Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$

Quenc hed path limits and p erio dization stabilit y for tilted Bro wnian motion in P oissonian p oten tials on H d Miklos Ab ert, Adam Arras, Jaelin Kim F ebruary 23, 2026 Abstract W e analyze the existence of Bro wnian motion tilted b y a p oten- tial of full supp ort on h yp erbolic spaces H d . On compact spaces, it is classical that these path limits, called Q-processes, exist and can b e di- rectly defined using the ground state of the corresp onding Sc hr¨ odinger op erator. On non-compact spaces lik e H d , the existence fails in gen- eral. W e show that for stationary r andom p otentials on H d with suitable sp ectral and sup norm b ounds, the Q-pro cesses exist a.s. F or p oten- tials that are factors of a P oisson p oint pro cess, the method w orks up to sup norm ( d − 1) 2 / 8. In this case, we also sho w that the path limit can b e approximated b y p erio dic potentials. As a to ol, we use the foliated space defined by the p oint pro cess. It turns out that the global ground state of this foliated space serv es as a substitute for the non-existing L 2 ground states on the lea ves of the foliation. Restricting the global ground state to a leaf giv es a generalized eigen w av e that can b e plugged into the usual machinery to get the Q-process. Keyw ords: Brownian motion; Q-process; Poisson point process; random p o- ten tial; hyperb olic space; Sc hr¨ odinger op erator; Do ob transform; Benjamini– Sc hramm conv ergence; foliated Laplacian. 1 1 In tro duction Let P x denote the Wiener measure of Bro wnian motion ( X t ) t ∈ R ≥ 0 on a smooth Riemannian manifold M , starting at x ∈ M . Giv en a contin uous p oten tial V : M → R ≥ 0 , w e consider the V -tilte d p ath me asur e of time T b y setting d Q x T d P x := 1 Z x T exp  − Z T 0 V ( X s ) d s  , (1) where Z x T is the normalizing factor. This measure describ es the tra jectory of a particle in a soft trap mo del, conditioned to surviv al un til time T . W e are in terested in the existence and asymptotic b ehavior of the path limit (that is, the weak limit on the space of tra jectories) as T → ∞ . When this limit exists, it is called the Q-pr o c ess with r esp e ct to V . When M is compact, it is classical that the abov e family of path measures admits a unique w eak limit as T → ∞ . The limit is a diffusion process whose generator is the Doob transform asso ciated with the (unique) ground state of the Sc hr¨ odinger op erator − 1 2 ∆ M + V . In the non-compact case, the ab ov e op erator typically does not admit a low est eigen v alue, and there is no general guaran tee for the existence of the Q-pro cess. Our main result provides a general existence criterion for homogeneous random p oten tials on hyperb olic spaces M = H d . Theorem 1. L et ω b e a homo gene ous Poisson Point Pr o c ess (PPP) on H d and let V ω : H d → [0 , ∞ ) b e a c ontinuous factor-of- ω p otential with maximum norm ∥ V ω ∥ ∞ < ( d − 1) 2 8 . (2) Then for a.e. ω and for al l x ∈ H d , the tilte d me asur es Q x T c onver ge to a diffusion pr o c ess that is indep endent of the starting p oint x . W e refer to Definition 2.1 for a precise formulation of a factor of PPP p oten tial. A natural family of p oten tials that can be used ab o ve is V ω ( x ) = min ( V max , X y ∈ ω η ( d ( x, y )) ) where η : R ≥ 0 → R ≥ 0 is a fixed seed function and V max is a constan t. Note that the in tensity of the Poisson p oint process is irrelev ant here. 2 Theorem 1 brings together tw o well-studied ob jects in a nov el wa y . Pre- vious results on Q-pro cesses concen trate on confining p oten tials, or alterna- tiv ely , p erio dic p otentials where the existence of the path limit follo ws from the classical theory . On the other side, Bro wnian motion of time T tilted by P oissonian (soft or hard) traps has b een extensively studied in the literature, see e.g. [25] and references therein, but we did not find a discussion on the ex- istence of the path limit. These results also concentrate on Euclidean spaces, where our sp ectral metho d stops working automatically and one needs new ideas. W e use the sp ectral theory of the foliated space defined b y the p oint pro cess to get Theorem 1. Indeed, in situations lik e ours, individual instances of ( H d , V ω ) do not admit L 2 ground states. Ho wev er, it turns out that the global spectral theory of L 2 of the ambien t foliated space does admit a unique ground state that (when restricted bac k to lea ves) can b e used as a substitute in finding the limiting Q-process. As w e will see, Theorem 1 will also hold for an arbitrary stationary ran- dom p oten tial V , but the b ound on the maxnorm will dep end on the global sp ectral gap of the potential. In particular, if the underlying group action do es not hav e spectral gap, we get an empty statement (this is exactly the issue with Euclidean spaces as noted abov e). In Theorem 1 w e only use P oisson p oint pro cesses because the spectral gap of a factor of P oisson p o- ten tial is maximal p ossible, that is, P oisson p oint pro cesses are Raman ujan, see Theorem 4. Our next result addresses how the Q-pro cesses defined by a general sta- tionary random p otential can be approximated b y Q-pro cesses defined b y p erio dic p otentials . Theorem 2. L et ( M n ) b e a se quenc e of c omp act hyp erb olic d -manifolds, with uniform sp e ctr al gap λ := lim inf n →∞ λ 1  − 1 2 ∆ M n  > 0 such that M n Benjamini-Schr amm c onver ges to H d . L et V n b e a non-ne gative c ontinuous p otential on M n with maximum norm lim sup n →∞ ∥ V n ∥ ∞ < λ. such that the p air ( M n , V n ) Benjamini-Schr amm c onver ges to ( H d , V ) wher e V is a stationary r andom function on H d . Then the sp e ctr al gap of the foliate d 3 sp ac e ( H d , V ) is at le ast λ and the Q -pr o c esses of ( M n , V n ) c onver ge to the Q -pr o c ess of ( M , V ) . W e need to explain Benjamini-Sc hramm con v ergence of ( M n , V n ) to ( M , V ) whic h in the abov e case is quite simple (for a general definition see [4]). Here, w e lift the function V n to H d using a random root and frame, which giv es us a p erio dic, stationary random function e V n on H d . Benjamini-Sc hramm con vergence means w eak conv ergence of the law of the function V ′ n to the la w of V . W e will clarify the exact space and setup in Section 5. It is not clear, ho w ev er, which stationary random p oten tials can be ap- pro ximated b y p erio dic ones with a goo d sp ectral gap. This is the p oint where factor of P oisson P oin t Pro cess potentials shine against an arbitrary stationary random p oten tial. Indeed, any factor of PPP p otential can b e ap- pro ximated b y p erio dic p oten tials whic h allo ws us to pro ve a general stabilit y result there. Theorem 3. Ther e exists λ d > 0 ( d ≥ 2) such that the fol lowing hold. L et ω b e a homo gene ous Poisson Point Pr o c ess (PPP) on H d and let V ω : H d → [0 , ∞ ) b e a c ontinuous factor of ω p otential with maximum norm ∥ V ω ∥ ∞ < λ d . (3) Then the Q-pr o c ess of ( H d , V ) c an b e appr oximate d by p erio dic Q-pr o c esses, that is, Q-pr o c esses define d by p erio dic p otentials. The abov e theorem has t wo parts. First, w e use that for ev ery d ≥ 2 there exists an expander family of compact h yp erb olic d -manifolds that Benjamini- Sc hramm con v erges to H d . This follows from tw o results: 1) By a w ell-known result of Clozel [11], for any symmetric space X for a semisimple Lie group (including H d ), there exists a constant ϵ d > 0, dep ending only on X , suc h that for ev ery congruence lattice Γ in G , the spectral gap of the orbifold Γ \ X is at least ϵ d . 2) Fixing an y co-compact arithmetic lattice Γ on H d , b y [3], w e hav e that any sequence of congruence subgroups of Γ, Γ \ H d Benjamini- Sc hramm conv erges to H d . W e set λ d to b e the b est asymptotic sp ectral gap for whic h this can b e achiev ed. With the exception of d = 2, the v alue of λ d is not known: for d = 2, b y the recen t work of Hide and Magee [15], we hav e λ 2 = 1 / 8 whic h is the optim um. Second, we use that for any suc h sequence of manifolds, the PPP of the compact manifolds will conv erge to the PPP of H d , and this extends to an y 4 factor p otential. T ogether these yield a sequence of ( M n , V n ) that satisfies the assumptions of Theorem 2. A limiting case of the ab o ve mo del is the so-called har d obstacle mo del , where the particle is conditioned to av oid an r -neighborho o d of the PPP { x : d ( x, ω ) < r } = [ y ∈ ω B ( y , r ) . If the radius is sufficien tly small, the complement of the r -neighborho o d con- tains an infinite cluster with p ositive probabilit y [7, 26], raising the follo wing question. Question 1.1. On an infinite connected comp onen t of the supercritical hard P oisson obstacle mo del, do es the limiting Q -pro cess exist? F ormally , the hard obstacle corresp onds to the limit V max → ∞ in the soft p oten tial V ω ( x ) = V max 1 { d ( x,ω ) 0 , ⟨ z , z ⟩ J = − 1  , where J = diag ( − 1 , 1 , . . . , 1) and ⟨· , ·⟩ J denotes the associated Mink o wski bilinear form. Additional structure will b e defined relative to a p ositiv ely orien ted orthonormal frame ( o, u ) ∈ F ( H d ). W e fix the canonical choice o = e 0 ∈ H d and u = ( e 1 , . . . , e d ) ∈ ( T o H d ) d , where ( e 0 , . . . , e d ) denotes the standard basis of R 1+ d . In this mo del, the group of orientation-preserving isometries of H d coincides with the identit y comp onent of the group of real linear transformations preserving the form, that is, A T J A = J . W e denote it b y G = Isom + ( H d ) = PSO(1 , d ) ⊂ GL d +1 ( R ) . 5 An y isometry g ∈ G acts naturaly on the bundle of (p ositively-orien ted) orthonormal frames b y g · ( o, u ) := ( g · o, d g o ( u )) . Since this action is simply transitiv e, w e obtain the iden tification G ≃ F ( H d ). W e denote by K the stabilizer of the origin o , so that H d ≃ G/K. W e refers to [13] for more on the h yp erb oloid mo del. Bro wnian motion among soft traps. Let P x denote the Wiener measure on C ( R ≥ 0 , H d ) of the Brownian motion ( X t ) t ≥ 0 with X 0 = x and generator 1 2 ∆ H d , and let E x b e the asso ciated exp ectation. F or f ∈ C ∞ c ( H d ), E x [ f ( X t )] =  e t 2 ∆ H d f  ( x ) . Giv en a b ounded contin uous p oten tial V : H d → R , w e define the tilted path measure Q x T b y d Q x T d P x = 1 Z x T exp  − Z T 0 V ( X s ) d s  , Z x T = E x  exp  − Z T 0 V ( X s ) d s  . (4) Assume that that ( Q x T ) T > 0 con verges w eakly on C ( R ≥ 0 , H d ) endow ed with the topology of uniform con vergence on compact time in terv als. More pre- cisely , we assume that for ev ery t ≥ 0 and ev ery Borel set B ∈ F t (the canonical filtration), Q x ( B ) := lim T →∞ Q x T ( B ) exists. These finite-dimensional marginals define a unique Mark ov probabil- it y measure Q x , called the Q -pr o c ess asso ciated with the killing p oten tial V . The p erio dic case. W e say that V is p erio dic if it is in v arian t under a torsion-free co compact lattice Γ ≤ G , so that M = Γ \ H d is a closed h yp erb olic manifold. The Schr¨ odinger op erator H = − 1 2 ∆ M + V 6 defines a self-adjoin t operator on L 2 ( M , v ol M ) with discrete sp ectrum. Its lo west eigen v alue ρ is simple [23], with a p ositiv e φ (normalized) eigenv ec- tor. Let T φ b e the diagonal op erator acting b y m ultiplication b y φ . The conjugated op erator (see [5, Section 1.15.8]) L = T ∗ φ ( ρ − H ) T φ = 1 2 ∆ M + ∇ log φ · ∇ (5) generates a diffusion pro cess whose transition density p L ( t, x, y ) = e tL xy with resp ect to v ol M is p L ( t, x, y ) = p 1 2 ∆ M ( t, x, y ) e tρ E x  exp  − Z t 0 V ( X s ) d s      X t = y  φ ( y ) φ ( x ) , where E x denotes exp ectation with resp ect to Brownian motion started from X 0 = x ∈ M . W e refers to [23, 20, 6]; and [24, 9] for renormalization b y p ositiv e harmonic functions. Lifting this diffusion to H d yields the existence of the Q -pro cess in the p erio dic setting. Casimir operator. In the h yp erb oloid model, the Lie algebra so (1 , d ) ≃ T ( o,u ) F ( H d ) is generated b y the orthonormal basis E k := e 0 e ∗ k − e k e ∗ 0 , E i,j := e i e ∗ j − e j e ∗ i , k , i, j ∈ { 1 , . . . , d } , i < j. F or A ∈ so (1 , d ) and F ∈ C ∞ ( G ), w e recall the (right) Lie deriv ativ e L A F ( g ) := d d t     t =0 F  g e xp( tA )  . The Casimir op er ator ∆ G on G is the bi-in v ariant second-order differen tial op erator ∆ G = d X k =1 L 2 E k − X 1 ≤ i 0, the Laplace transforms con verge p oint wise. By the L ´ evy con tin uity theorem for Laplace transforms [12, Ch. XI I I], we obtain w eak conv ergence of the sp ectral measures. By the sp ectral gap assumption, the ground eigenv alue ρ n ∈ [0 , ∥ V n ∥ ∞ ] is isolated from the remainder of sp ectrum of H n con tained in [ λ, ∞ ). It follo ws that ρ n − → ρ, ν 1 M n H n ( { ρ n } ) − → ν 1 b Ω H ( { ρ } ) . 17 Since Z x n n,t := E x n  exp  − Z t 0 V n ( X s ) d s  is B-S con tinuous and by sp ectral decomp osition, φ n ( · ) = lim t →∞ e − tρ n Z n,t ( · ) µ n ( { ρ n } ) , with con trol of the conv ergence uniformly in n , ∥ φ n ( · ) − e − tρ n Z n,t ( · ) µ n ( { ρ n } ) ∥ L 2 ( M n ) ≤ e − ϵt , where ϵ is a lo wer bounds for the spectral gap. W e obtain the conv erge of p V ol( M n ) φ n ( · ) to φ ( · ) in the Benjamini–Sc hramm sense. Finally , the Q -pro cess on M n has transition densit y p L n ( t, x, y ) = p ∆ M n 2 ( t, x, y ) e tρ n φ n ( y ) φ n ( x ) E x  exp  − Z t 0 V n ( X s ) d s      X t = y  . Eac h terms in B-S concluding the pro of. W e end the pap er with the p erio dization stabilit y . Pr o of of The or em 3. It is known that there exists a sequence of compact h yp erb olic d -manifolds ( M n ) that Benjamini–Schramm conv erges to H d and has a uniform sp ectral gap λ := lim inf n →∞ λ 1  − 1 2 ∆ M n  > 0 . This sp ectral gap follows from Clozel’s theorem [11] that states that for an y symmetric space X of a semisimple Lie group (in particular H d ), there exists a constant ϵ d > 0, dep ending only on X , such that ev ery congruence lattice Γ in G yields an orbifold Γ \ X with spectral gap at least ϵ d . F or cocompact arithmetic lattice Γ < Isom( H d ), b y [3], an y sequence of congruence subgroups of Γ Benjamini–Schramm con v erges to H d . W e define λ d to b e the maximal v alue that can b e achiev ed, and fix suc h a sequence ( M n ). Let ω n b e a P oisson p oint process on M n . Since V ω is a factor of the PPP , it can b e approximated by finite-range factors V R n ,ω , where R n → ∞ 18 sufficien tly slo wly compared to the injectivit y radius at a uniformly c hosen p oin t of M n . Using the realization ω n of the PPP on M n , we define a perio dic p otential V n on M n b y cop ying the finite-range rule defining V R n ,ω . By construction, the sequence ( M n , V n ) satisfies the assumptions of Theorem 2. Hence the asso ciated Q -pro cesses on M n con verge in the Benjamini–Schramm sense to the limiting Q -pro cess on ( H d , V ω ). Finally , lifting each Q -pro cess on M n to the univ ersal cov er H d pro duces a p erio dic Q -pro cess. This yields the desired p erio dic approximation of the limiting Q -pro cess. References [1] M. Ab ´ ert and I. Biringer, Unimo dular me asur es on the sp ac e of al l rie- mannian manifolds , Geometry & T op ology 26 (2022), no. 5, 2295–2404. 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