Anomalous diffusion properties of stochastic transport by heavy-tailed jump processes

Anomalous diffusion prop erties of sto c hastic transp ort b y hea vy-tailed jump pro cesses P aolo Cifani, F ranco Flandoli, Lorenzo Marino 25 th F ebruary , 2026 Abstract In this w ork, we in vestigate the large-scal e transp ort properties of a passiv e scalar adv ected by a turbule nt fluid, mo delled as a sup erp osi tion of div ergence- free v ector fields, eac h weigh ted b y an i ndep enden t symmetric α -stable-lik e pro cess. Motiv ated b y recen t wo rks [ CF25 , L T25 ] sho wing that complex small- scale s patial structures often lead to Bro wnian disp ersion, w e study if this principle p ersists when the driving noise exhibits hea vy-t ailed jump statistics. Our n umerical results sho w a clear dic hotom y link ed with the tail b eha viour of the noise. When considering standard α -stable pro cesses, ve ry large jumps surviv e the in teraction with the spatial complexit y and y ield anomalous, sup er- diffusiv e transp ort. In contr ast, when the α -stable noise is either truncated or exp onen tially temp ered, suppressing extremely long jumps, the transp ort undergoes a transition to a classical diffusiv e regime. 1 In tro duction The problem of the diffusiv e prop erties of turbulen t transp ort ∂ t T + u · ∇ T = 0 (1) (p ossibly with a molec ular diffusion term κ ∆ T ) is a basic one in Ph ysics (from the in tuition of Boussinesq [ Bou77 ], used so o f ten in fluid and plasma mo dels), Engi- neering and Mathematical Ph ysics [ BIL06 , MK99 ]. Sto c hastic mo de ls of turbulen t v elo cit y field s, of the fo rm u ( x , t ) = X k σ k ( x ) ξ k t 1 where σ k ( x ) are ve ctor fields, ξ k t are scalar (p ossibly g eneralized) sto c hastic pro c esses and the sum is extended to suitable w av e num b ers k , ha v e b een used for long time [ Kra94 ]. In recen t y ears, these mo dels ha v e sparke d renew ed inte rest as a suitable description of small-scales of turbulen t fluids [ MTVE01 , Gal20 , FGL22 , FL23 , CM23 , HLP21 , RMC17 , Luo2 1 , BMM25 , ECLG23 , EJP26 ], and many more authors and w orks. Most of t he classical w orks cited a b o v e and o t hers consider the case when ξ k t is a white noise (deriv ative of a Bro wnian motion) or an Ornstein-Uhlen b ec k pro cess, a ppro ximating white noise in the small time correlation limit. The diffusiv e prop erties o f suc h turbulen t transp o rt mo de ls ha v e b een thoroughly in v estigated in man y of the ab ov e mentione d w orks. V ery recen t ly , there has b een an increasing in terest in the case when ξ k t are either pro cesses with memory or pro c esses with large jumps. The reason is the kind of fluid regimes one aims to describ e. In 3D turbulence, or quasi-t w o dimensional turbulence when the small eddies b ehav e lik e in 3D, the turbulen t v ortex structures are small with short relaxation time, so that an approximation by white noise or Or nstein- Uhlen b ec k pro cess with small correlation time lo oks reas onable. But in the case of 2D turbulence, lo nger correlations app e ar and larger v ort ex structures start to dominate the energetics, due to the in v erse cascade. Therefore the description by white noise or close to white noise pro cesses ma y miss importa n t transport information. Up to no w, it is not exactly clear whic h kind of pro cesses is more suitable for describing suc h phenomena and th us it b ecame impo rtan t to in v estigate the diffusion prop e rties of other pro cesses, with memory or long jumps (mimic king long coheren t excursions). Concerning the case where ξ k t are related to F ractional Gaussian Noises (F GN) with Hurst parameter H > 1 / 2 (the persistence case ), let us men tion t wo w orks: in [ FR26 ] a theoretical result is prov ed, whic h giv es a strong indication of the fact that F GN with H > 1 / 2 pro duces stronger diffusion than classical white noise. Unfor- tunately , such result has been obtained in the case u ( x , t ) has a v ery simple space structure: a single field σ k ( x ) or when the σ k ( x ) are constan t. Therefore it w as not clear if the resu lt w as stable b y space-complexit y of u ( x , t ) , as it happ ens in turbule nt fluids. This w as in v estigated n umerically in [ CF25 ], where sev eral small-scale σ k ( x ) fields (similar to those used in the white noise case) m ultiplied b y independen t FGN pro cesses with H > 1 / 2 ha v e b een studied. T he result has b een quite unexpected: the b eha viour is similar to the white noise case. A tracer in suc h a v elo cit y field b eha v es lik e a Brow nian motion, since it jumps from one component to the other of the family of σ k and th us its displacemen ts are gov erned b y independen t pro cesses ξ k t . The presen t pap er aims to in v estigate the same regime as in [ CF25 ] but in the case when ξ k t are α - stable pro ces ses with large jumps. Namely , w e w an t to see whether 2 the long-tails structure is preserv ed in the presence of a complex space structure. W e shall in v estigate this problem nume rically , after a proper theoretical formulation. The same problem has b een studied theoretically recen tly [ L T25 ], pro viding a rigorous result on the diffusion prop erties; but the α -stable pro ces ses ξ k t in [ L T25 ] satisfy a certain condition o f small or mo derate jumps, differen t f rom t he one w e hav e in mind; the final result of [ L T25 ], under that condition, is a Bro wnian b eh aviour (namely a phenomenon similar to [ CF25 ], but rigorously pro v ed). W e shall in v estigate three differen t cases , also motiv ated b y the comparison with the theoretical result of [ L T25 ]: ( J 1 ) a cylindrical alpha stable noise, namely indep enden t alpha-stable pro c esses along eac h mo de of the space-dep enden t noise; ( J 2 ) truncation of the previous alpha-stable pro cesses; ( J 3 ) expo nen t ia l temp ering of them. Our results ab out these t hree cases are summarized in Section 2.3 , with the corre- sp onding n umerical sim ulations presen ted in Section 3 . The pro ces ses and equations in v estigated here are intro duced in Sections 2.1 and 2.2 . 2 Mo del form ulation and Main results Our aim is to inv estigate n umerically a mo dification of t he mo del app eared in [ CF25 ], in whic h the original fractional Bro wnian random we ights B H,k t are replaced by some hea vy-tailed ( non- Gaussian) noises. More precisely , w e w ould like to understand under whic h assumptions on the time scales and the considered noise, one should expect an anomalous diffusion in the large wa v e-nu mber limit, as opp osed to the diffusiv e b eha viour observ ed in [ CF25 ] and [ L T25 ]. W e are intere sted in the fo llo wing transp ort equation on R 2 : ( ∂ t T + u · ∇ x T = 0; T | t =0 = T 0 , (2) where u = ( u 1 ( x , t ) , u 2 ( x , t )) represen ts a "turbulen t" incompressible fluid. Here, w e enco de the turbulence as: u ( x , t ) := uC ( η , τ , α ) X k ∈ K η σ k ( x ) ˙ Z α, k t , 3 where u is the aver age velo city constan t (with dimension [ L ] / [ T ] ), C ( η , τ , α ) is a normalizing constan t whose precise expression will b e giv en later, η indicates the sp ac e sc ale with dimension [ L ] and the par a meter τ app earing in C ( η , τ , α ) represen ts the r elaxation time of the fluid with dimension [ T ] . The index set K η amoun ts for the wa ve num b ers k ∈ Z 2 whic h are activ ated at a fix ed space scale η > 0 and it is giv en by K η :=  k ∈ Z 2 : | k | ∈  1 2 η , 1 η  . Note in particular that the limit for η → 0 correspo nds indeed to the larg e w av e n um b er limit. The div ergence-free ve ctor field σ k are dimensi onless and define d as σ k ( x ) := ( √ 2 cos ( k · x ) k ⊥ | k | , if k ∈ K + η √ 2 sin ( k · x ) k ⊥ | k | , if k ∈ K − η where K + η is the set of k = ( k 1 , k 2 ) ∈ K η suc h that either k 1 > 0 or { k 1 = 0 , k 2 > 0 } , and K − η = − K + η . Ab o v e, each ˙ Z α, k t denotes the fo rmal time deriv ativ e of ( Z α, k t ) t ≥ 0 , a symmetric " α -stable-lik e" Lévy pro cess on R . Suc h pro cesses a r e c haracterized b y discon tin uous tra jectories. The parameter α ∈ (1 , 2) , know n as the stability index , quan tifies the frequency and relev a nce of larg e jumps: smaller v alues of α correspond to dynamics dominated b y rare but v ery large jumps whe reas in the formal limit as α → 2 , one reco v ers the con tin uous paths of a Brown ian motion. More precisely , we will test our mo del under three p ossible scenarios: ( J 1 ) ( Z α, k t ) t ≥ 0 are indep enden t, standard α -stable pro cesses; ( J 2 ) ( Z α, k t ) t ≥ 0 are indep enden t, truncated α -stable pro cesses with cut-off ǫ > 0 ; ( J 3 ) ( Z α, k t ) t ≥ 0 are indep enden t, exp o nen t ia lly temp ered α -stable pro cesses with tem- p ering p ar ame ter A > 0 . The ab ov e cases represen t "protot ypical" examples of jump-driv en noises and we ll illustrate how v ariations in the intens ity and/or frequency of the jumps influence the mo del limit b ehaviour. R oughly sp eaking, a standard α - stable pro cess exhibits a p o w er-law jump statistics: jumps of arbitrarily large size ma y o ccur with a frequency determined by a pow er la w distribution. Its dimension is [ T ] 1 /α , whic h is su ggested b y its α - self-similarit y . In case ( J 2 ) , one starts a g ain from the standard α -stable pro cess ab ov e but then remov es a ll jumps larger than the cut-off ǫ > 0 . Finally , the expo nen tia lly temp ered α -stable pro ce ss can be view ed as an interpolation betw een 4 the t w o previous mo dels: it retains jump activit y at all scales but larg e jumps b ecome expo nen tia lly ra re, with a ra te con trolled by the parameter A . In b o th cases ( J 2 ) and ( J 3 ) , the dynamics preserv es hea vy-tail features at small scales, while exhibiting Gaussian-lik e b eha viour a t large ones. This suggests that the dimension of ˙ Z α, k t is giv en b y [ T ] ( α − 1) /α . He nce, a dimensional analysis implies that the normalizing constan t C ( η , τ , α ) m ust ha v e dimension [ T ] ( α − 1) /α . More pre cisely , it is giv en b y C ( η , τ , α ) = [ Car d ( K η )] − 1 /α τ ( α − 1) /α . Note that when η is small enough, Car d ( K η ) ∼ π 1 η 2 − π 1 4 η 2 = π 3 4 η 2 . W e are thus interes ted in the follo wing SPDE on R 2 : ( dT + uC ( η , τ , α ) P k ∈ K η σ k ( x ) · ∇ x T ⋄ d Z α, k t = 0; T | t =0 = T 0 , (3) The sym b o l ⋄ ab ov e means the Marcus sto c hastic integral whic h is a generalization of the more well-kn own Stratono vic h in tegral for discon tinuous paths pro cesses . As the Stratono vic h one, the Marcus integral incorp orat es in its definition some corrector terms in order to reco v er the standard differen tiation properties. Precise definitions of the ab ov e sto c hastic pro cesses, together with the me aning of all parameters, a nd the corresp onding sto chas tic in tegration theory , are provide d b elo w. 2.1 Some bac kgrounds on stable Lévy pro cesses and their mo difications Let (Ω , F , P ) b e a probabilit y space and d ∈ N a generic dimension. An R d -v alued sto c hastic pro ce ss ( Z t ) t ≥ 0 is called a L évy pr o c ess if it starts from zero almost surely , has indep enden t and stationary incremen ts, and is càdlàg (i.e. it has righ t con tin uous paths ha ving left limits P -almost surely). A fundamen tal to ol in the analysis of Lévy pro cesses is giv en by the Lévy-Kin tc hine form ula (see for instance [ Jac01 ]). It allo ws to represen t the Lévy sym b ol Φ of ( Z t ) t ≥ 0 , giv en b y E [ e iξ · Z t ] = e t Φ( ξ ) , ξ ∈ R d in terms of the generating triplet ( b, Σ , ν ) as: Φ( ξ ) = ib · ξ − 1 2 Σ ξ · ξ + Z R d 0  e iξ · z − 1 − iξ · z 1 B (0 , 1) ( z )  ν ( dz ) , ξ ∈ R d , 5 where b is a v ector in R d , Σ is a symmetric, non-negative definite matrix in R d ⊗ R d and ν is a Lévy measure on R d 0 := R d r { 0 } , i.e. a σ - finite measure on B ( R d 0 ) , the Borel σ - algebra on R d 0 , suc h t ha t R (1 ∧ | z | 2 ) ν ( dz ) is finite. In particular, the la w of an y Lévy pro cess is completely determined by its generating triplet ( b, Σ , ν ) . In this w ork, w e restrict our atten tion to symmetric pur e-jump L évy pro cesses, namely pro cesses for whic h Σ = 0 , b = 0 and the Lévy measure ν is symmetric. The first class (i.e. case ( J 1 ) ) of noises considered in this w ork consists of α -stable Lévy pro cesses, with stabilit y index α ∈ (1 , 2) . These a re symmetric pure-jump Lévy pro cesses whose incremen ts follow an α - stable distributions. Their Lévy measure tak es the form: ν α ( dz ) := µ ( dθ ) dr r d + α , z = r θ ∈ R d , θ ∈ S d , r > 0 where µ is a finite Borel measure on S d , t he unitary sphere on R d , known as the sp e ctr al me asur e of Z t . In particular, w e will talk ab out an isotr opic α -stable pro cess when its sp ectral measure is rota tional inv arian t . In suc h a case, its c haracteristic function reduces to E [exp( iZ t · ξ )] = exp [ − σ α | ξ | α t ] , ξ ∈ R , for some constan t σ α > 0 , called the sc ale p ar ameter . Indeed, if Z t is an isotropic α - stable process with scale parameter σ α , then for an y λ > 0 , the pro ce ss λZ t remains isotropic α -stable, with scale parameter σ α λ α . In the particul ar case where σ α = 1 , the corresp o nding pro cess is usually called a standar d (isotr op i c ) α -stable process. As already noted ab ov e, α -stable pro cesses are self-similar with index 1 / α , i.e. ( Z ct ) t ≥ 0 d = ( c 1 /α Z t ) t ≥ 0 for any c > 0 . Unlik e t he Gaussian case, α - stable pro cesses do not p ossess finite momen ts of all orders. More precisely , they admit finite fractional momen t s only up to order β < α . In the isotropic case with scale parameter σ α , o ne can actually obtain an explicit form ula (cf. [ Sat13 , Eq. (25.6)]) for β ∈ (0 , α ) : E [ | Z t | β ] = 2 β Γ( d + β 2 )Γ(1 − β α ) Γ( d 2 )Γ(1 − β 2 ) ( σ α t ) β /α . (4) A fundamen tal distinction f rom the Bro wnian case is that understanding the global b ehav iour of an α -stable pro cess cannot b e ac hiev ed b y examining only its pro jections on to the co ordinate axes. This limitation arises from the fact that, in general, the comp onents of the pro ces s are not indep enden t. Consequen tly , α -stable pro cesses p ossess a substan tially riche r and more in tricate geometric structure. In order to analys e the geometry of the supp ort of the associated sp ectral measure, it 6 is therefore necess ary to consider pro jections along ar bitra ry directions (cf. [ ST94 , Thm. 2.1 .5 , Ex. 2.3.4] and [ Sat13 , Prop. 11.10]). If ( Z t ) t ≥ 0 is an α -stable pro ce ss with sp ectral measure µ and v ∈ S d is a direction, then Z t · v is again an α -stable pro cess (on R ) with scale parameter σ α ( v ) = Z S d | v · θ | α µ ( dθ ) . On the other hand, if for any v ∈ S d , Z t · v is an α - stable pro cess with scale para meter σ α ( v ) and suc h a constant is a ctually indep enden t fr o m v , then one can conclude that Z t is an isotropic α - stable pro cess on R d with scale parameter σ α . Starting from a giv en Lévy pro cess, one can construct new Lévy pro cesses b y means of sev eral standard mo difi cations, suc h as linear transformations or sub ordi- nation. In this w ork, w e fo cus on tw o classical examples of s uc h mo difications, namely truncation and exponential temp ering, w hic h su ppress large jumps while preserving α -stable b eha viour a t small scales. A truncated α -stable pro cess ( Z t ) t ≥ 0 is obtained by remo ving large jumps from its Lévy measure. Give n a cut-off parameter ǫ > 0 , it has the form: ν ( dz ) := 1 {| z |≤ ǫ } ν α ( dz ) , z ∈ R d , where ν α is an α - stable Lévy measure. An exponen tially temp ered α - stable pro ces s mo difies the Lévy measure b y in tro- ducing an exp onen tial damping of large jumps. F or a temp ering parameter A > 0 , the Lévy measure is ν ( dz ) := e − A | z | ν α ( dz ) , z ∈ R d , where ν α is again some α - stable L évy me asure. One of the k ey prop erty of tempered and truncated α -stable pro cesses is that, unlik e α -stable pro cess, they ha v e fin ite momen ts. 2.2 Solution concept for transp ort equation In connection with t he metho d of characteris tics, it is natural in this context to in terpret the sto chastic transp ort equation ( 3 ) in the se nse of Marcus sto chastic in- tegration (cf. [ HP23 ]). Indeed, Itô in terpretation is inappropriate in t his con text, as it do es not preserv e the classical ch ain rule and in tro duces artificial drift corrections unrelated to the transp ort mec hanism. On the other hand, while Stratonovic h in te- gration prov ides a consisten t framew ork for sto c hastic transport eq uation driv en b y 7 con tin uous path no ises (cf. [ CF25 ]), it do es not correctly describ e the dynamics in- duced by discon tin uous tra jectories. Indeed, in the presen ce of jumps, Strato no vic h in tegral mo dels the effect o f eac h jump as a purely instantane ous displacemen t, with- out taking into accoun t the infinitesimal flo w generated b y the vec tor field during the jump. By contrast, the Marcus integral accoun ts for the in ternal dynamics o f jumps b y asso ciating eac h jump of the driving pro cess with the deterministic flow gener- ated b y the v ector field. Suc h in terpretation preserv es the geometric structure of the transp ort equation and it is consisten t with the c haracteristic metho d. Informally , a solution T to ( 3 ) is c haracterized as T ( X η t ( x ) , t ) = T 0 ( x ) , where X η t ( x ) is the flo w map, solution to the characteris tics equation: ( d X η t ( x ) = uC ( η , τ , α ) P k ∈ K η σ k ( X η t ( x )) ⋄ d Z α, k t ; X η 0 ( x ) = x . (5) The sym b ol ⋄ ab o ve means the Marcus sto c hastic in tegral whic h is a generaliza- tion of the more w ell-kno wn Stratono vic h integral for discon tin uous paths pro cesses. F or a precise formulation of this in tegral, see [ App09 , Section 6.10]. Under our a s- sumptions, i.e. smo oth, div ergence-free ve ctor fields σ k and symmetric pure-j ump Lévy driv ers with finite first moment (i.e. α > 1 or truncated/temp ered v ariants), it is we ll-know n that the characteris tic Equation ( 5 ) admits a unique global càdlàg solution defining a sto chas tic flow of C 1 -diffeomorphisms almost surely (see, ag a in [ App09 , Section 6.10]). F or the precise connection betw een the sto c hastic transport Equation ( 3 ) and the asso ciated c haracteristics Equation ( 5 ) under our assumptions, as w ell as the w ell-p osedness of suc h models, see [ HP23 ]. 2.3 Main Results W e here outline a brief desc ription of the main re sults. They will be describ ed in more details in Section 3 . Roughly sp eaking, our num erical results sho w that the large- scale transp ort behav iour is strongly determined by ho w ra re extremely long j umps are in the driving noise. When the v elo city field is drive n b y a standard α - stable pro cess (i.e. case ( J 1 ) ), large jumps surviv e the inte raction with the comple x spatial structure and lead to anomalous, sup er-diffusiv e transp ort: particle displacemen ts scale lik e t 1 /α , and the limiting dynamics is consisten t with an α -stable pro cess. In con trast, when large jumps are either truncated or exponentially temp ered (i.e. cases ( J 2 ) a nd ( J 3 )) , their effect is progressi v ely suppressed, and the tracer dynamics shift 8 to a classical diffusiv e regime. In these cases, aft er a short transien t p erio d dominated b y jump-lik e b eha viour, the particle motion b ec omes Gaussian with the standard t 1 / 2 mean displacemen t scaling, similarly to what is observ ed for w hite noise-driv en mo dels. Our results highligh t how the tail b ehaviour o f the driving noise, i.e. the presence or absence o f a p ow er-law jump distribution, plays a crucial role in selectin g b et w een anomalous and classical diffusiv e regimes. Conjecture 1 L et x ∈ R 2 . Then, as η go es to zer o, the pr o c esses ( X η t ( x )) t ≥ 0 c on- ver ge in l a w to the 2 -dimensional pr o c ess ( x + X t ) t ≥ 0 , wher e: ( J 1 ): ( X t ) t ≥ 0 is an isotr op i c α -stable pr o c ess; ( J 2 ): ( X t ) t ≥ 0 is a c en tr e d Br ownian motion; ( J 3 ): ( X t ) t ≥ 0 is a c en tr e d Br ownian motion. The v alidity of the ab o v e r esult remains an op en question but w e hop e our for- m ulation will motiv ate further theoretical and practical in v estigations. Only in case ( J 2 ) , Luo and T eng in [ L T25 ] obtained a theoretical result of the diffusiv e scaling limit. As a preliminary contribution, w e pro vide the follo wing n umerical v erification. In the previous w ork [ CF25 ], the main ob jectiv e was to characteriz e the rat e of information diffused b y the v elo cit y field, quan tified via the mean squared displace- men t (MSD) of the particle p osition. While suc h a metric remains applicable f o r cases ( J 2 ) a nd ( J 3 ) , it is unsuitable f o r case ( J 1 ) . Indeed, as the α -stable driving noise inheren tly lac ks a finite second momen t, the resulting particle v ariance is ex- p ected to div erge. In order to compare the three cases on the same ground, w e prop ose instead to analyse the dynamics of the first absolute momen t: t → E [ | X η t ( 0 ) | ] . As an in termediate step to w ard the v alidit y of Conjecture 1 , our n umerical re- sults sho w that under α -stable driving noise (i.e. case ( J 1 ) ), the par t icle tra jectory exhibits sup er-diffusiv e b e hav iour c haracterized by the scaling E [ | X η t ( 0 ) | ] ∼ t 1 /α . As explained in ( 4 ), this is the b ehaviour usually associated to α -stable pro cesses. Con- v ersely , for the truncated and temp ered noise regimes, w e observ e that the pro cess con v erges to classical diffusiv e b ehav iour, namely E [ | X η t ( 0 ) | ] ∼ t 1 / 2 , after a short transien t p erio d. While suc h information already p oints tow ards Conjecture 1 , w e pro vide addi- tional confirmation by computing the probabilit y densit y function (PDF) of the par- ticle displacemen t along the x -axis at time t = 1 . By visually comparing their graph with the exp ected ones, w e show that 9 ( J 1 ): the distribution L is α -stable; ( J 2 ): the distribution L is cen tred Gaussian; ( J 3 ): the distribution L is cen tred Gaussian. where L b e the probability distribution f unction of X η 1 ( 0 ) · e 1 . The ab ov e claim a lready yields strong evidence in supp ort of the diffusiv e limi ts stated in Conjecture 1 f or cases ( J 2 ) and ( J 3 ) . Consequen tly , we restrict o ur sub- sequen t analysis to case ( J 1 ) . T o b etter understand the prop erties of t he limiting pro cess, w e in v estigate the geometric struc ture of the a sso ciated sp ectral measure . More precisely , w e fix v arious directions v ∈ S 2 and compare the distribution of the tracer pro jected along these directions: X η 1 ( 0 ) · v . The n umerical results presen ted b elo w indicate that the limiting pro cess is isotropic. F urthermore, as part of a more detailed analysis, w e also aim to estimate the co efficien ts σ app earing in front of the limiting pro cesses . Conjecture 2 L et x ∈ R 2 . Then, as η go es to zer o, the pr o c esses ( X η t ( x )) t ≥ 0 c on- ver ge in l a w to the 2 -dimensional pr o c ess ( x + σ X t ) t ≥ 0 , wher e: • ( X t ) t ≥ 0 is a standa r d α -stable pr o c ess in c ase ( J 1 ) and a c entr e d Br ownian motion in the other one s ; • the sc a l i n g p ar ameter is g iven by σ ∼ λu α/H η 1 − α/H τ ( α − 1) /H , (6) wher e, as b efor e, H = α in c a se ( J 1 ) a n d H = 2 in the o ther c ases and the p ar a meter λ > 0 dep e nds on the fe atur es o f the mo del (i.e. A , ǫ , σ k ). The heuristic deriv ation of the scaling para meter σ is detailed in Appendix A and relies on a suitable time discretization of the tracer dynamics . As express ed ab ov e, the "approximate" formula for σ depends how ev er on an additional parameter λ > 0 . Belo w in Section 3 , suc h a parameter is computed from num erical sim ulations and then compared against the theoretical prediction. W e ha ve not found an argumen t to predict the co efficien t λ , but w e can sho w n umerically that there exists a v alue pro viding a go od fit b et w een this formula and n umerical exp erimen ts. 10 Remark It app ears plausible that, in order to obta in a mathematically rigo r o us scaling limit, similar to the ones obtained in [ Gal20 , L T25 ], an additional assumption on the in terpla y b et w een the para meters of the mo del is required. More precisely , one ma y assume that the mean v elo cit y u := u ( τ , η ) is c hosen in suc h a wa y t ha t u α/H η 1 − α/H τ ( α − 1) /H ∼ 1 , as η → 0 , (7) where H = α in cas e ( J 1 ) and H = 2 in cases ( J 2 ) − ( J 3 ) . This scaling condition is designed to ensure t hat the scaling parameter σ of the limit pro cess is indep enden t from η and thus non-trivial, in the limit as η go es to 0 . Without suc h balance condition, one w o uld not expect the emergence of a meaningful macroscopic transport b eha viour sinc e the limitin g dynamics w ould either collapse to a degenerate regime or div erge. 3 Numerical sim ulations In this section w e p erform nume rical simul ations o f transp ort equation ( 1 ) with adv ection ve lo city u ( x , t ) giv en b y the sto c hastic mo del 3 . Analog ously to our related w ork [ CF25 ], a Monte Carlo metho d is emplo y ed where particle tra jectories are sim ulated b y n umerical in tegration o f the equations of characte ristics ( 5 ). In all the results presen ted b el ow, sto c hastic differen tial equations are in terpreted in the sense of Marcus in tegrals a nd solv ed n umerically via in tegration of the induced flow ODE for jumps. Expected v alues and probabilities are appro ximated b y appropriate ensem ble a v erages ov er 10 4 realisations. W e consider the following three cases for the stochastic pro cess Z t : ( J 1 ): ∆ Z t ∼ ∆ t 1 /α S α (1 , 0 , 0) sampled from the Cham b ers-Mallo ws-Stuc k algorithm (See [ ST94 , Sect. 1.7] for additional discussions on the topic); ( J 2 ): ∆ Z t sampled from the temp ered distribution b y the Bauemer-Merc haer algo- rithm (see [ KM11 ] for the details on the sim ulation); 1. ∆ Z t sampled from the truncated distribu tion b y a Ga ussian appro ximation of the small jump s (see [ AR01 ] for the de tails of the algorithm us ed for sim ulation). As a preliminary step, w e sho w the scaling of E [ | Z t | ] as a function of time for the aforemen tioned cases. Exp onen tial temp ering is a dopted to construct an effectiv e Lévy measure whose tails a re tamed b y a f a ctor e − A . Strong jumps are b ecome more and more rare for large v alues of A . In the truncated measure a parameter ǫ is 11 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 10 1 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 10 1 10 -3 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 10 1 Figure 1: E [ | Z t | ] as a function of time for the α -stable pro cess (to p-left figure), the temp ered pro cess (top-righ t figure) and the trunc ated pro cess (b ottom figure). F or the tempered pro cess w e set A = 0 . 3 ; f or the truncated pro cess w e set ǫ = 10 − 3 . Slop es 1 /α and 1 / 2 a re represen ted b y the dashed lines. in tro duced to suppress all jumps of size larger than ǫ . This b ehaviour is summ arised in Fig. 1 . Clearly , the scaling t 1 /α is sho wn by the α -stable pro cess at all times, as exp ected from theory . F or the temp ered and the truncated pro cess, after an initial p ow er law with exp onen t 1 /α , a transition to Gaussianit y tak es place with its c haracteristic scaling t 1 / 2 at large enough t . The time a t whic h the latter scaling tak es o v er dep ends on the parameter A and ǫ of the temp ered and truncated measure, respectiv ely . The main observ ation here is that for small t , the pro cess is dominated b y jumps while for large t rar e ev en ts are either tamed or cut out suc h that diffusion is reco vere d. An illustration of the sample paths of Z t is sho wn in Fig. 2 . Next, w e simulate particle tra jectories b y n umerical in tegration of ( 5 ). The r ef- erence length scale η is set to 2 π / 20 a nd the tot a l sim ulation time to T = 1 . W e 12 0 1 2 3 4 5 6 7 8 9 10 -15 -10 -5 0 5 10 15 20 Figure 2: Sample paths of the α -stable pro ces s (solid line), the temp ered pro cess (dashed line) and the truncated pro cess (dot-dashed line). consider the same three cases detailed ab ov e, namely the driving sto c hastic pro ce sses are the α -stable pro ces s and its tempered and truncated coun terpart. Fig. 3 shows E [ | X t | ] as a function of time. Ov erall the particle p osition X t b eha v es a s the driv er sto c hastic pro cess Z t . When the tempered o r the truncated pro cesses are used in the v ector field, the same transi- tion from the scaling t 1 /α at small times to the scaling t 1 / 2 at large times is observ ed. W e remark here that ty pical time scale at whic h this transition tak es place is en tirely related to the c hoice of the t emp ering parameter A and of the truncation threshold ǫ and not to the ph ysical time t ∗ , defined in [ CF25 ], and link ed to the length η . What is remark able here is, rather, the fact that X t main tains the same scaling la w o f Z t when the latter is an α - stable pro ce ss, in spite of the complex in teractions among the v ector field comp onen ts σ k . An illustration of the sample paths of X t is sho wn in Fig. 4 . Scaling of the first absolute mome nt of t 1 /α and t 1 / 2 suggest α -stable and G a us- sian displacemen t o f the par t icle X t , respectiv ely . As a further confirmation of this h yp othesis, we presen t in Fig. 5 the probabilit y distribution function of X t at the final time t = 1 . When driv ing X t b y eith er the temp ered of the truncated pro cess, the computed p df v a lues (dots) closely adhere to the Gaussian (solid line) ha ving the 13 10 -4 10 -3 10 -2 10 -1 10 0 10 -2 10 -1 10 0 10 1 10 -4 10 -3 10 -2 10 -1 10 0 10 -2 10 -1 10 0 10 -4 10 -3 10 -2 10 -1 10 0 10 -2 10 -1 10 0 Figure 3: E [ | X t | ] a s a function of time for the α -stable process (top-left figure), the temp ered pro cess (top-rig ht figure) and the truncated pro cess (b ot t o m figure). The parameters o f the driving stochastic pro cesses are the same as in Fig. 1 . Slop es 1 /α and 1 / 2 are represen ted b y the dashed lines. same mean a nd standard deviation. Differen tly , when Z t is α - stable the nu merical p df approxi mate an α -stable distribution with parameter σ n umerically computed from E [ | X t | ] . T o suppo rt Conjecture 1 , w e compute the p df of X t , driv en b y the α -stable pro cess, pro jected along the direc tion e θ for θ = π / 6 , π / 3 , π / 2 . As show n in Fig. 6 , at all v alues of θ the n umerical p dfs closely matc h the α -stable distribution S 1 . 5 . As stated in Conjecture 2 , the truncated and the temp ered noise regimes con v erge to the classical (Brown ian) diffusion b eha viour at sufficien tly larg e time while the pure α -stable case remains stable. An express ion for the scaling parameter σ is pro vided in ( 6 ). A n umerical v erification of the v alidit y of the latter w ould require a series o f sim ulations in the parameter space u , η , τ . F urthermore, the parameter λ dep ends as w ell on para meters of the driving no ise, suc h as the temp ering and 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 0 2 4 6 8 10 Figure 4: Sample paths of X t for driver Z t b eing an α -stable pro ce ss (solid line), a temp ered pro cess (dashed line) and a truncated pro cess (dot-dashed line). truncation factors A and ǫ . While an analysis of this kind is in principle p ossible, it b ecomes quic kly computationally unfeasible. This is the reason wh y in this w ork, w e limit ourselv es to the v erification of the dep endence of σ on the length scale η . Fig. 7 depic ts σ as a function of η in the three noise regim es. A g o o d a g reemen t is found b et w een the n umerical v alues and the p ostulated theoretical function, further strengthening our claim. 4 Conclusions In this w ork we ha v e inv estigated t he diffusion effects of certain sto c hastic mo dels of turbulen t fluids , a problem of ma jor importa nce for applications lik e the anomalous diffusion in confined fusion plasma. These sto chastic mo dels, also in the works quoted b elo w, are made of the su- p erp osition of div ergence-free highly oscillating ve ctor fields, each weigh ted by an independen t sto c hastic pro cess. The difference b et w een the v arious pap ers is mostly in the c hoice of sto chas tic pro c esses. A main tec hnical motiv ation for this w ork comes from t w o previous pap ers, [ CF25 , L T25 ]. In b oth of them it is sho wn that a classical Bro wnian b eha viour of the diffusi on 15 -10 -5 0 5 10 10 -3 10 -2 10 -1 -6 -4 -2 0 2 4 6 10 -4 10 -3 10 -2 10 -1 10 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Figure 5: Probabilit y distribution function of X t at the final time t = 1 for driv er Z t b eing an α -stable pro cess (top-left figure),for driv er Z t b eing a temp ered pro cess (top-righ t figure) and for driv er Z t b eing a truncated pro cess (b ottom figure). arises in spite of the fa ct that t he time-structure of the sto c hastic turbulence mo del w ere not Bro wnian, but fractional Brown ian with H > 1 / 2 in one case and α -stable (with a b ound on the tails) in the second one. In the presen t w ork w e ha v e show n that this is not alw a ys the case. Our main resul t is that, if the driving processes of the turbulence mo del exhib it hea vy-tailed jump statistic s, then this prop ert y p ersists for the passiv e scalar adv ec- tion. The "Brown ianization" observ ed and pro v ed in [ CF25 , L T25 ] do es not ta ke place. F or comparison, we also in v estigated the cases when t he driving pro cesses, still α -stable-lik e, are either truncated or expo nen t ia lly temp ered. In this case Brownian- ization tak es place. W e may summarize the in tuitions b ehind the quoted results and the new ones presen ted here in the following wa y . T w o elemen ts of a turbulence mo del (of the 16 -10 -5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 -10 -5 0 5 10 10 -3 10 -2 10 -1 Figure 6: Probabilit y distribution function of X t · e θ , at the final t ime t = 1 , using an α - stable driv er Z t for θ = π / 6 , π / 3 , π / 2 . The solid blac k line is a reference S 1 . 5 distribution. form said ab o ve , namely sup erp osition of div ergence-free highly oscillating v ector fields w eigh ted b y independen t sto chastic pro cesses) concur t o pro duce a Brow nian b eha viour of tracers: • the high n um b er of highly oscillating space comp onen ts, com bined with the independence o f the sto c hastic mu ltipliers. In suc h a case, the tra cer is sub ject to the effect of sev eral indep enden t inputs, some stronger and some w eak er dep ending on its relativ e p osition with respect to the oscillation of the comp o- nen ts, c hanging v ery rapidly the relativ e strength of these comp onen ts, hence restoring some form of c ha o ticit y of the background; • some limitation in the strength or frequency of ve ry larg e inputs by the sto c has- tic m ultipliers. In the case of clas sical α -stable cylind rical turbulen t fields, the second property ab o v e do es not hold while the first one is not sufficien t to restore a Bro wnian b e- ha viour. W e observ e an α -stable b eha viour of the tracer, namely an anomalous, sup er- diffusiv e transp ort. A Scaling parameter for limiting pro cess The main aim of this s ection is to deriv e an heuristic form ula for the scaling co efficien t σ app earing in Conjecture 2 . 17 0.1 0.15 0.2 0.25 0.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5 0.55 0.6 0.65 0.7 0.75 Figure 7 : P arameter σ as a function of η for driv er Z t b eing a n α - stable pro cess (top- left), a temp ered one (top- rig h t) or a truncated one (bo ttom), computed n umerically (dots) and analytically (solid line) from ( 6 ). The co effi cien ts of the solid line are fitted from the n umerical data. W e recall that the par t icle dynamics, when starting from zero, is defined by ( 5 ): X η t = uC ( η , τ , α ) X k ∈ K η Z t 0 σ k ( X η s ) ⋄ d Z α, k s where, for simplicit y , w e omitted the dep endency fro m the initial condition 0 . W e denote b y t η the a v erage time the tracer is influenced mostly b y a certain F ourier comp onen t b efore c hanging to another one. It is clear that since for η ≪ 1 , more and more F ourier comp onen ts are added to the dynamics, the av erage time t η ≪ 1 is small. Then, it is w ell-kno wn that one can approximate: X η t + t η − X η t ∼ uC ( η , τ , α ) X k ∈ K η σ k ( X η t )  Z α, k t + t η − Z α, k t  . 18 The v alidit y of suc h W ong- Z ak ai approx imation of the Marcus sto c hastic in tegral has b een corrob orated b y v arious studies (cf. [ Mar78 , KKP19 ]). Since X η t w anders through space and the F ourier mo des σ k are highly oscillatory , the v alue of | σ k ( X η t ) | α is "t ypically" close to its spatial a v erage: h σ k i α := lim R → + ∞ Z [ − R 2 ; R 2 ] 2 | σ k ( x ) | α d x = 1 2 π Z 2 π 0 | sin( t ) | α dt ∼ α − 1 √ π α Γ( α − 1 2 ) Γ( α 2 ) . Hence, the tracer mean displacemen t can b e roughly appro ximated b y: E    X η t + t η − X η t    ∼ uτ ( α − 1) /α E h    Z α, k t + t η − Z α, k t    i ∼ uτ ( α − 1) /α t 1 α η . Notice no w that the incremen ts X η t η , X η 2 t η − X η t η , X η 3 t η − X η 2 t η , . . . are appro ximately indep enden t, since eac h F ourier mo de is affected b y an indepen- den t pro cess. Moreo v er, each incremen t has a length | X η ( i +1) t η − X η it η | of order λη , for a certain parameter λ > 0 . Indeed, these incremen ts measure the distance the tracer tra v el when affected b y a certain F ourier componen t, b efore "jumping" on a no t her one. Suc h a t ypical “distance” to tra v el in order to jump from o ne to the other is of the order of t he w a v e-length of the sin usoidal compo nen t s of the noise, p ossibly mo dified by a factor λ . The in tuition b ehind is that the tracer is on t he “top” of a cosine function, where the function tak es approxim ately the v alue ± 1 . Mov ing a little bit, just a p o rtion of the wa v e-length η , it will b e no more on the top of that co- sine comp onen t, but more near the top of another comp onen t. Summarizing, at eac h time step it η , w e can appro ximate X η it η b y a random w alk with mean displacemen t of size iλη . T o conclude o ur heuristic deriv ation, we now split the argumen t b et w een the differen t cases. When we claim a diffusiv e limit (i.e. cases ( J 2 ) and ( J 3 ) ) so that appro ximately X η t ∼ σ W t , for a Bro wnian motion { W t } t ≥ 0 on R 2 , the random w alk approx imation of X η it η has finite square-a v erage distance from the origin g iv en by i ( λη ) 2 . Since it should b e also equal to σ 2 it η , it follows that σ ∼ λη √ t η . By the argumen ts ab ov e, we also ha ve that λη ∼ uτ ( α − 1) /α t 1 α η 19 so that t η ∼  λη u  α τ 1 − α . Finally , w e obtain that σ ∼ λ 1 − α/ 2 u α/ 2 η 1 − α/ 2 τ ( α − 1) / 2 . Note that b y our assumption ( 7 ), the parameter σ is actually indep enden t fro m η . The heuristic deriv a tion for the case ( J 1 ) follo ws a similar line of reasoning, relying directly on F ormula 4 for the mean displaceme nt. A ck nowle dgemen t. The researc h o f P .C. and L.M. is funded by the Europ e an Union (ER C, No isyFluid, No. 101 053472). 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