∞ ( O )  → H implies that ∥ u ∥ H ≤ |O | 1 2 ∥ u ∥ L ∞ ( O ) for all u ∈ L ∞ ( O ) . 2.2. Assumptions. The following assumptions are in order throughout the presen t w ork. (A1) The spatial domain O is a b ounded smo oth domain in R d with d ∈ { 2 , 3 } . The parameter ε is strictly positive, and we set either α d = 0 and α n = 1 (homogeneous Neumann b oundary conditions) or α n = 0 and α d = 1 (homogeneous Dirichlet b oundary conditions). (A2) The function Ψ : R → [0 , + ∞ ] is conv ex, low er semicontin uous and Int( D (Ψ)) = (0 , L ) , where L ∈ (0 , + ∞ ] . Moreov er, the restriction Ψ | (0 ,L ) ∈ C 2 (0 , L ) and its sub diﬀeren tial is single-v alued, i.e., ∂ Ψ( s ) = { Ψ ′ ( s ) } ∀ s ∈ (0 , L ) , and there exist constants C 0 , C 1 > 0 such that Ψ ′ ( s ) s ≥ C 0 s 2 − C 1 ∀ r ∈ (0 , L ) . F urthermore, the second deriv ativ e Ψ ′′ is conv ex and we assume that Ψ ′′ (0) := lim s → 0 + Ψ ′′ ( s ) ∈ [0 , + ∞ ] , and, if L < + ∞ , Ψ ′′ ( L ) := lim s → L − Ψ ′′ ( s ) ∈ [0 , + ∞ ] . (A3) The function F : R → R is √ C F -Lipsc hitz-con tinuous for some C F > 0 . (A4) Setting Γ := { v ∈ L ∞ ( O ) : 0 ≤ v ≤ L a.e. in O } if L < + ∞ and Γ := { v ∈ H : v ≥ 0 a.e. in O } if L = + ∞ , the function G : Γ → L 2 ( U, H ) satisﬁes: (i) there exists a constan t C G > 0 such that ∥ G ( v 1 ) − G ( v 2 ) ∥ 2 L 2 ( U,H ) ≤ C G ∥ v 1 − v 2 ∥ 2 H ∀ v 1 , v 2 ∈ Γ; (ii) there exists a constan t, which without loss of generality we set equal to C G , such that + ∞ X k =1 ∥ p Ψ ′′ ( v ) G ( v )[ e k ] ∥ 2 H ≤ C G (1 + ∥ Ψ( v ) ∥ L 1 ( O ) ) ∀ v ∈ Γ , with the understanding that ∞ · 0 := 0 . (A5) Let ( Z , Z ) b e a Blackw ell space, ν is a σ -ﬁnite p ositive measure on ( Z , Z ) , and µ is a time- homogeneous Poisson random measure with intensit y ν . (A6) The function J : Γ × Z → H satisﬁes: (i) there exists a constan t C J > 0 such that Z Z ∥ J ( v 1 , z ) − J ( v 2 , z ) ∥ 2 H ν (d z ) ≤ C J ∥ v 1 − v 2 ∥ 2 H ∀ v 1 , v 2 ∈ Γ; (ii) there exists a constan t δ J ∈ (0 , 1] such that δ J v ≤ v + J ( v , z ) ≤ (1 − δ J ) L + δ J v a.e. in O , ∀ v ∈ Γ , ∀ z ∈ Z if L < + ∞ and suc h that δ J v ≤ v + J ( v , z ) ≤ C J (1 + v ) a.e. in O , ∀ v ∈ Γ , ∀ z ∈ Z if L = + ∞ ; AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 7 (iii) there exists a constan t, which without loss of generality we set equal to C J , such that Z Z ∥ p Ψ ′′ ( δ J v ) J ( v , z ) ∥ 2 H ν (d z ) ≤ C J (1 + ∥ Ψ( v ) ∥ L 1 ( O ) ) ∀ v ∈ Γ , Z Z ∥ p Ψ ′′ ((1 − δ J ) L + δ J v ) J ( v , z ) ∥ 2 H ν (d z ) ≤ C J (1 + ∥ Ψ( v ) ∥ L 1 ( O ) ) ∀ v ∈ Γ , if L < + ∞ , Z Z ∥ p Ψ ′′ ( C J (1 + v )) J ( v , z ) ∥ 2 H ν (d z ) ≤ C J (1 + ∥ Ψ( v ) ∥ L 1 ( O ) ) ∀ v ∈ Γ , if L = + ∞ , with the understanding that ∞ · 0 := 0 . Remark 2.1. Let us p oin t out some observ ations ab out the previous assumptions. Assumptions (A2) and (A3) are classical hypotheses in the in v estigation of diﬀuse in terface mo dels. They encompass a wide class of p oten tial by introducing a con v ex-conca ve splitting. The conv ex part, in particular, ma y exhibit singularities at the endp oin ts of (0 , L ) , allowing for thermo dynamical p oten tials lik e the Flory–Huggins one (see [ 21 , 29 ]). The regular p erturbation F is most signiﬁcan t when F is concav e (otherwise it could b e absorb ed in Ψ ). The comp etition of con v ex and concav e p oten tials results in t ypical double-well proﬁles that are resp onsible for bistable dynamics. Assumption (A4) includes prop erties that are not new in the con text of singular diﬀuse interface mo dels (see, for instance, [ 18 , 19 , 43 ] for further insights). Finally , in the same spirit, Assumption (A6) ensures the correct compatibilit y conditions to grant existence of solutions. In particular, Assumption (A6) - (iii) ensures that, after a jump, the v ariable do es not exit the ph ysically relev ant range (0 , L ) (in the case L = + ∞ , it also prescrib es a gro wth condition, in order not to hav e unnaturally “large” jumps). Remark 2.2. As a consequence of the Lipsc hitz contin uit y prop erties given in Assumptions (A3) , (A4) - (i) and (A6) - (i) , we infer the existence of p ositiv e constants e C F , e C G and e C J suc h that | F ( s ) | 2 ≤ e C F (1 + | s | 2 ) ∀ s ∈ R , ∥ G ( v ) ∥ 2 L 2 ( U,H ) ≤ e C G  1 + ∥ v ∥ 2 H  ∀ v ∈ Γ and Z Z ∥ J ( v , z ) ∥ 2 H ν (d z ) ≤ e C J  1 + ∥ v ∥ 2 H  ∀ v ∈ Γ . Remark 2.3. It will turn out to b e useful to introduce the function Φ : R → ( −∞ , + ∞ ] , Φ( r ) = Ψ( r ) + Z r 0 F ( s ) d s, r ∈ R . By p ossibly renominating the p ositiv e constan ts C 0 and C 1 , it is not restrictiv e to assume that Φ ′ ( r ) r ≥ C 0 r 2 − C 1 ∀ r ∈ (0 , L ) . Since Φ is b ounded from b elo w, w e ma y replace Φ by Φ + C for a suitable constant C > 0 and assume, without loss of generality , that Φ ≥ 0 . 2.3. Main results. In this subsection, we state the main results of this work. The ﬁrst is a w ell-posedness result, also precising a notion of solution for problem ( 1.2 ). Theorem 2.4. L et Assumption (A1) - (A6) hold. Fix p ≥ 2 and let r := min { 4 , p } . L et further u 0 ∈ L p (Ω , F 0 ; H ) b e such that Φ( u 0 ) ∈ L r 2 (Ω , F 0 ; L 1 ( O )) . Then, ther e exists a unique pr o gr essively me asur able H -value d c àd làg pr o c ess u such that (i) u ∈ L r (Ω; L ∞ (0 , T ; H ) ∩ L 2 (0 , T ; V )) ; (ii) u ( ω , t, x ) ∈ (0 , L ) P ⊗ d t ⊗ d x -almost sur ely; (iii) Ψ( u ) ∈ L r 2 (Ω; L ∞ (0 , T ; L 1 ( O )) ; (iv) Ψ ′ ( u ) ∈ L r (Ω; L 2 (0 , T ; H )) ; (v) u (0) = u 0 ; 8 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA (vi) for every v ∈ V , it holds ( u ( t ) , v ) H + ε Z t 0 Z O ∇ u ( s ) · ∇ v d x d s + Z t 0 Z O (Ψ ′ ( u ( s )) + F ( u ( s ))) v d x d s = ( u 0 , v ) H +  Z t 0 G ( u ( s )) d W ( s ) , v  H + Z (0 ,t ] Z Z J ( u ( s − ) , z ) µ (d s, d z ) , v ! H for al l t ≥ 0 , P -almost sur ely. Mor e over, given two initial c onditions u 01 and u 02 satisfying the same assumptions liste d ab ove, letting u 1 and u 2 denote the c orr esp onding solutions, the fol lowing c ontinuous dep endenc e estimate holds: ∥ u 1 − u 2 ∥ L r (Ω; L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V )) ≤ C ∥ u 01 − u 02 ∥ L r (Ω; H ) . The second result, instead, establishes existence, uniqueness and main properties of the inv arian t measure of the system. Theorem 2.5. L et Assumptions (A1) - (A6) hold. If L = + ∞ , let further min( ε, C 0 ) > K 2 e C G 2 + e C J ! . Then, ther e exists at le ast one er go dic invariant me asur e supp orte d on the set A str :=  v ∈ V : Ψ( v ) ∈ L 1 ( O ) , Ψ ′ ( v ) ∈ H  . Mor e over, if Assumption (A1) holds with α n = 0 and α d = 1 (Dirichlet b oundary c onditions) and ερ 1 > C G 2 + C J + p C F if L < + ∞ or min ( ερ 1 , ε, C 0 ) > max K 2 e C G 2 + e C J ! , C G 2 + C J + p C F ! if L = + ∞ , then, the invariant me asur e is unique, er go dic, str ongly and exp onentially mixing. Remark 2.6. It is exp ected that the actual supp ort of the in v ariant measures of the system is even a subset of A str . This can b e shown by using higher-order estimates (as in [ 7 ], for instance) that, ho w ev er, need further integrabilit y assumptions to hold on the parameters of the problem. 3. Pr oof of Theorem 2.4 This section is dev oted to proving Theorem 2.4 , i.e., a well-posedness result for system ( 1.2 ). Without loss of generality , we shall assume Neumann b oundary conditions, namely α d = 0 and α n = 1 . The pro of in the Dirichlet case is analogous. 3.1. An appro ximated problem. By virtue of Assumption (A2) , we can apply the theory of maximal monotone op erators (see for instance [ 3 , 9 ]) to introduce a suitable regularization of the single-v alued sub diﬀeren tial ∂ Ψ , whic h we shall denote by Ψ ′ , enabling in turn a regularization of Ψ . Indeed, since Ψ is prop er, conv ex and low er semicon tinuous, the sub diﬀeren tial Ψ ′ can b e iden tiﬁed with a maximal monotone graph in R × R . It is therefore p ossible to consider the so-called Y osida appro ximation of Ψ , namely a one-parameter family of globally-deﬁned and ev erywhere non-negative functions { Ψ λ } λ> 0 enjo ying the following prop erties (cf. [ 3 ]): (a) Ψ λ is conv ex and conv erges to Ψ p oint wise in R and monotonically increasing as λ → 0 + ; (b) Ψ λ ∈ C 1 , 1 ( R ) and the Lipschitz constan t of Ψ ′ λ is 1 λ ; (c) | Ψ ′ λ | conv erges to | Ψ ′ | p oint wise in R and monotonically increasing as λ → 0 + , (d) Ψ λ (0) = Ψ ′ λ (0) = 0 for all λ > 0 . AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 9 F or any λ > 0 , let R λ denote the non-expansive contin uous resolven t op erator R λ : R → R , R λ ( s ) := ( I + λ Ψ ′ ) − 1 ( s ) , s ∈ R . Then, the Y osida approximation Ψ λ of Ψ is precisely deﬁned by Ψ λ ( s ) = Ψ( R λ ( s )) + 1 2 λ | s − R λ ( s ) | 2 for all s ∈ R . In order to setup an approximated version of ( 1.2 ), it is necessary to account for the fact that the op erators G and J may not b e w ell deﬁned when ev aluated on regular solutions, i.e., that appro ximated solutions may not b elong to Γ . F or all λ > 0 , deﬁne then the regularized op erators G λ : H → L 2 ( U, H ) , G λ = G ◦ R λ as well as J λ : H × Z → H , J λ ( u, z ) = J ( R λ ( u ) , z ) − λ Ψ ′ ( R λ ( u )) ∀ ( u, z ) ∈ H × Z . The action of the resolven t op erator enables G λ and J λ ( · , z ) to b e globally deﬁned op erators ov er H (and not only on Γ ), for all z ∈ Z . Before moving on, w e pro ve an elementary , yet imp ortan t preliminary result. Lemma 3.1. The families of r e gularize d functions { G λ } λ> 0 and { J λ } λ> 0 satisfy the fol lowing pr op erties: (i) ther e exist two p ositive c onstants, that without loss of gener ality we stil l denote by C G and C J , such that the two Lipschitz c onditions ∥ G λ ( v 1 ) − G λ ( v 2 ) ∥ 2 L 2 ( U,H ) ≤ C G ∥ v 1 − v 2 ∥ 2 H Z Z ∥ J λ ( v 1 , z ) − J λ ( v 2 , z ) ∥ 2 H ν (d z ) ≤ C J ∥ v 1 − v 2 ∥ 2 H hold for al l λ > 0 and al l v 1 , v 2 ∈ V ; (ii) if L < + ∞ , then for the same c onstant δ J app e aring in (A6) - (ii) it holds δ J R λ ( v ) ≤ v + J λ ( v , z ) ≤ (1 − δ J ) L + δ J R λ ( v ) a.e. in O , ∀ v ∈ H , ∀ z ∈ Z , for al l λ > 0 . Analo gously, if L = + ∞ , we have δ J R λ ( v ) ≤ v + J λ ( v , z ) ≤ C J (1 + R λ ( v )) a.e. in O , ∀ v ∈ H , ∀ z ∈ Z , for al l λ > 0 ; (iii) ther e exist two p ositive c onstants, that without loss of gener ality we stil l denote by C G and C J , such that the inte gr al b ounds + ∞ X k =1 ∥ q Ψ ′′ λ ( v ) G λ ( v )[ e k ] ∥ 2 H ≤ C G (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) , Z Z    q Ψ ′′ λ ( δ J R λ ( v )) J λ ( v , z )    2 H ν (d z ) ≤ C J (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) , Z Z    q Ψ ′′ λ ((1 − δ J ) L + δ J R λ ( v )) J λ ( v , z )    2 H ν (d z ) ≤ C J (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) if L < + ∞ , Z Z    q Ψ ′′ λ ( C J (1 + R λ ( v )) J λ ( v , z )    2 H ν (d z ) ≤ C J (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) if L = + ∞ , hold for al l λ > 0 and v ∈ H . Pr o of. Let λ > 0 b e ﬁxed and let us start from the Lipsc hitz conditions given in item (i). Using Assumption (A4) - (i) , for any v 1 and v 2 in H , w e ha v e ∥ G λ ( v 1 ) − G λ ( v 2 ) ∥ 2 L 2 ( U,H ) = ∥ G ( R λ ( v 1 )) − G ( R λ ( v 2 )) ∥ 2 L 2 ( U,H ) ≤ C G ∥ R λ ( v 1 ) − R λ ( v 2 ) ∥ 2 H ≤ C G ∥ v 1 − v 2 ∥ 2 H , 10 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA since the resolv ent op erator is non-expansiv e in H . An analogous argumen t shows the claim for J λ , indeed Assumption (A6) - (i) gives Z Z ∥ J λ ( v 1 , z ) − J λ ( v 2 , z ) ∥ 2 H ν (d z ) ≤ 2 Z Z ∥ J ( R λ ( v 1 ) , z ) − J ( R λ ( v 2 ) , z ) ∥ 2 H ν (d z ) + 2 λ 2 ∥ Ψ ′ λ ( v 1 ) − Ψ ′ λ ( v 2 ) ∥ 2 H ≤ 2( C J + 1) ∥ v 1 − v 2 ∥ 2 H as Ψ ′ λ is 1 λ -Lipsc hitz-con tinuous and the resolv en t op erator is non-expansiv e. The prop ert y in item (ii) can b e shown as follo ws. Without loss of generalit y , we sho w the case L < + ∞ , as the other is analogous. Let v ∈ V and z ∈ Z b e arbitrary . Then, v + J ( R λ ( v ) , z ) = v − R λ ( v ) + R λ ( v ) + J ( R λ ( v ) , z ) = λ Ψ ′ λ ( v ) + R λ ( v ) + J ( R λ ( v ) , z ) , o wing to the deﬁnition of the resolven t op erator R λ . Observ e that, by Assumption (A6) - (ii) , w e hav e δ J R λ ( v ) ≤ R λ ( v ) + J ( R λ ( v ) , z ) ≤ (1 − δ J ) L + δ J R λ ( v ) and therefore δ J R λ ( v ) ≤ v + J ( R λ ( v ) , z ) − λ Ψ ′ λ ( v ) ≤ (1 − δ J ) L + δ J R λ ( v ) . The claim then holds recalling the deﬁnition of J λ . Finally , w e show the b ounds listed in item (iii). Let v ∈ H . First, using Assumption (A4) - (ii) , we ha ve + ∞ X k =1    q Ψ ′′ λ ( v ) G λ ( v )[ e k ]    2 H = + ∞ X k =1    q Ψ ′′ ( R λ ( v )) R ′ λ ( v ) G ( R λ ( v ))[ e k ]    2 H ≤ + ∞ X k =1 ∥ p Ψ ′′ ( R λ ( v )) G ( R λ ( v ))[ e k ] ∥ 2 H ≤ C G (1 + ∥ Ψ( R λ ( v )) ∥ L 1 ( O ) ) ≤ C G (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) , since, giv en the deﬁnition of Ψ λ , w e ha ve Ψ λ ( s ) ≥ Ψ( R λ ( s )) for all s ∈ R . F or the remaining estimates, once again, we show one of the claimed inequalities, as the three arguments are v ery similar. Indeed, observing that δ J R λ ( v ) ∈ dom Ψ ′′ almost everywhere in O , Z Z    q Ψ ′′ λ ( δ J R λ ( v )) J λ ( v , z )    2 H ν (d z ) ≤ 2 Z Z ∥ p Ψ ′′ ( δ J R λ ( v )) J ( R λ ( v ) , z ) ∥ 2 H ν (d z ) + 2 λ 2    q Ψ ′′ λ ( δ J R λ ( v ))Ψ ′ λ ( v )    2 H ≤ 2 C J (1 + ∥ Ψ( R λ ( v )) ∥ L 1 ( O ) ) + 2 λ ∥ Ψ ′ λ ( v ) ∥ 2 H ≤ 2 C J (1 + ∥ Ψ λ ( v ) ∥ L 1 ( O ) ) + 2 λ ∥ Ψ ′ λ ( v ) ∥ 2 H . Recalling the deﬁnition of the Y osida appro ximation, w e then hav e Ψ λ ( v ) = Ψ( R λ ( v )) + | v − R λ ( v ) | 2 2 λ = Ψ( R λ ( v )) + λ 2 | Ψ ′ λ ( v ) | 2 . In tegrating this equality o ver O , recalling that Ψ ◦ R λ ≤ Ψ λ and applying the triangular inequality yields then λ ∥ Ψ ′ λ ( v ) ∥ 2 H ≤ 4 ∥ Ψ λ ( v ) ∥ L 1 ( O ) and the claim follows. □ W e are no w in a p osition to consider the appro ximated problem        d u λ − ε ∆ u λ d t + Ψ ′ λ ( u λ ) d t + F ( u λ ) d t = G λ ( u λ ) d W + Z Z J λ ( u − λ , z ) µ (d t, d z ) in (0 , T ) × O , α d u λ + α n ∂ n u λ = 0 on (0 , T ) × ∂ O , u λ ( · , 0) = u 0 in O , (3.1) AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 11 and hence, we show that the ab o v e problem is well-posed. T o this end, w e set up a ﬁxed p oin t argument. F or any ﬁxed λ > 0 , let X := L r (Ω; L ∞ (0 , T ; H ) ∩ L 2 (0 , T ; V )) where r := min { 4 , p } and p is giv en in Theorem 2.4 , and consider for eac h v ∈ X the sto chastic diﬀerential problem        d w λ − ε ∆ w λ d t = − Ψ ′ λ ( v ) d t − F ( v ) d t + G λ ( v ) d W + Z Z J λ ( v − , z ) µ (d t, d z ) in (0 , T ) × O , α d w λ + α n ∂ n w λ = 0 on (0 , T ) × ∂ O , w ( · , 0) = u 0 in O , (3.2) whic h, as a linear equation driv en by additive noise, is w ell-p osed, for instance, b y chec king the assumptions listed in [ 34 ]. In particular, the solution map S : X → X , v 7→ w λ is well deﬁned. Let now v 1 and v 2 b e tw o elemen ts of X , and set w i,λ = S ( v i ) ∀ i ∈ { 1 , 2 } . Then, the sto c hastic pro cess w := w 1 ,λ − w 2 ,λ formally satisﬁes the sto c hastic partial diﬀerential system              d w − ε ∆ w d t = − [Ψ ′ λ ( v 1 ) − Ψ ′ λ ( v 2 )] d t − [ F ( v 1 ) − F ( v 2 )] d t + [ G λ ( v 1 ) − G λ ( v 2 )] d W + Z Z  J λ ( v − 1 , z ) − J λ ( v − 2 , z )  µ (d t, d z ) in (0 , T ) × O , α d w + α n ∂ n w = 0 on (0 , T ) × ∂ O , w ( · , 0) = 0 in O . (3.3) F or some t ∈ (0 , T ) , let us apply the Itô form ula to the squared H -norm of w ( t ) , i.e., precisely , to the functional w 7→ 1 2 ∥ w ∥ 2 H deﬁned ov er H . Due to the presence of a semimartingale noise, the ordinary Itô formula for twice diﬀer- en tiable functionals (as for instance in [ 16 , Theorem 4.32]) is not complete: for the jump part we indeed resort to [ 10 , Theorem B.1] and get 1 2 ∥ w ( t ) ∥ 2 H + ε Z t 0 ∥∇ w ( s ) ∥ 2 H d s = − Z t 0  w ( s ) , Ψ ′ λ ( v 1 ( s )) − Ψ ′ λ ( v 2 ( s ))  H d s − Z t 0 ( w ( s ) , F ( v 1 ( s )) − F ( v 2 ( s ))) H d s + Z t 0 ( w ( s ) , G λ ( v 1 ( s )) − G λ ( v 2 ( s ))) H d W ( s ) + 1 2 Z t 0 ∥ G λ ( v 1 ( s )) − G λ ( v 2 ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( w ( s − ) , J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z  ∥ w ( s − ) + J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z ) ∥ 2 H − ∥ w ( s − ) ∥ 2 H − 2( w ( s − ) , J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z )) H  ν (d z ) d s. (3.4) Let us handle the v arious terms app earing to the right hand side of ( 3.4 ). First of all, by the Lipsc hitz prop erties listed in Assumptions (A2) and (A3) , we ha v e by the Cauc hy–Sc h w arz and Y oung inequalities     Z t 0  w ( s ) , Ψ ′ λ ( v 1 ( s )) − Ψ ′ λ ( v 2 ( s ))  H d s + Z t 0 ( w ( s ) , F ( v 1 ( s )) − F ( v 2 ( s ))) H d s     ≤ σ sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ 2 H + C σ  1 λ 2 + C F  Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s (3.5) 12 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA for an y σ > 0 and for some p ositiv e constan t C σ only dep ending on σ . As for the Itô trace term, we emplo y Lemma 3.1 to get 1 2 Z t 0 ∥ G λ ( v 1 ( s )) − G λ ( v 2 ( s )) ∥ 2 L 2 ( U,H ) d s ≤ C G 2 Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s. (3.6) Finally , the second-order term in the jump term, i.e., the last term in ( 3.4 ), after some elemen tary manip- ulations, equals Z t 0 Z Z  ∥ w ( s − ) + J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z ) ∥ 2 H − ∥ w ( s − ) ∥ 2 H − 2( w ( s − ) , J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z )) H  ν (d z ) d s = Z t 0 Z Z ∥ J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z ) ∥ 2 H ν (d z ) d s ≤ C J Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s. (3.7) T aking into accoun t ( 3.5 )-( 3.7 ) into ( 3.4 ) (after selecting an y σ > 0 small enough), letting r = min { 4 , p } and then taking r 2 -th p ow ers, suprem ums in time and P -exp ectations, we hav e, E sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H + E     Z t 0 ∥∇ w ( s ) ∥ 2 H d s     r 2 ≤ C " E     Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 ( w ( s ) , G λ ( v 1 ( s )) − G λ ( v 2 ( s ))) H d W ( s )     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 Z Z ( w ( s − ) , J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z )) H µ (d s, d z )     r 2 # (3.8) for a constant C > 0 dep ending on the structural parameters of the problem. W e are only left with the sto c hastic i ntegrals to the righ t hand side of ( 3.8 ). Indeed, they can b e handled b y means of a suitable Burkholder–Da vis–Gundy inequalit y . In the ﬁrst case, the version of the inequalit y for con tin uous real-v alued martingales (see, for instance, [ 40 , Theorem 3.49]) gives E sup τ ∈ [0 ,t ]     Z τ 0 ( w ( s ) , G λ ( v 1 ( s )) − G λ ( v 2 ( s ))) H d W ( s )     r 2 ≤ C E     Z t 0 ∥ w ( s ) ∥ 2 H ∥ G λ ( v 1 ( s )) − G λ ( v 2 ( s )) ∥ 2 L 2 ( U,H ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r 2 H     Z t 0 ∥ G λ ( v 1 ( s )) − G λ ( v 2 ( s )) ∥ 2 L 2 ( U,H ) d s     r 4 # ≤ C E " sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r 2 H     Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s     r 4 # ≤ 1 4 E sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H + C E     Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s     r 2 , (3.9) while the similar result for the more general case of càdlàg pro cesses [ 35 , Theorem 1] giv es, for the second in tegral, as r = min { 4 , p } ≤ 4 , w e hav e AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 13 E sup τ ∈ [0 ,t ]     Z τ 0 Z Z ( w ( s − ) , J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z )) H µ (d t, d z )     r 2 ≤ C E     Z t 0 ∥ w ( s − ) ∥ 2 H Z Z ∥ J λ ( v 1 ( s − ) , z ) − J λ ( v 2 ( s − ) , z ) ∥ 2 H ν (d z ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ] ∥ w ( τ − ) ∥ r 2 H     Z t 0 ∥ v 1 ( s − ) − v 2 ( s − ) ∥ 2 H d s     r 4 # ≤ 1 4 E sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H + C E     Z t 0 ∥ v 1 ( s − ) − v 2 ( s − ) ∥ 2 H d s     r 2 , (3.10) where we hav e also used the fact that sup τ ∈ [0 ,t ] ∥ w ( τ − ) ∥ r H = sup τ ∈ [0 ,t ] lim y → τ − ∥ w ( y ) ∥ r H ≤ sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H , tacitly extending w to w (0) = 0 in a small left neighborho o d of 0. Finally , o wing to ( 3.9 ) and ( 3.10 ), ( 3.8 ) no w reads E sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H + E     Z t 0 ∥∇ w ( s ) ∥ 2 H d s     r 2 ≤ C " E     Z t 0 ∥ v 1 ( s ) − v 2 ( s ) ∥ 2 H d s     r 2 + E     Z t 0 ∥ v 1 ( s − ) − v 2 ( s − ) ∥ 2 H d s     r 2 # whic h ev en tually leads to E sup τ ∈ [0 ,t ] ∥ w ( τ ) ∥ r H + E     Z t 0 ∥∇ w ( s ) ∥ 2 H d s     r 2 ≤ C t r 2 E sup τ ∈ [0 ,t ] ∥ v 1 ( τ ) − v 2 ( τ ) ∥ r H , where C > 0 only dep ends on the structural parameters of the problem and, in particular, is indep endent of the time t ≥ 0 . Recalling the deﬁnition of w , we ha v e indeed ∥S ( v 1 ) − S ( v 2 ) ∥ r X ≤ C t r 2 ∥ v 1 − v 2 ∥ r X . As the Lipsc hitz constant dep ends contin uously on t and v anishes in the limit t → 0 + for all ﬁxed p ≥ 1 , there exists some ﬁnal time T ∗ = T ∗ ( p ) (recall that r = r ( p ) ) dep ending also on the structural parameters of the problem such that C 1 r T 1 2 ∗ < 1 , and, therefore, the classical Banach ﬁxed p oin t theorem for contractions applies, yielding existence and uniqueness of a ﬁxed p oin t for the map S , i.e., existence and uniqueness of a lo cal-in-time probabilistically- strong solution to problem ( 3.1 ). In order to retrieve a global solution, exploiting uniqueness of solutions, w e can apply a standard patc hing argumen t (up to P -indistinguishabilit y), since the maximal time T ∗ only dep ends on univ ersal constan ts. 3.2. Uniform estimates. Having established the existence of approximated solutions, we now present an ensemble of uniform estimates with resp ect to the Y osida regularization parameter λ > 0 . First estimate. First of all, we w an t to apply the Itô form ula to the H -norm of u λ . F ollowing the procedure illustrated in the previous subsection, inv oking again b oth [ 16 , Theorem 4.32]) and [ 10 , Theorem B.1] we 14 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA ha v e 1 2 ∥ u λ ( t ) ∥ 2 H + ε Z t 0 ∥∇ u λ ( s ) ∥ 2 H d s = 1 2 ∥ u λ (0) ∥ 2 H − Z t 0  u λ ( s ) , Ψ ′ λ ( u λ ( s ))  H d s − Z t 0 ( u λ ( s ) , F ( u λ ( s ))) H d s + Z t 0 ( u λ ( s ) , G λ ( u λ ( s )) d W ( s )) H + 1 2 Z t 0 ∥ G λ ( u λ ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( u λ ( s − ) , J λ ( u λ ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z  ∥ u λ ( s − ) + J λ ( u λ ( s − ) , z ) ∥ 2 H − ∥ u λ ( s − ) ∥ 2 H − 2( u λ ( s ) , J λ ( u λ ( s − ) , z )) H  ν (d z ) d s. (3.11) In particular, recalling the monotonicit y of Ψ ′ λ due to the conv exit y of Ψ λ , we clearly hav e Z t 0  u λ ( s ) , Ψ ′ λ ( u λ ( s ))  H d s ≥ − C 1 , (3.12) and by Assumption (A3) and Lemma 3.1 we ha v e     Z t 0 ( u λ ( s ) , F ( u λ ( s ))) H d s     ≤ q e C F Z t 0 (1 + ∥ u λ ( s ) ∥ H ) 2 d s (3.13) as well as 1 2 Z t 0 ∥ G λ ( u λ ( s )) ∥ 2 L 2 ( U,H ) d s ≤ C  t + Z t 0 ∥ u λ ( s ) ∥ 2 H d s  . (3.14) F or the last term in ( 3.11 ), w e obtain Z t 0 Z Z  ∥ u λ ( s − ) + J λ ( u λ ( s − ) , z ) ∥ 2 H − ∥ u λ ( s − ) ∥ 2 H − 2( u λ ( s − ) , J λ ( u λ ( s − ) , z )) H  ν (d z ) d s = Z t 0 Z Z ∥ J λ ( u λ ( s − ) , z ) ∥ 2 H ν (d z ) d s ≤ C  t + Z t 0 ∥ u λ ( s ) ∥ 2 H d s  . (3.15) Collecting ( 3.12 )-( 3.15 ) in ( 3.11 ), then taking r 2 -th p o w ers, with r = min { 4 , p } , supremums in time and P -exp ectations, we ha v e, E sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r H + E     Z t 0 ∥∇ u λ ( s ) ∥ 2 H d s     r 2 ≤ C t r 2 + E     Z t 0 ∥ u λ ( s ) ∥ 2 H d s     r 2 ! + E sup τ ∈ [0 ,t ]     Z τ 0 ( u λ ( s ) , G λ ( u λ ( s ))) H d W ( s )     r 2 + E sup τ ∈ [0 ,t ]     Z t 0 Z Z ( u λ ( s − ) , J λ ( u λ ( s − ) , z )) H µ (d s, d z )     r 2 . (3.16) AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 15 In order to handle the sto c hastic terms in ( 3.16 ), w e mimic the computations in ( 3.9 ) and ( 3.10 ), namely E sup τ ∈ [0 ,t ]     Z τ 0 ( u λ ( s ) , G λ ( u λ ( s ))) H d W ( s )     r 2 ≤ C E     Z t 0 ∥ u λ ( s ) ∥ 2 H ∥ G λ ( u λ ) ∥ 2 L 2 ( U,H ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r 2 H     Z t 0 ∥ G λ ( u λ ( s )) ∥ 2 L 2 ( U,H ) d s     r 4 # ≤ C E " sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r 2 H t r 4 +     Z t 0 ∥ u λ ( s ) ∥ 2 H d s     r 4 !# ≤ 1 4 E sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r H + C t r 2 + E     Z t 0 ∥ u λ ( s ) ∥ 2 H d s     r 2 ! , (3.17) and, by the same tok en, using also the fact that r ≤ 4 , E sup τ ∈ [0 ,t ]     Z t 0 Z Z ( u λ ( s − ) , J λ ( u λ ( s − ) , z )) H µ (d s, d z )     r 2 ≤ C E     Z t 0 ∥ u λ ( s − ) ∥ 2 H Z Z ∥ J λ ( u λ ( s − ) , z ) ∥ 2 H ν (d z ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ] ∥ u λ ( τ − ) ∥ r 2 H t r 4 +     Z t 0 ∥ u λ ( s ) ∥ 2 H d s     r 4 !# ≤ 1 4 E sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r H + C t r 2 + E     Z t 0 ∥ u λ ( s ) ∥ 2 H d s     r 2 ! , (3.18) Com bining ( 3.16 ), ( 3.17 ) and ( 3.18 ) and exploiting the Gronw all lemma, we obtain that there exists a constan t C 1 > 0 indep enden t of λ > 0 such that ∥ u λ ∥ L r (Ω; L ∞ (0 ,T ; H )) ∩ L r (Ω; L 2 (0 ,T ; V )) ≤ C 1 (3.19) for all λ > 0 and r = min { 4 , p } . Se c ond estimate. A second useful estimate is given b y applying the Itô lemma to H : H → R H ( v ) := Z O Φ λ ( v ) d x ev aluated at v = u λ ( s ) for any s ≥ 0 , where, in turn, w e set Φ λ : R → R Φ λ ( r ) = Ψ λ ( r ) + Z r 0 F ( s ) d s for all r ∈ R . Without loss of generalit y , w e assume that Φ λ ≥ 0 . Indeed, it is well kno wn that for λ ∈ (0 , λ 0 ) it holds Ψ λ ( s ) ≥ 1 4 λ 0 s 2 − C λ 0 , and therefore, as Assumption (A3) implies | F ( s ) | ≤ q e C F (1 + s ) for all s ∈ R , w e also hav e Φ λ ( s ) ≥  1 4 λ 0 − q e C F  s 2 − C λ 0 − e C F 16 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA for all s ∈ R and all λ ∈ (0 , λ 0 ) . Cho osing an y λ 0 > 0 suﬃcien tly small yields a b ound from b elo w for Φ λ . The application of the Itô lemma yields (compare with [ 7 , Subsection 4.2]), thanks to [ 10 , Theorem. B.1] Z O Φ λ ( u λ ( t )) d x + ε Z t 0 (Φ ′′ λ ( u λ ( s )) , |∇ u λ ( s ) | 2 ) H d s + Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s = Z O Φ λ ( u λ (0)) d x + 1 2 Z t 0 + ∞ X k =0 (Φ ′′ λ ( u λ ( s )) , | G λ ( u λ ( s ))[ e k ] | 2 ) H d s + Z t 0 (Φ ′ λ ( u λ ( s )) , G λ ( u λ ( s ))) H d W ( s ) + Z t 0 Z Z (Φ ′ λ ( u λ ( s − )) , J λ ( u λ ( s − ) , z )) H µ (d t, d z ) + Z t 0 Z Z  Z O Φ λ ( u λ ( s − ) + J λ ( u λ ( s − ) , z )) d x − Z O Φ λ ( u λ ( s − )) d x − Z O Φ ′ λ ( u λ ( s − )) J λ ( u λ ( s − ) , z ) d x  ν (d z ) d s. (3.20) On the left hand side, we observe that Z t 0 (Φ ′′ λ ( u λ ( s )) , |∇ u λ ( s ) | 2 ) H d s = Z t 0 (Ψ ′′ λ ( u λ ( s )) , |∇ u λ ( s ) | 2 ) H d s + Z t 0 ( F ′ ( u λ ( s )) , |∇ u λ ( s ) | 2 ) H d s ≥ − C F Z t 0 ∥∇ u λ ( s ) ∥ 2 H d s (3.21) b y con vexit y of Ψ λ and the fact that F ′ is w ell-deﬁned almost ev erywhere and bounded, as a consequence of Assumption (A3) and Rademacher’s theorem. On the righ t hand side, arguing similarly and o wing to Lemma 3.1 we hav e 1 2 Z t 0 + ∞ X k =1 (Φ ′′ λ ( u λ ( s )) , | G λ ( u λ ( s ))[ e k ] | 2 ) H d s ≤ C Z t 0 + ∞ X k =1 ∥ q Ψ ′′ λ ( u λ ( s )) | G λ ( u λ ( s ))[ e k ] |∥ 2 H d s + Z t 0 ∥ G λ ( u λ ( s )) ∥ 2 L 2 ( U,H ) d s ! ≤ C  t + Z t 0 ∥ Ψ λ ( u λ ( s )) ∥ L 1 ( O ) d s + Z t 0 ∥ u λ ( s ) ∥ 2 H d s  . (3.22) Moreo v er, o wing once again to Lemma 3.1     Z t 0 Z Z  Z O Φ λ ( u λ ( s − ) + J λ ( u λ ( s − ) , z )) d x − Z O Φ λ ( u λ ( s − )) d x − Z O Φ ′ λ ( u λ ( s − )) J λ ( u λ ( s − ) , z ) d x  ν (d z ) d s     ≤ C     Z t 0 Z Z Z O Z 1 0 Φ ′′ λ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     ≤ C     Z t 0 Z Z Z O Z 1 0 Ψ ′′ λ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     + C     Z t 0 Z Z Z O Z 1 0 F ′ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     ≤ C     Z t 0 Z Z Z O Z 1 0 Ψ ′′ λ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     + C     Z t 0 Z Z Z O | J λ ( u λ ( s − ) , z ) | 2 d x ν (d z ) d s     . (3.23) Without loss of generality , we shall assume that L is ﬁnite. The pro of in the case L = + ∞ is analogous. It is clear how to handle the second term in ( 3.23 ). F or the ﬁrst one, w e inv ok e Lemma 3.1 once more. Indeed, observe that u λ ( s − ) + θ J λ ( u λ ( s − ) , z ) = (1 − θ ) u λ ( s − ) + θ [ u λ ( s − ) + J λ ( u λ ( s − ) , z )] for any θ ∈ [0 , 1] , and in turn, Lemma 3.1 implies there exists a constant σ ∈ [0 , 1] suc h that u λ ( s − ) + θ J λ ( u λ ( s − ) , z ) = (1 − θ ) u λ ( s − ) + σ θ δ J R λ ( u λ ( s − )) + (1 − σ ) θ  (1 − δ J ) L + δ J R λ ( u λ ( s − ))  AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 17 and using the conv exit y of Ψ ′′ λ w e ev en tually arriv e at     Z t 0 Z Z Z O Z 1 0 Ψ ′′ λ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     ≤     Z t 0 Z Z Z O Ψ ′′ λ ( u λ ( s − )) | J λ ( u λ ( s − ) , z ) | 2 d x ν (d z ) d s     +     Z t 0 Z Z Z O Ψ ′′ λ ( δ J R λ ( u λ ( s − ))) | J λ ( u λ ( s − ) , z ) | 2 d x ν (d z ) d s     +     Z t 0 Z Z Z O Ψ ′′ λ ((1 − δ J ) L + δ J R λ ( u λ ( s − ))) | J λ ( u λ ( s − ) , z ) | 2 d x ν (d z ) d s     The last tw o terms can b e con trolled directly by means of Lemma 3.1 . As for the ﬁrst one, recall that δ J R λ ( v ) ≤ R λ ( v ) = (1 − δ J ) R λ ( v ) + δ J R λ ( v ) ≤ (1 − δ J ) L + δ J R λ ( v ) as the range of R λ lies in the eﬀectiv e domain of Ψ ′ , namely (0 , L ) . Since, by deﬁnition of Y osida appro ximation, Ψ ′′ λ ≤ Ψ ′′ ◦ R λ and Ψ ′′ is conv ex, we deduce that     Z t 0 Z Z Z O Ψ ′′ λ ( u λ ( s − )) | J λ ( u λ ( s − ) , z ) | 2 d x ν (d z ) d s     ≤ C      Z t 0 Z Z Z O Ψ ′′ λ ( u λ ( s − )) | J ( R λ ( u λ ( s − )) , z ) | 2 d x ν (d z ) d s     + λ 2     Z t 0 Z O Ψ ′′ λ ( u λ ( s − )) | Ψ ′ λ ( u λ ( s − )) | 2 d x d s      ≤ C      Z t 0 Z Z Z O Ψ ′′ ( R λ ( u λ ( s − ))) | J ( R λ ( u λ ( s − )) , z ) | 2 d x ν (d z ) d s     + λ     Z t 0 Z O | Ψ ′ λ ( u λ ( s − )) | 2 d x d s      ≤ C      Z t 0 Z Z Z O Ψ ′′ ( δ J R λ ( u λ ( s − ))) | J ( R λ ( u λ ( s − )) , z ) | 2 d x ν (d z ) d s     +     Z t 0 Z Z Z O Ψ ′′ ((1 − δ J ) L + δ J R λ ( u λ ( s − ))) | J ( R λ ( u λ ( s − )) , z ) | 2 d x ν (d z ) d s     + λ     Z t 0 Z O | Ψ ′ λ ( u λ ( s − )) | 2 d x d s      On account of all of the ab o v e, w e conclude by Lemma 3.1 and Assumption (A6) that     Z t 0 Z Z Z O Z 1 0 Ψ ′′ λ ( u λ ( s − ) + θ J λ ( u λ ( s − ) , z )) | J λ ( u λ ( s − ) , z ) | 2 d θ d x ν (d z ) d s     ≤ C  1 + Z t 0 ∥ Ψ λ ( u λ ( s − )) ∥ L 1 ( O ) d s + λ Z t 0 ∥ Ψ ′ λ ( u λ ( s − )) ∥ 2 H d s  . Collecting the ab o v e estimates in ( 3.20 ), raising the result to the p o w er r 2 , with r = min { p, 4 } , taking suprem ums in time and P -exp ectations, we get E sup τ ∈ [0 ,t ]     Z O Φ λ ( u λ ( t )) d x     r 2 + (1 − C λ ) r 2 E     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 2 ≤ 6 r 2 C r 2 " 1 + E     Z O Φ λ ( u λ (0)) d x     r 2 + E     Z O ∥∇ u λ ( s ) ∥ 2 H d s     r 2 + E     Z t 0 ∥ Φ λ ( u λ ( s )) ∥ L 1 ( O ) d s     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 (Φ ′ λ ( u λ ( s )) , G λ ( u λ ( s ))) H d W ( s )     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 Z Z (Φ ′ λ ( u λ ( s − )) , J λ ( u λ ( s − ) , z )) H µ (d t, d z )     r 2 # , (3.24) where we reconstructed Φ λ from Ψ λ b y algebraic manipulations and using the First estimate (recall that an y primitive of F is quadratically b ounded). In ( 3.24 ), w e kept C indep enden t of b oth λ and r for b etter clarit y . The v alue of C > 0 is then the same on b oth the left and the righ t hand side. Finally , we need to 18 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA con trol the sto chastic terms, following the same strategy employ ed in the First estimate, namely E sup τ ∈ [0 ,t ]     Z τ 0  Φ ′ λ ( u λ ( s )) , G λ ( u λ ( s ))  H d W ( s )     r 2 ≤ C E     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H ∥ G λ ( u λ ( s )) ∥ 2 L 2 ( U,H ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ] ∥ G λ ( u λ ( τ )) ∥ r 2 L 2 ( U,H )     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 4 # ≤ C E " sup τ ∈ [0 ,t ]  1 + ∥ u λ ( τ ) ∥ r 2 H      Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 4 # ≤ σ E     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 2 + C 1 + E sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r H ! , (3.25) where σ > 0 is arbitrarily small (but ﬁxed). Analogously , owing still to the fact that r ≤ 4 , we hav e E sup τ ∈ [0 ,t ]     Z τ 0 Z Z (Φ ′ λ ( u λ ( s − )) , J λ ( u λ ( s − ) , z )) H µ (d s, d z )     r 2 ≤ C E     Z t 0 ∥ Φ ′ λ ( u λ ( s − )) ∥ 2 H Z Z ∥ J λ ( u λ ( s − ) , z ) ∥ 2 H ν (d z ) d s     r 4 ≤ C E " sup τ ∈ [0 ,t ]     Z Z ∥ J λ ( u λ ( τ − ) , z ) ∥ 2 H ν (d z )     r 4     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 4 # ≤ σ E     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 2 + C 1 + E sup τ ∈ [0 ,t ] ∥ u λ ( τ ) ∥ r H ! . (3.26) Let us remark that, in all Burkholder–Davis–Gundy-t ype estimates for jump terms, in order to allow for r > 4 , controls in L p (Ω; L p (0 , T ; H )) are generally needed. In particular, this would require some strengthened form of Assumption (A6) . In the ab o v e computations, we used the prop erties of G λ and J λ giv en by Lemma 3.1 . Consequen tly , for suﬃciently small v alues of σ and λ , ( 3.25 ) and ( 3.26 ) jointly with ( 3.24 ) imply that E sup τ ∈ [0 ,t ]     Z O Φ λ ( u λ ( t )) d x     r 2 + E     Z t 0 ∥ Φ ′ λ ( u λ ( s )) ∥ 2 H d s     r 2 ≤ C " 1 + E     Z O Φ λ ( u λ (0)) d x     r 2 + E     Z t 0 ∥ Φ λ ( u λ ( s )) ∥ L 1 ( O ) d s     r 2 # , o wing also to the First estimate. The Gronw all lemma then implies that there exists a constant C 2 > 0 indep enden t of λ suc h that ∥ Φ ′ λ ( u λ )) ∥ L r (Ω; L 2 (0 ,T ; H )) ≤ C 2 . It is easy to c heck b y comparison that ∥ Ψ ′ λ ( u λ )) ∥ L r (Ω; L 2 (0 ,T ; H )) ≤ C 2 for a p ossibly diﬀerent constant that w e do not relab el. Thir d estimate. The ﬁnal estimate aims to sho w that the sequence { u λ } λ> 0 is a Cauc h y sequence in the space L r (Ω; L ∞ (0 , T ; H )) ∩ L r (Ω; L 2 (0 , T ; V )) where r = min { 4 , p } , along v alues of λ that con verge to 0. Therefore, it must b e con vergen t to some limit. This estimate follows closely the computations done in the First estimate (see also Subsection 3.4 ) and AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 19 exploits the resolven t identit y to get a dep endence on λ . Let λ 1 > 0 and let λ 2 > 0 such that λ 1 > λ 2 . Applying the Itô lemma to the H -norm of u λ 2 − u λ 1 w e get 1 2 ∥ u λ 1 ( t ) − u λ 2 ( t ) ∥ 2 H + Z t 0 ∥∇ u λ 1 ( s ) − ∇ u λ 2 ( s ) ∥ 2 H d s = − Z t 0  u λ 1 ( s ) − u λ 2 ( s ) , Ψ ′ λ 1 ( u λ 1 ( s )) − Ψ ′ λ 2 ( u λ 2 ( s ))  H d s − Z t 0 ( u λ 1 ( s ) − u λ 2 ( s ) , F ( u λ 1 ( s )) − F ( u λ 2 ( s ))) H d s + Z t 0 ( u λ 1 ( s ) − u λ 2 ( s ) , ( G λ 1 ( u λ 1 ( s )) − G λ 2 ( u λ 2 ( s ))) d W ( s )) H + 1 2 Z t 0 ∥ G λ 1 ( u λ 1 ( s )) − G λ 2 ( u λ 2 ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( u λ 1 ( s − ) − u λ 2 ( s − ) , J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z  ∥ u λ 1 ( s − ) − u λ 2 ( s − ) + J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z ) ∥ 2 H − ∥ u λ 1 ( s − ) − u λ 2 ( s − ) ∥ 2 H − 2( u λ 1 ( s ) − u λ 2 ( s ) , J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z )) H  ν (d z ) d s. (3.27) Recalling the resolven t identit y λ Ψ ′ λ ( s ) + R λ ( s ) = s for all s ∈ R and λ > 0 , w e observ e that  u λ 1 ( s ) − u λ 2 ( s ) , Ψ ′ λ 1 ( u λ 1 ( s )) − Ψ ′ λ 2 ( u λ 2 ( s ))  H =  λ 1 Ψ ′ λ 1 ( u λ 1 ( s )) − λ 2 Ψ ′ λ 2 ( u λ 2 ( s )) , Ψ ′ λ 1 ( u λ 1 ( s )) − Ψ ′ λ 2 ( u λ 2 ( s ))  H +  R λ 1 ( u λ 1 ( s )) − R λ 2 ( u λ 2 ( s )) , Ψ ′ λ 1 ( u λ 1 ( s )) − Ψ ′ λ 2 ( u λ 2 ( s ))  H ≥ − ( λ 1 + λ 2 )(Ψ ′ λ 1 ( u λ 1 ( s )) , Ψ ′ λ 2 ( u λ 2 ( s ))) H ≥ − λ 1 + λ 2 2 ∥ Ψ ′ λ 1 ( u λ 1 ( s )) ∥ 2 − λ 1 + λ 2 2 ∥ Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 . The second term, thanks to the Lipschitz contin uit y of F , satisﬁes   ( u λ 1 ( s ) − u λ 2 ( s ) , F ( u λ 1 ( s )) − F ( u λ 2 ( s ))) H   ≤ C ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H , with C > 0 indep endent of λ . Thanks to the Lipschitz contin uit y of G and recalling that G λ = G ◦ R λ , w e get ∥ G λ 1 ( u λ 1 ( s )) − G λ 2 ( u λ 2 ( s )) ∥ 2 L 2 ( U,H ) ≤ ∥ R λ 1 ( u λ 1 ( s )) − R λ 2 ( u λ 2 ( s )) ∥ 2 H and the only deterministic term left is the last one. By the same tok en, the jump term satisﬁes Z Z  ∥ u λ 1 ( s − ) − u λ 2 ( s − ) + J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z ) ∥ 2 H − ∥ u λ 1 ( s − ) − u λ 2 ( s − ) ∥ 2 H − 2( u λ 1 ( s ) − u λ 2 ( s ) , J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z )) H  ν (d z ) = Z Z ∥ J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z ) ∥ 2 ν (d z ) ≤ C ∥ R λ 1 ( u λ 1 ( s − )) − R λ 2 ( u λ 2 ( s − )) ∥ 2 H . 20 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA Emplo ying the abov e estimates in ( 3.27 ), raising to the p o w er r 2 , taking suprem ums in time and P - exp ectations, we arriv e at E sup τ ∈ [0 ,t ] ∥ u λ 1 ( t ) − u λ 2 ( t ) ∥ r H + E     Z t 0 ∥∇ u λ 1 ( s ) − ∇ u λ 2 ( s ) ∥ 2 H d s     r 2 ≤ C " E     Z t 0 ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H d s     r 2 + E     Z t 0 ∥ R λ 1 ( u λ 1 ( s )) − R λ 2 ( u λ 2 ) ∥ 2 H d s     r 2 + λ 1 + λ 2 2 E     Z t 0 ∥ Ψ ′ λ 1 ( u λ 1 ( s )) ∥ 2 H d s     r 2 + λ 1 + λ 2 2 E     Z t 0 ∥ Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 H d s     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 ( u λ 1 ( s ) − u λ 2 ( s ) , ( G λ 1 ( u λ 1 ( s )) − G λ 2 ( u λ 2 ( s ))) d W ( s )) H     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 Z Z ( u λ 1 ( s − ) − u λ 2 ( s − ) , J λ 1 ( u λ 1 ( s − ) , z ) − J λ 2 ( u λ 2 ( s − ) , z )) H µ (d s, d z )     r 2 # (3.28) F or the sto c hastic terms in ( 3.28 ), we follow the same computations of the First estimate so that, even tually , E sup τ ∈ [0 ,t ] ∥ u λ 1 ( t ) − u λ 2 ( t ) ∥ r H + E     Z t 0 ∥∇ u λ 1 ( s ) − ∇ u λ 2 ( s ) ∥ 2 H d s     r 2 ≤ C " E     Z t 0 ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H d s     r 2 + E     Z t 0 ∥ R λ 1 ( u λ 1 ( s )) − R λ 2 ( u λ 2 ) ∥ 2 H d s     r 2 + λ 1 + λ 2 2 E     Z t 0 ∥ Ψ ′ λ 1 ( u λ 1 ( s )) ∥ 2 H d s     r 2 + λ 1 + λ 2 2 E     Z t 0 ∥ Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 H d s     r 2 # (3.29) The last tw o exp ectations in ( 3.29 ) are uniformly b ounded in λ 1 and λ 2 . F or the resolven t diﬀerence, observ e that by summing and subtracting R λ 1 ( u λ 2 ( s )) ∥ R λ 1 ( u λ 1 ( s )) − R λ 2 ( u λ 2 ( s )) ∥ 2 H ≤ 2 ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H + 2 ∥ R λ 1 ( u λ 2 ( s )) − R λ 2 ( u λ 2 ( s )) ∥ 2 H ≤ C  ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H + ∥ λ 1 Ψ ′ λ 1 ( u λ 2 ( s )) − λ 2 Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 H  ≤ C  ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H + λ 2 1 ∥ Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 H + λ 2 2 ∥ Ψ ′ λ 2 ( u λ 2 ( s )) ∥ 2 H  where we used the fact that the Y osida appro ximation conv erges monotonically from b elo w as λ → 0 + (recall that λ 1 > λ 2 ). Then, we arrive at E sup τ ∈ [0 ,t ] ∥ u λ 1 ( t ) − u λ 2 ( t ) ∥ r H + E     Z t 0 ∥∇ u λ 1 ( s ) − ∇ u λ 2 ( s ) ∥ 2 H d s     r 2 ≤ C " E     Z t 0 ∥ u λ 1 ( s ) − u λ 2 ( s ) ∥ 2 H d s     r 2 + λ 1 + λ 2 + λ 2 1 + λ 2 2 # (3.30) The Gronw all lemma implies the result, taking the limits λ 1 → 0 + and λ 2 → 0 + . 3.3. P assage to the limit λ → 0 + . In this section, w e infer conv ergence prop erties from the estimates sho wn in the previous Subsection. All conv ergences illustrated hereafter are alwa ys intended as λ → 0 + along an arbitrary decreasing sequence. First of all, since { u λ } λ> 0 is a Cauc hy sequence in a Banac h space, w e conclude that u λ → u in L r (Ω; L ∞ (0 , T ; H )) ∩ L r (Ω; L 2 (0 , T ; V )) where r = min { 4 , p } . By reﬂexivity and the uniform b ound given in the Se c ond estimate , we hav e Φ ′ λ ( u λ )  ξ in L r (Ω; L 2 (0 , T ; H )) . AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 21 It is our aim now to pro v e that ξ = Ψ ′ ( u ) + F ( u ) . Observe that since F is a Lipschitz-con tin uous function w e immediately achiev e that E     Z t 0 ∥ F ( u λ ( s )) − F ( u ( s )) ∥ 2 H d s     r 2 ≤ C E     Z t 0 ∥ u λ ( s )) − u ( s ) ∥ 2 H d s     r 2 giving F ( u λ ) → F ( u ) in L r (Ω; L 2 (0 , T ; H )) . In particular, then, by comparison it is immediate to deduce that Ψ ′ λ ( u λ )  ξ − F ( u ) in L r (Ω; L 2 (0 , T ; H )) . Since Ψ ′ λ = Ψ ′ ◦ R λ , we now study the con vergence of the sequence R λ ( u λ ) . The triangle inequality and the resolven t identit y give ∥ R λ ( u λ ( s )) − u ( s ) ∥ 2 H ≤ 2 ∥ R λ ( u λ ( s )) − u λ ( s ) ∥ 2 H + 2 ∥ u λ ( s ) − u ( s ) ∥ 2 H = 2 λ 2 ∥ Ψ ′ λ ( u λ ( s )) ∥ 2 H + 2 ∥ u λ ( s ) − u ( s ) ∥ 2 H implying that R λ ( u λ ) → u in L r (Ω; L 2 (0 , T ; H )) , thanks to the Se c ond estimate . Now, observe that E Z t 0 (Ψ ′ ( R λ ( u λ ( s ))) − Ψ ′ ( y ( s )) , R λ ( u λ ( s )) − y ( s )) H d s ≥ 0 for all y ∈ L 2 (Ω; L 2 (0 , T ; H )) suc h that Ψ ′ ( y ) ∈ L 2 (Ω; L 2 (0 , T ; H )) and Ψ ′ is maximal monotone. Passing to the limit yields ( ξ − F ( u ) − Ψ ′ ( y ) , u − y ) L 2 (Ω; L 2 (0 ,T ; H ) ≥ 0 and maximality implies ξ − F ( u ) = Ψ ′ ( u ) , as claimed. Observe, that, in turn, this implies that u ∈ V ∩ Γ a.e. in Ω × O × (0 , T ) Finally , w e are left to deal with the stochastic in tegrals. Let us start from the Itô term. The k ey to ol is, once again, the Burkholder-Davis-Gundy inequalit y , recalling that G λ = G ◦ R λ . Indeed, E sup τ ∈ [0 ,t ]     Z τ 0 G ( R λ ( u λ ( s ))) − G ( u ( s )) d W ( s )     r H ≤ C E     Z t 0 ∥ G ( R λ ( u λ ( s ))) − G ( u ( s )) ∥ 2 L 2 ( U,H ) d s     r 2 ≤ C E     Z t 0 ∥ R λ ( u λ ( s )) − u ( s ) ∥ 2 H d s     r 2 ≤ C " λ 2 E     Z t 0 ∥ Ψ ′ λ ( u λ ( s )) ∥ 2 H d s     r 2 + E     Z t 0 ∥ u λ ( s ) − u ( s ) ∥ 2 H d s     r 2 # giving G λ ( u λ ) · W → G ( u ) · W in L r (Ω; L ∞ (0 , T ; H )) . A similar argument works for the jump term, as, since r ≤ 4 , the Burkholder-Da vis-Gundy inequality w orks similarly . Therefore, Z Z J λ ( u − λ , z ) µ (d t, d z ) → Z Z J ( u − , z ) µ (d t, d z ) in L r (Ω; L ∞ (0 , T ; H )) . The prov en con vergences are also enough to infer that the limit pro cess u is indeed a probabilistically-strong solution to ( 1.2 ). The pro of of existence is complete. 22 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA 3.4. Uniqueness of solutions. Finally , we prov e that the solution to problem ( 1.2 ) is path wise unique. This is, essentially , a third iteration of the computations shown in Subsection 3.1 . In particular, let u 01 and u 02 b e elements of L p (Ω; H ) complying with the assumptions of Theorem 2.4 . F or i ∈ { 1 , 2 } , let u i denote the solution to ( 1.2 ) having u 0 i as initial state. Setting u := u 1 − u 2 , u 0 := u 01 − u 02 w e ha v e that formally              d u − ε ∆ u d t = − [Ψ ′ ( u 1 ) − Ψ ′ ( u 2 )] d t − [ F ( u 1 ) − F ( u 2 )] d t + [ G ( u 1 ) − G ( u 2 )] d W + Z Z  J ( u − 1 , z ) − J ( u − 2 , z )  µ (d t, d z ) in (0 , T ) × O , α d u + α n ∂ n u = 0 on (0 , T ) × ∂ O , u ( · , 0) = u 0 in O . (3.31) F or simplicit y , in the follo wing we set without loss of generality α d = 0 and α n = 1 . Arguing as in Subsection 3.1 , we apply the Itô lemma to the functional v 7→ 1 2 ∥ v ∥ 2 H ev aluated at v = u ( t ) for any t > 0 . This yields 1 2 ∥ u ( t ) ∥ 2 H + ε Z t 0 ∥∇ u ( s ) ∥ 2 H d s = 1 2 ∥ u 01 − u 02 ∥ 2 H − Z t 0  u ( s ) , Ψ ′ ( u 1 ( s )) − Ψ ′ ( u 2 ( s ))  H d s − Z t 0 ( u ( s ) , F ( u 1 ( s )) − F ( u 2 ( s ))) H d s + Z t 0 ( u ( s ) , ( G ( u 1 ( s )) − G ( u 2 ( s ))) d W ( s )) H + 1 2 Z t 0 ∥ G ( u 1 ( s )) − G ( u 2 ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( u ( s − ) , J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z  ∥ u ( s − ) + J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z ) ∥ 2 H − ∥ u ( s − ) ∥ 2 H − 2( u ( s ) , J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z )) H  ν (d z ) d s. (3.32) In the same fashion, we employ all the known assumptions as well as the conv exity prop erties of Ψ to deduce that Z t 0  u ( s ) , Ψ ′ ( u 1 ( s )) − Ψ ′ ( u 2 ( s ))  H d s ≥ 0 (3.33) as well as     Z t 0 ( u ( s ) , F ( u 1 ( s )) − F ( u 2 ( s ))) H d s     ≤ C Z t 0 ∥ u ( s ) ∥ 2 H d s (3.34) and 1 2 Z t 0 ∥ G ( u 1 ( s )) − G ( u 2 ( s )) ∥ 2 L 2 ( U,H ) d s ≤ C Z t 0 ∥ u ( s ) ∥ 2 H d s. (3.35) Once again, for the last term of ( 3.32 ) we hav e Z t 0 Z Z  ∥ u ( s − ) + J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z ) ∥ 2 H − ∥ u ( s − ) ∥ 2 H − ( u ( s − ) , J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z )) H  ν (d z ) d s = Z t 0 Z Z ∥ J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z ) ∥ 2 H ν (d z ) d s ≤ C Z t 0 ∥ u ( s ) ∥ 2 H d s. (3.36) AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 23 Therefore, collecting ( 3.33 )-( 3.36 ) in ( 3.32 ), taking r 2 -th p ow ers, with r = min { 4 , p } , supremums in time and P -exp ectations, w e get E sup τ ∈ [0 ,t ] ∥ u ( τ ) ∥ r H + E     Z t 0 ∥∇ u ( s ) ∥ 2 H d s     r 2 ≤ C E     Z t 0 ∥ u ( s ) ∥ 2 d s     r 2 + E sup τ ∈ [0 ,t ]     Z τ 0 ( u ( s ) , G ( u 1 ( s )) − G ( u 2 ( s ))) H d W ( s )     r 2 + E sup τ ∈ [0 ,t ]     Z t 0 Z Z ( u ( s − ) , J ( u 1 ( s − ) , z ) − J ( u 2 ( s − ) , z )) H µ (d t, d z )     r 2 . (3.37) On account of the computations in ( 3.9 ) and ( 3.10 ), we can conclude b y the Gronw all lemma that ∥ u 1 − u 2 ∥ L r (Ω; L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V )) ≤ C ∥ u 01 − u 02 ∥ L r (Ω; H ) and the claim follows setting u 01 = u 02 P -almost surely . 4. Pr oof of Theorem 2.5 This section is devoted to sho wing Theorem 2.5 , i.e., to p erforming a long-time analysis of the sto c hastic equation ( 1.2 ) in terms of existence and uniqueness of inv arian t measures and mixing prop erties. As we shall see, it will b e useful to introduce some ad-ho c notation that w e illustrate hereafter. 4.1. Preliminaries. First, we consider the set A := { v ∈ H : Ψ( v ) ∈ L 1 ( O ) } = [ n ∈ N  v ∈ H : Z O Ψ( v ) ≤ n  , whic h implies that A is a conv ex Borel subset of H since Ψ is con vex and lo wer semicon tinuous. Let us endo w the set A with the structure of metric space inherited by the space H . The resulting separable metric space will b e denoted by ( A , d ) . The sym b ol B ( A ) denotes the σ -algebra of all Borel subsets of A ; moreov er, for a set A ∈ B ( A ) , we denote by A c its complement. By P ( A ) we denote the set of all probabilit y measures on ( A , B ( A )) . Moreov er, B b ( A ) and C b ( A ) will stand for the space of all measurable b ounded functions from A to R and the space of contin uous b ounded functions from A to R , respectively . Instead, b y Lip( A ) we denote the space of Lipschitz con tinuous functions from A to R , endow ed with its classical norm ∥ · ∥ Lip . W e are now in a p osition to rigorously introduce the family of transition op erators asso ciated to equation ( 1.2 ). F or every x ∈ A , we denote by u x the unique solution to ( 1.2 ) with initial datum x . Accordingly , for every t ≥ 0 , we set u x ( t ) := u ( t, x ) to denote its v alue at time t . Observ e that for all x ∈ A , Theorem 2.4 establishes that u x ( t ) ∈ A for every t > 0 , P -almost surely, and that u x ( t ) is F t / B ( A ) -measurable. Hence, given t > 0 , the law of u x ( t ) on A is deﬁned as the pushforward measure L t : A × B ( A ) → [0 , 1] , L t ( x, A ) = La w P ( u x ( t ))( A ) = P ( u x ( t ) ∈ A ) , ∀ x ∈ A , A ∈ B ( A ) . F or every A ∈ B ( A ) one has that the map L 2 (Ω , F t ; H ) → [0 , 1] given by X 7→ Law P ( X )( A ) , X ∈ L 2 (Ω , F t ; H ) , is measurable: this can b e prov ed by density arguments, by approximating 1 A through Lipsc hitz-con tinuous functions and by the dominated conv ergence theorem. Hence, recalling that x 7→ u x ( t ) is contin uous from A to X ∈ L 2 (Ω , F t ; H ) , we ha v e that the map x 7→ L t ( x, A ) is B ( A ) -measurable for any A ∈ B ( A ) . This implies that the map L t is a transition kernel from ( A , B ( A )) into itself for ev ery t ≥ 0 . Hence, we can deﬁne the family of op erators P = ( P t ) t ≥ 0 as P t : B b ( A ) → B b ( A ) , ( P t ϕ )( x ) := Z A ϕ ( y ) L t ( x, d y ) = E [ ϕ ( u x ( t ))] , ∀ x ∈ A , ϕ ∈ B b ( A ) . The family of op erators P is w ell deﬁned: indeed, P t ϕ ∈ B b ( A ) for ev ery ϕ ∈ B b ( A ) since L t is a transition kernel. Let us p oin t out once more that, due to the nonlinear nature of the problem, the solution of equation ( 1.2 ) b elongs to A . Therefore, the transition semigroup can only make sense as a family of op erators acting on B b ( A ) and not on B b ( H ) as in the more classical cases. F urthermore, thanks 24 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA to [ 40 , Theorem 9.30] one has that the unique solution of ( 1.2 ) is also a Mark ov pro cess. Therefore, we deduce that the family of op erators P is a Mark o v semigroup, namely P t + s = P t P s for every s and t ≥ 0 . The following deﬁnition precises ﬁnally the notion of in v ariant measure. Deﬁnition 4.1. An inv ariant measure for the transition semigroup P is a probability measure γ ∈ P ( A ) suc h that Z A ϕ ( x ) γ (d x ) = Z A P t ϕ ( x ) γ (d x ) for all t ≥ 0 and for all ϕ ∈ C b ( A ) . Throughout this section, in tegration with resp ect to x will t ypically denote in tegration o v er A with resp ect to some probability measure, instead of integration o ver the space domain O . 4.2. Existence of an in v ariant measure. First, we sho w that the transition semigroup P admits indeed at least one inv ariant measure, in the sense of Deﬁnition 4.1 . The pro of relies on an adaptation of the Krylo v–Bogoliub o v theorem to the case of complete separable metric spaces, see [ 44 , Theorem 3.3]. As customary when applying the Krylov–Bogoliubov theorem, we need to sho w that the semigroup P is F eller and the tigh tness prop erty for time-a veraged laws. The F eller prop ert y follo ws directly from the con tin uous dep endence of the solution from the initial data. Indeed, let t > 0 and ϕ ∈ C b ( A ) b e ﬁxed. Giv en a sequence { x n } n ∈ N ⊂ A which conv erges in A to x ∈ A as n → + ∞ , we need to pro v e that the sequence { P t ϕ ( x n ) } n ∈ N con v erges to P t ϕ ( x ) as n → + ∞ . Owing to the con tin uous dep endence estimate in Theorem 2.4 , we ha v e that ∥ u x n ( t ) − u x ( t ) ∥ L 2 (Ω; H ) ≤ ∥ u x n − u x ∥ L 2 (Ω; L ∞ (0 ,t ; H )) ≤ C t ∥ x n − x ∥ H . It follows that, as n → + ∞ , for every t ≥ 0 , the con v ergence u x n ( t ) → u x ( t ) holds in L 2 (Ω; H ) , hence also in probabilit y . In turn, this implies that ϕ ( u x n ( t )) → ϕ ( u x ( t )) in probability by the contin uit y of ϕ . The b oundedness of ϕ and the Vitali theorem yield in particular that ϕ ( u x n ( t )) → ϕ ( u x ( t )) even in L 1 (Ω) , and th us | ( P t ϕ )( x n ) − ( P t ϕ )( x ) | ≤ E [ | ϕ ( u x n ( t )) − ϕ ( u x ( t )) | ] → 0 , as n → + ∞ . This shows that P is F eller. Next, w e prov e that P satisﬁes the tigh tness prop ert y given, for instance, in [ 44 , Theorem 3.3]. T o this end, we consider the solution to ( 1.2 ) stemming from a ﬁxed initial condition x 0 ∈ A . Consider then the family of measures ( γ t ) t> 0 ⊂ P ( A ) deﬁned by γ t : B ( A ) → [0 , 1] , A 7→ 1 t Z t 0 ( P s 1 A )( x 0 ) d s = 1 t Z t 0 P ( u x 0 ( t ) ∈ A ) d s, ∀ A ∈ B ( A ) , t > 0 . Let us prov e that the family is tight. Let B V n b e the closed ball in V of radius n ∈ N and centered at the origin, and set b B V n := B V n ∩ A to denote the part of the ball in A . Since the embedding V  → H is compact, the set b B V n is a compact subset of A . Exploiting Lemma A.1 and the Chebyc hev inequalit y we infer, for any t ≥ 1 , γ t  b B V n  c  = 1 t Z t 0 ( P s 1 ( b B V n ) c )( x 0 ) d s = 1 t Z t 0 P  ∥ u x 0 ( s ) ∥ 2 V ≥ n 2  d s ≤ 1 tn 2 Z t 0 E  ∥ u x 0 ( s ) ∥ 2 V  d s ≤ C n 2 , where C is a p ositiv e constant dep ending on the parameters e C G , e C J , C 1 and O in the case L < + ∞ and on the parameters e C G , e C J , C 1 , C 0 , ε and K 2 in the case L = + ∞ . Cho osing n ∈ N suﬃcien tly large yields the claim and completes the pro of. AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 25 4.3. Supp ort of inv ariant measures. Ha ving established the existence of in v ariant measures, w e fo cus here on some of their qualitativ e prop erties. In particular, we shall pro ve integrabilit y prop erties that in turn pro vide information on their supp ort, which is contained in a strict subset of A . T o this end we in tro duce the set A str :=  x ∈ A ∩ V : Φ ′ ( x ) ∈ H  ⊂ A . (4.1) In the deﬁnition ab ov e, recall that Φ is deﬁned as in Remark 2.3 . Exploiting the low er semicontin uit y of | Φ ′ | , it is p ossible to show that A str is a Borel subset of H , hence of A . Let γ ∈ P ( A ) b e an inv ariant measure for the transition semigroup P . F or the sake of clarity , we divide the argument into four steps. Step 1. First, w e sho w that there exists a p ositiv e constant C indep enden t of γ such that Z A ∥ x ∥ 2 H γ (d x ) ≤ C (4.2) for any inv arian t measure γ . T o this end, w e consider the mapping Θ : A → [0 , + ∞ ) , x 7→ ∥ x ∥ 2 H , ∀ x ∈ A and its approximations { Θ n } n ∈ N , deﬁned for every n ∈ N as Θ n : A → [0 , n 2 ] , Θ n : x 7→ ( ∥ x ∥ 2 H if x ∈ B H n ∩ A , n 2 otherwise , x ∈ A , where w e set B H n as the closed ball of radius n in H cen tered at zero. It is immediate to show that Θ n ∈ B b ( A ) for ev ery n ∈ N . Moreo v er, for an y n ∈ N , the in v ariance of the measure γ and the b oundedness of Θ n imply Z A Θ n ( x ) γ (d x ) = Z A P t Θ n ( x ) γ (d x ) ∀ t ≥ 0 . By deﬁnition of Θ n , we also get P t Θ n ( x ) := E [Θ n ( u x ( t ))] ≤ E  ∥ u x ( t ) ∥ 2 H  for all t ≥ 0 and n ∈ N . Owing to Lemma A.1 , we hav e E ∥ u ( t ) ∥ 2 H ≤ ∥ x ∥ 2 H exp ( − 2 K 2 " min( ε, C 0 ) 1 L< + ∞ + min( ε, C 0 ) − K 2 e C G 2 + e C J !! 1 L =+ ∞ #! t ) + C 1 +  e C G 2 + e C J  (1 + 1 L< ∞ |O | L 2 ) 1 K 2 h min( ε, C 0 ) 1 L< + ∞ +  min( ε, C 0 ) − K 2  e C G 2 + e C J  1 L =+ ∞ i . (4.3) The additional assumption on the structural parameters yields that, b y letting t → + ∞ in ( 4.3 ), the exp onen tial term v anishes. Letting C > 0 denote the constant to the right hand side of ( 4.3 ), we therefore ha v e lim sup t → + ∞ P t Θ n ( x ) ≤ C, and, as a consequence, Z A Θ n ( x ) γ (d x ) ≤ C for all n ∈ N . Since C do es not dep end on n , the claim ( 4.2 ) is then a consequence of the monotone con v ergence theorem as Θ n → Θ p oin twise from b elow. Notice that the condition min( ε, C 0 ) > K 2 e C G 2 + e C J ! in the case L = + ∞ is needed b oth to ensure the existence of inv arian t measures and to grant that the constan t C in ( 4.2 ) is indeed p ositiv e. 26 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA Step 2. In a similar fashion, we no w sho w that there exists a p ositive constan t C such that Z A ∥ x ∥ 2 V γ (d x ) ≤ C (4.4) for any inv arian t measure γ . Again, w e consider the mapping Λ : A → [0 , + ∞ ] , x 7→ ∥ x ∥ 2 V 1 A∩ V ( x ) ∀ x ∈ A ∩ V , extended with v alue + ∞ outside A ∩ V , and its appro ximating sequence { Λ n } n ∈ N , where for every n ∈ N , Λ n : A → [0 , n 2 ] , Λ n : x 7→ ( ∥ x ∥ 2 V if x ∈ B V n ∩ A , n 2 otherwise , x ∈ A , where B V n is the closed ball of radius n in V . It is clear that we hav e Λ n ∈ B b ( A ) for every n ∈ N . Exploiting the inv ariance of γ , the b oundedness of Λ n , the deﬁnition of P and the F ubini–T onelli theorem w e ha v e that Z A Λ n ( x ) γ (d x ) = Z 1 0 Z A Λ n ( x ) γ (d x ) d s = Z 1 0 Z A P s Λ n ( x ) γ (d x ) d s = Z 1 0 Z A E [Λ n ( u x ( s ))] γ (d x ) d s = Z A Z 1 0 E [Λ n ( u x ( s ))] d s γ (d x ) ≤ Z A Z 1 0 E  ∥ u x ( s ) ∥ 2 V  d s γ (d x ) . By Lemma A.1 we infer that Z A Z 1 0 E  ∥ u x ( s ) ∥ 2 V  d s γ (d x ) ≤ C  1 + Z A ∥ x ∥ 2 H γ (d x )  , and therefore estimate ( 4.2 ) yields Z A Λ n ( x ) γ (d x ) ≤ C , for a p ositiv e constan t C indep enden t of n ∈ N and for all n ∈ N . Since Λ n con v erges p oin t wise and monotonically from b elo w to Λ , the monotone con v ergence theorem yields the second claim ( 4.4 ). Step 3. A third approximation argument yields that there exists a p ositiv e constant C such that Z A ∥ Ψ( x ) ∥ L 1 ( O ) γ (d x ) ≤ C (4.5) for all inv arian t measures γ . Deﬁne the mapping Ξ : A → [0 , + ∞ ) , x 7→ ∥ Ψ( x ) ∥ L 1 ( O ) , ∀ x ∈ A and its approximations { Ξ n } n ∈ N giv en b y Ξ n : A → [0 , n ] , x 7→ ( ∥ Ψ( x ) ∥ L 1 ( O ) if ∥ Ψ( x ) ∥ L 1 ( O ) ≤ n, n otherwise , ∀ x ∈ A . AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 27 It holds that Ξ n ∈ B b ( A ) for every n ∈ N . Exploiting the inv ariance of γ , the b oundedness of Ξ n , the deﬁnition of P , the F ubini-T onelli theorem and Lemma A.2 we infer Z A Ξ n ( x ) γ (d x ) = Z 1 0 Z A Ξ n ( x ) γ (d x ) d s = Z 1 0 Z A P s Ξ n ( x ) γ (d x ) d s = Z A Z 1 0 E [Ξ n ( u x ( s ))] d s γ (d x ) ≤ Z A Z 1 0 E [ ∥ Ψ( u x ( s )) ∥ L 1 ( O ) ] d s γ (d x ) ≤ C  1 + Z A ∥ x ∥ 2 H γ (d x )  . F rom ( 4.2 ) w e th us infer the existence of a p ositiv e constan t C , indep endent of n ∈ N , such that Z A Ξ n ( x ) γ (d x ) ≤ C . As in the previous steps, since Ξ n con v erges p oint wise and monotonically from b elo w to Ξ , the monotone con v ergence theorem yields ( 4.5 ). Step 4. Ev entually , through a ﬁnal iteration of an approximation argumen t, we show that there exists a p ositiv e constant C such that Z A ∥ Φ ′ ( x ) ∥ 2 H γ (d x ) ≤ C . (4.6) for all inv arian t measures γ . Let Π : A → [0 , + ∞ ] , x 7→ ( ∥ Φ ′ ( x ) ∥ 2 H if Φ ′ ( x ) ∈ H, + ∞ otherwise , ∀ x ∈ A , and its approximations { Π n } n ∈ N , deﬁned for every n ∈ N by Π n : A → [0 , n 2 ] , Π n : x 7→ ( ∥ Φ ′ ( x ) ∥ 2 H if ∥ Φ ′ ( x ) ∥ H ≤ n, n 2 otherwise , x ∈ A . Once again, it holds that Π n ∈ B b ( A ) for every n ∈ N . Arguing similarly as ab ov e, thanks to Lemma A.2 , w e get Z A Π n ( x ) γ (d x ) = Z 1 0 Z A Π n ( x ) γ (d x ) d s = Z 1 0 Z A P s Π n ( x ) γ (d x ) d s = Z A Z 1 0 E [Π n ( u x ( s ))] d s γ (d x ) ≤ Z A Z 1 0 E [ ∥ Φ ′ ( u x ( s )) ∥ 2 H ] d s γ (d x ) ≤ C  1 + Z A  ∥ x ∥ 2 H + ∥ Ψ( x ) ∥ L 1 ( O )  γ (d x )  . It follo ws then from ( 4.2 ) and ( 4.5 ) the existence of a p ositiv e constan t C , indep enden t of n ∈ N , such that Z A Π n ( x ) γ (d x ) ≤ C for all n ∈ N . Since Π n con v erges p oint wise and monotonically from below to Π , the monotone conv ergence theorem yields ( 4.6 ). On account of ( 4.4 ) and ( 4.6 ), we conclude that γ is indeed supp orted on A str . 28 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA 4.4. Existence of an ergo dic in v ariant measure. Let us recall ﬁrst the deﬁnition of ergo dicit y for the transition semigroup P . In this direction, note that for every inv ariant measure γ , b y densit y and b y deﬁnition of inv ariance, the semigroup P can b e extended (with the same symbol for conv enience) to a strongly contin uous linear semigroup of con tractions on L p ( A , µ ) for every p ∈ [1 , + ∞ ) . Deﬁnition 4.2. An in v ariant measure γ ∈ P ( A ) for the semigroup P is said to b e ergo dic if lim t →∞ 1 t Z t 0 P s ϕ d s = Z A ϕ ( x ) γ (d x ) in L 2 ( A , γ ) ∀ ϕ ∈ L 2 ( A , γ ) . The pro of of existence of ergo dic in v ariant measure is analogous to the one of [ 44 , Prop osition 3.7], and therefore we omit the details. 4.5. Uniqueness of the inv ariant measure. Finally , we sho w that, for Dirichlet b oundary conditions, pro vided the system is suﬃcien tly dissipative, the inv arian t measure is unique, strongly and exp onen tially mixing, according to the follo wing deﬁnition. Deﬁnition 4.3. An in v ariant measure γ ∈ P ( A ) for the semigroup P is said to b e strongly mixing if lim t → + ∞ P t ϕ = Z A ϕ ( x ) γ (d x ) in L 2 ( A , γ ) ∀ ϕ ∈ L 2 ( A , γ ) , and it is said to b e exp onen tially mixing if for any ϕ ∈ Lip ( A )     P t ϕ ( x ) − Z A ϕ ( y ) γ (d y )     ≤ C ( x ) ∥ ϕ ∥ Lip e − ct , ∀ x ∈ A , t > 0 , for some p ositiv e constan t c and function C ( x ) . In order to prov e uniqueness of inv ariant measures, we pro ceed as follo ws, recalling that w e ask for the additional assumption ερ 1 > C G 2 + C J + p C F if L < + ∞ , or min ( ερ 1 , min( ε, C 0 )) > max K 2 e C G 2 + e C J ! , C G 2 + C J + p C F ! if L = + ∞ . Let x, y ∈ A and let u x , u y b e the corresponding solutions admitting x and y as initial conditions, resp ectiv ely. Setting w := u x − u y , the Itô formula for ∥ w ∥ 2 H yields, P -almost surely , for every t ≥ 0 , 1 2 ∥ w ( t ) ∥ 2 H + ε Z t 0 ∥∇ w ( s ) ∥ 2 H d s = 1 2 ∥ x − y ∥ 2 H − Z t 0  w ( s ) , Ψ ′ ( u x ( s )) − Ψ ′ ( u y ( s ))  H d s − Z t 0 ( w ( s ) , F ( u x ( s )) − F ( u y ( s ))) H d s + Z t 0 ( w ( s ) , ( G ( u x ( s )) − G ( u y ( s ))) d W ( s )) H + 1 2 Z t 0 ∥ G ( u x ( s )) − G ( u y ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( w ( s − ) , J ( u x ( s − ) , z ) − J ( u y ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z ∥ J ( u x ( s − ) , z ) − J ( u y ( s − ) , z ) ∥ 2 H ν (d z ) d s. Reasoning as in Subsection 3.4 , exploiting the Poincaré inequalit y and b earing in mind that the sto c hastic in tegrals are martingales, we obtain the estimate E  ∥ w ( t ) ∥ 2 H  + 2 α E Z t 0 ∥ w ( s ) ∥ 2 H d s ≤ ∥ x − y ∥ 2 H , AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 29 where α := ερ 1 −  C G 2 + C J + √ C F  is a p ositiv e constant by assumption. By the Gron wall lemma we obtain E ∥ ( u x − u y )( t ) ∥ 2 H ≤ e − 2 αt ∥ x − y ∥ 2 H , ∀ t ≥ 0 . (4.7) Consequen tly , let γ b e an inv ariant measure for the transition semigroup P . F or an y ϕ ∈ C 1 b ( H ) and x ∈ A , by deﬁnition of inv ariance and the estimate ( 4.7 ) w e ha v e     P t ( ϕ | A )( x ) − Z A ϕ ( y ) γ (d y )     2 ≤ ∥ D ϕ ∥ 2 C b ( H ) Z A E  ∥ u x ( t ) − u y ( t ) ∥ 2 H  γ (d y ) ≤ ∥ D ϕ ∥ 2 C b ( H ) e − 2 αt Z A ∥ x − y ∥ 2 H γ (d y ) , (4.8) and the right hand side conv erges to zero as t → + ∞ since Z A ∥ y ∥ 2 H γ (d y ) < + ∞ b y virtue of the argument sho wn in Subsection 4.3 . Since C 1 b ( H ) | A is dense in L 2 ( A , γ ) , we infer that     P t ϕ ( x ) − Z A ϕ ( y ) γ (d y )     2 → 0 as t → + ∞ , ∀ ϕ ∈ L 2 ( A , γ ) , from whic h we deduce that the strong mixing prop ert y holds true, again b y the argumen ts in Subsection 4.3 . B y computations similar to ( 4.8 ) and the argument sho wn in Subsection 4.3 , for any ϕ ∈ Lip ( A ) , we also infer     P t ϕ ( x ) − Z A ϕ ( y ) γ (d y )     ≤ C (1 + ∥ x ∥ H ) ∥ ϕ ∥ Lip e − αt , ∀ x ∈ A , t > 0 . for a p ositive constan t C , that is, the exp onen tial mixing prop ert y holds true. The ab o v e computation easily implies also the uniqueness of the inv ariant measure. Indeed, let π b e another in v arian t measure, then for all ϕ ∈ C 1 b ( H ) w e ha ve     Z A ϕ ( y ) γ (d y ) − Z A ϕ ( x ) π (d x )     =     Z A Z A ( P t ( ϕ | A )( y ) − P t ( ϕ | A )( x )) π (d x ) γ (d y )     ≤ ∥ D ϕ ∥ 2 C b ( H ) e − 2 αt Z A Z A ∥ x − y ∥ 2 H π (d x ) γ (d y ) → 0 as t → + ∞ . The fact that the unique inv ariant measure is also ergo dic follows from the existence of ergo dic inv ariant measures, and this concludes the pro of. 5. Appr o xima tion and simula tion 5.1. Appro ximation. W e approximate ( 1.2 ) by a semi-implicit Euler–Maruy ama metho d in time and a conforming ﬁnite element discretization in space. The Laplacian and the singular monotone part of the drift are treated implicitly , while the sto c hastic incremen ts are handled explicitly . This is natural in the presence of singular conv ex p otentials, where an implicit step preven ts spurious excursions outside the ph ysically admissible range and mirrors the monotonicity structure exploited in the analysis. In con trast, for p olynomial double-w ell p oten tials the drift ma y grow superlinearly , and standard explicit sc hemes ma y div erge [ 5 , 13 ]; in that setting, tamed v arian ts are often used [ 23 , 28 , 32 ]. Here, since the constrain t is enforced through a singular conv ex con tribution, a semi-implicit treatmen t is the appropriate stabilization mec hanism. Time discr etization. Let 0 = t 0 < t 1 < · · · < t N = T with t n +1 − t n = τ for all n ∈ { 0 , .., N − 1 } . F ollowing the previous discussion, we discretize ( 1.2 ) by the implicit–explicit Euler–Maruyama scheme u n +1 − τ ε ∆ u n +1 + τ Ψ ′ λ ( u n +1 ) + τ F 1 ( u n +1 ) = u n + τ F 2 ( u n ) + G ( u n ) ∆ W n + ∆ J n , (5.1) where F ( u n +1 , u n ) = F 1 ( u n +1 ) + F 2 ( u n ) is divided into its implicit and explicit comp onen ts, ∆ W n = W ( t n +1 ) − W ( t n ) and ∆ J n is a comp ensated jump increment. 30 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA In the following, we discuss each comp onen t: • W e split the p otential function as in the analysis: ∂ Ψ + F 1 , with Ψ conv ex and singular at the endp oin ts. Concretely , w e tak e as conv ex singular contribution the logarithmic entrop y Ψ( u ) = θ  u ln u + (1 − u ) ln(1 − u )  , u ∈ (0 , 1) , whic h yields a strong barrier at u = 0 and u = 1 , while the p erturbation F 1 ( u ) = − 4 θ 0 ( u − 1 2 ) is Lipsc hitz con tin uous and ﬁts assumption (A3) , with θ < θ 0 as sp eciﬁed b elo w. • The cylindrical Wiener term mo dels contin uous background ﬂuctuations. At eac h time step we generate a mean-zero spatial random ﬁeld by a truncated sine expansion, η n ( x ) = α K X k =1 L X l =1 ξ n k,l sin( k π x 1 ) sin( lπ x 2 ) , ξ n k,l i.i.d. ∼ N (0 , 1) , and enforce R Ω η n d x = 0 by subtracting its spatial mean. The multiplicativ e prefactor G ( u ) = c noise u (1 − u ) is chosen interface-localized so that noise v anishes in the pure phases. • The jump term represen ts abrupt lo calized damage ev ents and is mo deled by a (p ossibly comp en- sated) Poisson integral. In the comp ensated case w e discretize the increment Z t n +1 t n Z Z J ( u − , z ) ¯ µ (d t, d z ) , ¯ µ = µ − ν (d z ) d t, b y the standard Euler–Maruyama appro ximation ∆ J n = N n J X k =1 J ( u n , z n k ) − τ Z Z J ( u n , z ) ν (d z ) , (5.2) where N n J ∼ P oisson( τ ν ( Z )) and the marks z n k are sampled i.i.d. from ν /ν ( Z ) . By construction, E [∆ J n | F t n ] = 0 . In an uncomp ensated model one instead replaces ¯ µ by µ (equiv alen tly , drops the drift correction term in ( 5.2 ) or chooses F 2 as the comp ensator); this corresp onds to a systematic accum ulation of damage. Each even t is centered at a random lo cation z ∈ O and acts through a smo oth track k ernel, J ( u, z )( x ) = A jump ( u ( x )) κ ( x ; z ) , κ ( x ; z ) = exp  − ∥ x − z ∥ 2 2 σ 2 track  . (5.3) The c hoice of the f unction A jump ensures that jumps are lo calized to in termediate states and negligible in the pure phases. Since 0 ≤ κ ≤ 1 , this ﬁts the uniform b oundedness assumptions on J used in the analysis. Sp ac e-time discr etization. Let T h b e a triangulation of O with mesh size h and V h ⊂ H 1 (Ω) the space of con tin uous piecewise-linear functions. The fully discrete problem reads: ﬁnd u n +1 h ∈ V h suc h that for all v h ∈ V h , ( u n +1 h , v h ) + τ ε ( ∇ u n +1 h , ∇ v h ) + τ (Ψ ′ λ ( u n +1 h ) , v h ) + τ ( F 1 ( u n +1 h ) , v h ) = ( u n h , v h ) + τ ( F 2 ( u n h ) , v h ) + ( G ( u n h )∆ W n , v h ) + (∆ J n h , v h ) , (5.4) where ∆ J n h is the ﬁnite elemen t representation of ( 5.2 ). At eac h time step w e solve the resulting nonlinear system by Newton’s metho d. The implementation is carried out in the Firedrak e framework [ 41 ]. 5.2. Sim ulation setup. W e solve the sto c hastic Allen–Cahn mo del with the semi-implicit v ariational sc heme from ( 5.4 ) to illustrate mesoscopic dynamics under contin uous ﬂuctuations and sudden, lo calized damage hits. In this n umerical section the phase ﬁeld u ( t, x ) ∈ [0 , 1] denotes the lo cal damage level, with u = 0 corresp onding to an undamaged (health y) state and u = 1 to fully damaged, irreparable material. W e inv estigate t wo diﬀerent cases under diﬀeren t regimes by tuning the jump intensit y λ jump , whic h is related to the exp ected num ber of radiation ev en ts p er unit area and time. W e refer to the four cases of λ jump ∈ { 0 , 10 , 50 , 100 } as "None", "F ew", "Some", and "Many", resp ectiv ely , AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 31 The ﬁxed parameters are τ = 0 . 05 , T = 10 , h = 1 / 128 , O = (0 , 1) 2 , ε = 1 / 1600 , θ = 1 / 2 , θ 0 = 1 , and σ track = 1 / 10 . F or the P oisson jump counter N n ∼ P oisson( ξ ) p er time step we hav e ξ = E [ N n ] = λ jump τ |O | , P ( N n = k ) = e − ξ ξ k k ! . In particular (since |O | = 1 ), ξ = 0 . 05 λ jump , so for λ jump = 10 one has P ( N n = 0) = e − 0 . 5 ≈ 0 . 61 , whereas for λ jump ≥ 50 jumps o ccur in almost ev ery step ( P ( N n = 0) = e − 2 . 5 ≈ 0 . 082 for λ jump = 50 ). 5.3. Case 1: Random initial datum and comp ensated jump. T o emphasize damage nucleation from a nearly health y conﬁguration, w e initialize the system close to u ≡ 0 . 5 with a small smo oth random p erturbation: u ( x, 0) = 0 . 5 + η ( x ) , x ∈ O , (5.5) where η is a smo oth mean-zero random ﬁeld generated by a truncated sine series and scaled to hav e small amplitude. W e choose c noise = 1 / 2 . The jump forcing is implemented in compensated form, consisten tly with ( 5.2 ), and w e c ho ose A jump ( u ) = 1 2 u (1 − u ) . Therefore, v arying the jump parameter λ jump primarily mo diﬁes the in termittency , spatial clustering and v ariance of damage patterns. Figure 1 illustrates the evolution of the damage ﬁeld for increasing jump in tensities. Starting from a small random p erturbation around u ≡ 0 . 5 , the dynamics rapidly separates into near-pure phases, and the spatial organization at the ﬁnal time dep ends visibly on the jump activit y: higher jump frequencies lead to a denser pattern of lo calized nucleation even ts and thus to a diﬀerent coarsening morphology . The rightmost column shows the sampled jump lo cations, highlighting the increasingly clustered track pattern as λ jump gro ws. Figure 2 rep orts the evolution of the total damage R O u ( t ) d x (ensemble mean with uncertaint y bands). In the comp ensated setting, the total damage remains approximat ely constan t in the regimes “None” and “Many”, while the intermediate regime “F ew” ma y displa y a more noticeable deviation: when jumps are r are, single realizations can produce a stronger intermitten t bias, and the ensem ble is more sensitive to ﬁnite-sample ﬂuctuations. Finally , Figure 2 also sho ws that u min ( t ) and u max ( t ) sta y strictly inside (0 , 1) for all jump intensities, conﬁrming that the logarithmic barrier eﬀectively prev en ts excursions to the endp oin ts at the discrete lev el. 5.4. Case 2: Circular initial datum and uncomp ensated jump. T o mo del a single, initially healthy cell n ucleus, w e initialize the viability ﬁeld u ( x, 0) as a circular region of health y tissue ( u = 0 ) with radius 0 . 4 , surrounded by an extracellular environmen t considered lethally damaged ( u = 1 ). A smo oth h yp erbolic tangent transition zone with thickness prop ortional to ε is used to regularize the interface: u ( x, 0) = 1 2 1 − tanh 0 . 4 − ∥ x − ( 1 2 , 1 2 ) ∥ √ 2 ε !! . This simpliﬁed geometry allows for a clear analysis of radiation damage within a well-deﬁned biological unit. This time, we choose c noise ∈ { 1 2 , 5 } and inv estigate the inﬂuence of the increase in the random ﬂuctuations due to the Wiener noise. Moreo v er, we use an uncomp ensated jump by c ho osing F 2 as the comp ensator and we c ho ose A jump = 1 2 (1 − u ) to allow damage ev en ts ev en in the region with u = 0 , so fully healthy regions may b e damaged. In this interpretation, the uncomp ensated jump forcing describ es abrupt, spatially lo calized radiation hits (trac ks) that increase damage on a verage, whereas the Wiener p erturbation represents p ersistent micro-scale v ariability . This p ersp ectiv e is in line with phase-ﬁeld mo deling approac hes in oncology , where therap eutic interv entions (chemotherap y , radiotherap y , imm unotherap y) are incorp orated as additional forcing terms and one is interested not only in tumor control but also in the impact on neighboring health y structures; see, e.g., the discussion of treatment eﬀects in [ 22 , Sec. 2.7]. In particular, the track k ernel in ( 5.3 ) provides a simple mechanism to model the fact that radiation is not p erfectly focused: even when the target is lo calized, sto c hastic energy deposition may induce collateral damage in the surrounding region. 32 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA t = 0 t = 0 . 25 t = 1 t = 25 Jumps None F ew Some Man y 0.45 0.55 0.4 0.7 0.2 0.8 0 1 Figure 1. Case 1 (random initial condition, comp ensated jumps): snapshots of u ( t, · ) at t = 0 , 0 . 25 , 1 , 25 for increasing jump intensities (None/F ew/Some/Many). The last column visualizes the jump lo cations up to ﬁnal time. 0 2 4 6 8 10 0 . 46 0 . 48 0 . 5 0 . 52 0 . 54 t ! ! u ( t ) d x None F ew Some Man y 0 . 5 1 1 . 5 2 2 . 5 0 0 . 2 0 . 4 0 . 6 0 . 8 1 t u min u max None F ew Some Man y Fi g u r e 2. Ca s e 1 ( co m p e n s a t ed j u m p s ) : l ef t : ev o l u t i o n o f t h e t o t a l d a m a g e ! Figure 2. Case 1 (comp ensated jumps): left: evolution of the total damage R O u ( t ) d x for diﬀerent jump in tensities (ensemble mean with uncertain ty bands); righ t: ev olution of u min ( t ) and u max ( t ) for diﬀerent jump intensities. The v alues remain strictly within (0 , 1) , indicating eﬀective enforcement of the logarithmic barrier. AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 33 t = 0 t = 0 . 25 t = 1 t = 4 Jumps None F ew Some Man y 0 1 0 1 0 1 0 1 Figure 3. Case 2 (circular initial condition, uncomp ensated jumps): snapshots of u ( t, · ) at t = 0 , 0 . 25 , 1 , 4 for increasing jump intensities (None/F ew/Some/Many), c noise = 1 / 2 . The last column visualizes the jump lo cations up to ﬁnal time. 0 2 4 6 8 10 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 t ! ! u ( t ) d x None F ew Some Man y 0 2 4 6 8 10 0 0 . 2 0 . 4 0 . 6 0 . 8 1 t u min u max None F ew Some Man y Figure 4. Case 2 (uncomp ensated jumps): left: evolution of the total damage ! Figure 4. Case 2 (uncomp ensated jumps): left: evolution of the total damage R O u ( t ) d x for diﬀeren t jump in tensities (ensem ble mean with uncertain ty bands). Increasing jump in tensit y accelerates the growth of total damage; righ t: ev olution of u min ( t ) and u max ( t ) for diﬀerent jump intensities. The solution remains strictly within (0 , 1) for all regimes. 34 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA Figure 3 sho ws the ev olution of a healthy circular region em b edded in a damaged background, choosing c noise = 1 / 2 In contrast to Case 1, the jump term is uncomp ensated and thus injects damage on a v erage. Consequen tly , the total damage R O u ( t ) d x increases monotonically in Figure 4 , and the destruction of the health y region accelerates as the jump in tensit y increases: for “Man y” jumps the cell is rapidly ero ded and the domain b ecomes fully damaged. The corresp onding min/max curves in Figure 4 again remain strictly within (0 , 1) , indicating robust enforcement of the physical b ounds across all regimes. Lastly , w e depict in Figure 5 the case of c noise = 5 . It is visible that the interface is deformed in contrast to the case of c noise = 1 / 2 from b efore. t = 0 t = 0 . 25 t = 1 t = 4 Jumps None F ew Some Man y 0 1 0 1 0 1 0 1 Figure 5. Case 2 (circular initial condition, uncomp ensated jumps): snapshots of u ( t, · ) at t = 0 , 0 . 25 , 1 , 4 for increasing jump in tensities (None/F ew/Some/Man y), with c noise = 5 . The last column visualizes the jump lo cations up to ﬁnal time. Appendix A. Fur ther useful estima tes In this app endix, we include t wo useful estimates that are needed in the inv estigation of the longtime b eha vior of the system, i.e., in Section 4 . Let x ∈ H b e such that Ψ( x ) ∈ L 1 ( O ) and let u b e the unique corresp onding solution to 1.2 given by Theorem 2.4 . Deﬁne also the functional H 0 : D ( H 0 ) ⊂ V → R , H 0 ( v ) = Z O Φ( v ) d y , with Φ given in Lemma 2.3 . The ﬁrst estimate reads as follows. AN AC EQUA TION WITH JUMP-DIFFUSION NOISE FOR BIOLOGICAL DAMA GE AND REP AIR PROCESSES 35 Lemma A.1. L et Assumptions (A1) - (A6) hold. Then, for every x ∈ A and t ≥ 0 it holds that E ∥ u ( t ) ∥ 2 H ≤ ∥ x ∥ 2 H exp ( − 2 K 2 " min( ε, C 0 ) 1 L< + ∞ + min( ε, C 0 ) − K 2 e C G 2 + e C J !! 1 L =+ ∞ #! t ) + C 1 +  e C G 2 + e C J  (1 + 1 L< ∞ |O | L 2 ) 1 K 2 h min( ε, C 0 ) 1 L< + ∞ +  min( ε, C 0 ) − K 2  e C G 2 + e C J  1 L =+ ∞ i and " min( ε, C 0 ) 1 L< + ∞ + min( ε, C 0 ) − K 2 e C G 2 + e C J !! 1 L =+ ∞ # E Z t 0 ∥ u ( s ) ∥ 2 V d s ≤ 1 2 ∥ x ∥ 2 H + C 1 + e C G 2 + e C J ! (1 + 1 L< ∞ |O | L 2 ) ! t. Pr o of. W e apply the Itô form ula to the squared H -norm of the solution u at time t . F or every t ≥ 0 , it holds 1 2 ∥ u ( t ) ∥ 2 H + ε Z t 0 ∥∇ u ( s ) ∥ 2 H d s + Z t 0  u ( s ) , Φ ′ ( u ( s ))  H d s = 1 2 ∥ x ∥ 2 H + Z t 0 ( u ( s ) , G ( u ( s )) d W ( s )) H + 1 2 Z t 0 ∥ G ( u ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ( u ( s − ) , J ( u ( s − ) , z )) H µ (d s, d z ) + Z t 0 Z Z ∥ J ( u ( s − ) , z ) ∥ 2 H ν (d z ) d s, (A.1) P -almost surely. F or ev ery t ≥ 0 from Remark 2.3 we infer Z t 0  u ( s ) , Φ ′ ( u ( s ))  H d s ≥ C 0 Z t 0 ∥ u ( s ) ∥ 2 H − C 1 t, whereas Remark 2.2 yields 1 2 Z t 0 ∥ G ( u ( s )) ∥ 2 L 2 ( U,H ) d s + Z t 0 Z Z ∥ J ( u ( s − ) , z ) ∥ 2 H ν (d z ) d s ≤ e C G 2 + e C J !  t (1 + 1 L< ∞ |O | L 2 ) + 1 L =+ ∞ Z t 0 ∥ u ( s ) ∥ 2 H d s  . (A.2) Therefore, we obtain the estimate 1 2 ∥ u ( t ) ∥ 2 H + ε Z t 0 ∥∇ u ( s ) ∥ 2 H d s + C 0 Z t 0 ∥ u ( s ) ∥ 2 H d s ≤ 1 2 ∥ x ∥ 2 H + C 1 + e C G 2 + e C J ! (1 + 1 L< ∞ |O | L 2 ) ! t + e C G 2 + e C J ! 1 L =+ ∞ Z t 0 ∥ u ( s ) ∥ 2 H d s + Z t 0 ( u ( s ) , G ( u ( s )) d W ( s )) H + Z t 0 Z Z ( u ( s − ) , J ( u ( s − ) , z )) H µ (d s, d z ) . 36 ANDREA DI PRIMIO, MAR VIN FRITZ, LUCA SCARP A, AND MARGHERIT A ZANELLA T aking expectations on b oth sides of the ab o v e inequality and bearing in mind that the sto c hastic integrals are martingales, we get 1 2 E ∥ u ( t ) ∥ 2 H + " min( ε, C 0 ) 1 L< + ∞ + min( ε, C 0 ) − K 2 e C G 2 + e C J !! 1 L =+ ∞ # E Z t 0 ∥ u ( s ) ∥ 2 V d s ≤ 1 2 ∥ x ∥ 2 H + C 1 + e C G 2 + e C J ! (1 + 1 L< ∞ |O | L 2 ) ! t, whic h prov es the second statemen t. By the contin uous Sob olev embedding V  → H , from the ab ov e estimate we infer 1 2 E ∥ u ( t ) ∥ 2 H + 1 K 2 " min( ε, C 0 ) 1 L< + ∞ + min( ε, C 0 ) − K 2 e C G 2 + e C J !! 1 L =+ ∞ # E Z t 0 ∥ u ( s ) ∥ 2 H d s ≤ 1 2 ∥ x ∥ 2 H + C 1 + e C G 2 + e C J ! (1 + 1 L< ∞ |O | L 2 ) ! t and the Gronw all lemma yields the ﬁrst claim. This concludes the pro of. □ The second estimate concerns in tegrabilit y prop erties of the p ossibly singular nonlinearit y of the problem. Lemma A.2. L et Assumptions (A1) - (A6) hold. Then, for every t ≥ 0 ther e exists a c onstant C > 0 such that, for al l x ∈ A , E Z t 0 ∥ Ψ( u ( s )) ∥ L 1 ( O ) d s ≤ C (1 + ∥ x ∥ 2 H ) and E Z t 0 ∥ Φ ′ ( u ( s )) ∥ 2 H d s ≤ C (1 + ∥ x ∥ 2 H + ∥ Ψ( x ) ∥ L 1 ( O ) ) , Pr o of. T o prov e the ﬁrst claim, we start from ( A.1 ). F or any function f : R → R , let f ∗ denote its con v ex conjugate. Recalling that for any r and s ∈ R , the F enchel–Y oung inequality states that r s = Ψ( r ) + Ψ ∗ ( s ) ⇔ s ∈ ∂ Ψ( r ) , c ho osing s = Ψ ′ ( r ) yields r Ψ ′ ( r ) = Ψ( r ) + Ψ ∗ (Ψ ′ ( r )) for all r ∈ D (Ψ ′ ) , since in our case the sub diﬀerential is single-v alued. Using the fact that the sto c hastic int egrals are martingales, thanks to estimate ( A.2 ) and Assumption (A3) , we thus infer 1 2 E ∥ u ( t ) ∥ 2 H + ε E Z t 0 ∥∇ u ( s ) ∥ 2 H d s + E Z t 0 ∥ Ψ( u ( s )) ∥ L 1 ( O ) d s + E Z t 0 Z O Ψ ∗ (Ψ ′ ( u ( s ))) d s ≤ 1 2 ∥ x ∥ 2 H + e C G 2 + e C J ! (1 + 1 L< ∞ |O | L 2 ) t + q e C F + e C G 2 + e C J ! 1 L =+ ∞ ! E Z t 0 (1 + ∥ u ( s ) ∥ H ) 2 d s. Since Ψ ∗ is b ounded from b elow, from Lemma A.1 we infer the ﬁrst claim. Let us no w prov e the second claim. W e apply the Itô formula to the functional H 0 ev aluated at v = u ( t ) for any t ≥ 0 . Arguing as in the Se c ond estimate of Section 3.2 , we obtain, for any t ≥ 0 E sup τ ∈ [0 ,t ] ∥ Φ( u ( τ )) ∥ L 1 ( O ) + E Z t 0 ∥ Φ ′ ( u ( s )) ∥ 2 H d s ≤ C  1 + ∥ Φ( x ) ∥ L 1 ( O ) + E Z t 0 ∥ Φ( u ( s )) ∥ L 1 ( O ) d s  . F rom the Gron w all lemma and the ﬁrst claim we thus infer E Z t 0 ∥ Φ ′ ( u ( s )) ∥ 2 H d s ≤ C (1 + ∥ Φ( x ) ∥ L 1 ( O ) ) , and the pro of is complete. □ REFERENCES 37 A c kno wldegmen ts. The present research has b een supp orted by MUR, grant Dipartimento di Ec- cellenza 2023-2027. A.D.P ., L.S. and M.Z. are mem b ers of Grupp o Nazionale p er l’Analisi Matemat- ica, la Probabilità e le loro Applicazioni (GNAMP A), Istituto Nazionale di Alta Matematica (INdAM). These authors gratefully ac kno wledge the ﬁnancial support of the pro ject “Prot. P2022TX4FE_02 - Sto c hastic particle-based anomalous reaction-diﬀusion mo dels with heterogeneous interaction for radia- tion therapy” ﬁnanced by the Europ ean Union - Next Generation EU, Missione 4-Comp onen te 1-CUP: D53D23018970001. F urthermore, M.F. ackno wledges the support of the state of Upper Austria and A.D.P . ac kno wledges supp ort from the Europ ean Union (ER C, NoisyFluid, No. 101053472). References [1] S. M. Allen and J. 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Classe di Scienze, Scuola Normale Superiore, Piazza dei Ca v alieri 7, 56126 Pisa, It al y Email addr ess : andrea.diprimio@sns.it Johann Radon Institute for Comput a tional and Applied Ma thema tics, Al tenberger Str. 69, 4040 Linz, A ustria Email addr ess : marvin.fritz@ricam.oeaw.ac.at Dip ar timento di Ma tema tica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, It al y Email addr ess : luca.scarpa@polimi.it Dip ar timento di Ma tema tica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, It al y Email addr ess : margherita.zanella@polimi.it

An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

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