The cutoff phenomenon for the stochastic heat and the wave equation subject to small L\'evy noise

This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and L\'evy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, H\"ogele and Pardo (JSP2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.


Introduction
Stochastic partial differential equations with Gaussian and non-Gaussian Lévy noise are ubiquitous in the applications and have produced a vast literature in mathematics in recent years, standards texts include [1,7,23,25,29,30,32,33,34,35,40,48,54,59,60,61,64,69]. This article generalizes the small noise cutoff phenomenon from finite-dimensional ergodic Ornstein-Uhlenbeck systems which was established in [8] for the Wasserstein distance to a class of linear stochastic partial differential equations with Wiener and Lévy noise.
More precisely, we study the asymptotically abrupt (as ε → 0) ergodic convergence of the strong solutions of the linear stochastic partial differential equations of the following type where A is either the Dirichlet Laplacian in an appropriate Hilbert space H or the respective matrix-valued damped wave operator. We treat additive and multiplicative noise. For additive noise the process η = L with L = (L t ) t 0 is either a Q-Brownian motion or a Lévy process with values in H. The generalization of the results in [8] relies on the explicit knowledge of the eigensystem of A. In case of multiplicative noise dη t is replaced by X ε t (h)dL t in an appropriate sense. For the stochastic heat equation with multiplicative noise we still have a detailed knowledge of the stochastic flow under commutativity assumptions, which allows to establish the cutoff phenomenon. This method seems to break down for the stochastic wave equation with multiplicative noise due to infeasibly strong commutativity requirements on the noise coefficients.
In a series of articles, [8,9,10,11,12,13], the authors have studied the cutoff phenomenon of finite dimensional stochastic differential equations. Their setting covers linear or smooth coercive nonlinear dynamical systems close to an asymptotically exponentially stable fixed point subject to different small additive white and red noises in the (renormalized) Wasserstein distance and the total variation distance. The idea of this article is to establish the cutoff phenomenon to the most famous and elementary class of stochastic partial differential equations, that is, the stochastic heat and the stochastic wave equation.
The concept of the cutoff phenomenon was coined by Aldous and Diaconis in the context of card-shuffling in [4]. It roughly states the asymptotically abrupt convergence of the marginals to the dynamical equilibrium as a function of the deck size. As an introduction to the cutoff phenomenon in discrete time and space we recommend to [43] and Chapter 18 in [50]. Since the seminal paper [4], this threshold behavior has been shown to be present in many discrete stochastic systems and most of the results are stated in terms of the total variation distance. However, citing [14] "This precision comes at a high technical price and is largely responsible for the variety of treatments found in the literature." For standard texts illustrating the mentioned mathematical richness of cutoff thermalization in discrete cutoff parameter we refer to [2,3,4,14,15,16,17,19,26,36,37,38,39,45,46,47,49,50,51,53,55,58,65,67,70]. Most recent developments in this large field of active research include topics ranging from sparse random graphs, sparse random digraphs, sparse Markov chains, sparse bistochastic random matrices, square plaquette model, zero-range process to asymmetric simple exclusion processes (ASEP) are found in [18,20,21,22,27,41,44]. For a precise review on the results for stochastic differential equations and its embedding in the literature of the discrete cutoff phenomenon we refer to the introductions in [8,10,13].
Stronger notions of the cutoff phenomenon come in two formulations. Roughly speaking, the weaker and virtually universally valid window cutoff phenomenon describes limits of type (1.2), where the time scale divergence in The profile cutoff phenomenon conceptualizes the stronger statement that the limit exists for any fixed ̺ ∈ R. It is the main result of [8] to characterize the necessity and sufficiency of the limit where K h , Λ h are explicit positive spectral constants associated to the operator A and the initial condition h.
In the spirit of [8] this article combines i) the precise knowledge of the spectrum and the eigenvectors of the Dirichlet Laplacian (and the respective wave operator on the product space of position times velocity), ii) the analogous results of Lemma 2.1 in [8] correspondingly adapted to the infinite dimensional setting (Lemma 3.1 and Proposition 4.1) and iii) the properties of the Wasserstein distance, in particular, the shift linearity (item d) in Lemma 2.1). In case of multiplicative noise, instead of items iii), item i) and ii) are combined with iv) the precise representation of the stochastic flow for multiplicative noise and the triviality of the invariant measure µ ε . As a consequence we establish the following three main results. In Theorem 3.1 and Corollary 3.1 we establish the profile cutoff phenomenon for the stochastic heat equation with additive noise, with the help of the selfadjointness and the negativity of the spectrum of the Dirichlet-Laplacian. The profile is calculated explicitly. This is a generalization of the finite dimensional case. In Theorem 4.1 it is the precise knowledge of the Dirichlet Laplacian and its extension to the respective wave operator which allows to establish a nontrivial extension of the results of the stochastic damped linear oscillator in [8] Subsection 4.2. The case of the stochastic heat equation with multiplicative noise is completely new even in finite dimensions. In Theorem 5.1 and Corollary 5.1 the profile cutoff phenomenon is shown for multiplicative Q-Brownian motion with the help of the explicit representation of the associated Brownian flow which retains the precise knowledge of the initial value. In Theorem 5.2 and Corollary 5.5 the analogous results are shown for a pure jump Lévy noise. The case of the stochastic wave equation with multiplicative noise seems not to be feasible for cases of interest due to the lack of commutativity.

Stochastic heat and wave equation with additive noise
2.1. Ornstein-Uhlenbeck process. Let H be a separable Hilbert space and (S(t)) t 0 be a C 0 -semigroup having infinitesimal generator A, and acting in H. That is, (S(t)h) t 0 is given as the mild solution of the Cauchy problem Let ε ∈ (0, 1] and L = (L t ) t 0 be a Lévy process taking values on H. For any t > 0 the law of L t is uniquely determined by its characteristics (b, Q, ν). We refer to Section 2 in [5] for details. For any h ∈ H we consider (X ε t (h)) t 0 the unique mild solution of the linear stochastic differential equation The variation of constants formula yields In this article we assume that H is infinite-dimensional. The finite dimensional case is completely discussed in [8].

2.2.
Wasserstein metric of order p-th on H. Since our main results are stated in the Wasserstein distance we gather its most important properties. For any two probability distributions µ 1 and µ 2 on H with finite p-th moment for some p > 0, the Wasserstein distance of order p between them is given by where the infimum is taken over all couplings Π of marginals µ 1 and µ 2 , see for instance [56,68]. The main properties used in this manuscript are gathered for convenience in the following lemma.

Lemma 2.1 (Properties of the Wasserstein distance).
Let p > 0, u 1 , u 2 ∈ H, c ∈ R be given and U 1 , U 2 random vectors in H with finite p-th moment. Then we have: (a) The Wasserstein distance defines a metric.
(c) Homogeneity: (d) Shift linearity: For p 1 it follows For p ∈ (0, 1) equality (2.5) is false in general, see Remark 2.4 in [8]. However, for any p > 0 we have the following inequality (e) Shift continuity: For any deterministic sequence (h n ) n∈N ⊂ H such that h n → h ∈ H as n → ∞ it follows that Property (a) is standard, see for instance [68]. The properties (b) and (c) hold in any Banach space, see for instance Lemma 2.2 in [8] for H = R d . The proof of item (d) is given in Appendix A. Property (e) is a direct consequence of (2.6) and item (b).

2.3.
Existence of an invariant probability measure µ ε . Since our results are stated in terms of W p for some p > 0, we need the following moment condition.
We note that µ ε d = ε L ∞ , where L ∞ is the limit as t → ∞ (in distribution) of In particular, µ 1 d = L ∞ . In the sequel this result is strengthened to the convergence with respect to W p .
Lemma 2.2 (Ergodic convergence in Wasserstein distance). Assume Hypotheses 2.1 and 2.2 are satisfied for some p > 0. Then for any h ∈ H and t 0 Proof. Let h, h 0 ∈ H and t 0. The natural coupling and (2.4) imply Due to (2.2) the Wasserstein distance is dominated by the L 2 -distance which yields By the Markov property (Theorem 5.1 in [5]) and disintegration we have the desired result, that is,

2.4.
Cutoff inequality for Ornstein-Uhlenbeck processes. In case of additive noise the cutoff phenomenon for d ε = ε −1 W p , p > 0, relies on the following estimate.
For p 1 the shift linearity simplifies the cutoff phenomenon as follows.
Corollary 2.1 (Cutoff semigroup approximation for p 1). Assume Hypothesis 2.1 and Hypothesis 2.2 for some p 1. For any h ∈ H and t 0 (2.14) where L t and L ∞ are given in (2.7). In particular, for any (t ε ) ε>0 such that t ε → ∞ as ε → 0, we have Proof. It follows directly from inequality (2.9) in Lemma 2.3 and the shift linearity for p 1 given in Lemma 2.1(d).
By reparametrization the small noise cutoff phenomenon for fixed initial data is equivalent to the cutoff phenomenon for initial data and constant noise intensity.
Corollary 2.2 (Large initial data). Assume Hypothesis 2.1 and Hypothesis 2.2 for some p > 0. Then for any h ∈ H and t 0 it follows that Proof. It follows directly from the homogeneity given in item (c) of Lemma 2.1 and the fact that

Profile cutoff for the heat equation with additive noise
In this section we establish the profile cutoff phenomenon for a slightly generalized heat equation (2.2) subject to small additive noise. For the linear operator A we assume the following hypothesis.
Hypothesis 3.1. We assume that A is a self-adjoint, strictly negative operator with domain D(A) ⊂ H such that there is a fixed orthonormal basis (e n ) n∈N0 in H verifying: (e n ) n∈N0 ⊂ D(A), Ae n = −λ n e n , with 0 < λ 0 . . . λ n λ n+1 for all n such that λ n → ∞.
The standard example we have in mind is the standard Dirichlet Laplacian ∆ = being an open bounded connected domain O ⊂ R d with Lipschitz boundary, see Chapter 10 in [24]. The second example in mind is the Stokes operator, see for instance Section 4.5 on thermohydraulics in [66]. The following lemma is the infinite dimensional version of Lemma 2.1 in [8] for the heat semigroup S.
Proof. By Hypothesis (3.1) we have for any h ∈ H and t 0 For h = 0 we define and obtain the representation the orthogonality of the family (e n ) n∈N0 yields and conclude (3.1) by Remark 3.1. In the case of 0 < λ 0 < · · · < λ n < λ n+1 for all n such that λ n → ∞, the norm is given by |v h | 2 = e k , h 2 , where k is the smallest non-negative integer such that e k , h = 0.
We state the first main result which provides an abstract cutoff profile for any p > 0.
Theorem 3.1 (Abstract heat profile cutoff phenomenon for any p > 0). Let Hypotheses 2.2 and 3.1 be satisfied for some p > 0. We keep the notation of Lemma 3.1. Then for h ∈ H, h = 0 there exists a unique cutoff profile P p,h : R → (0, ∞) given by where L ∞ is given in (2.7), and v h is given explicitly in (3.3).
We stress that in general the right-hand side of (3.6) is abstract. For H = R d the result is given in Theorem 3.3 in [8].
For p 1 the cutoff profile can be calculated explicitly as follows.
Corollary 3.1 (Heat cutoff profile for p 1). Let Hypotheses 2.2 and 3.1 be satisfied for some p 1. We keep the notation of Lemma 3.1. Then for h ∈ H, h = 0 there exists a unique cutoff profile P h : R → (0, ∞) given by where t ε is given in (3.7) and |v h | is given explicitly in (3.4).
Proof. The proof is a direct consequence of the inequality in item (d) of Lemma 2.1 applied to right-hand side of (3.6) in Theorem 3.1.
The profile error can be quantified explicitly.
for any h ∈ H, where C * and λ * are given in (2.4), and N h , N * h are defined in (3.2). Proof. The triangle inequality combined with (2.14) in Corollary 2.1, (3.9) and Lemma 2.2 implies This completes the proof.
The following corollary shows that the profile cutoff phenomenon is the strongest notion of cutoff in this article.
where t ε is given in (3.7).
The proof is virtually identical to the proof of Corollary 3.2 in [8] with e −Qt being replaced by S(t). By reparametrization the small noise cutoff phenomenon for fixed initial data is equivalent to the cutoff phenomenon for initial data and constant noise intensity.
The analogues of Corollary 3.1, Corollary 3.2 and Corollary 3.3 are valid when . Proof. It follows directly from Corollary 2.2.

Window and profile cutoff for the wave equation with additive noise
In this section we establish the profile and window cutoff phenomenon for the wave equation (2.2) subject to small additive noise. The linear operator A in (2.2) is given by  We assume that ∆ is a self-adjoint, strictly negative operator with domain D(∆) ⊂ H such that there is a fixed orthonormal basis (e n ) n∈N in H verifying: (e n ) n∈N ⊂ D(∆), ∆e n = −λ n e n with Dirichlet boundary conditions, that is, e n ∂O = 0 and a simple growing point spectrum 0 < λ 0 < · · · < λ n < λ n+1 for all n such that λ n → ∞.
The operator ι denotes the natural embedding operator of D((−∆) 1/2 ) in H. In this setting, we consider the Hilbert space H = D((−∆) 1/2 ) × H, with the inner product Let (S γ (t)z) t 0 be the solution of the Cauchy problem and consider the stochastic linear damped wave equation [62] yields that the semigroup S γ is asymptotically exponentially stable, that is, there are C * , λ * > 0 such that This implies the existence of a unique invariant probability measure µ ε for the dynamics (4.3).

4.1.
Explicit computations of the eigensystem. In the sequel, we calculate the spectrum and the eigenfunctions of A in terms of the well-known spectrum and the eigenfunctions of −∆. Denote the spectrum of −∆ by 0 < λ 1 < · · · < λ n < · · · . Formally, the characteristic equations of A are given by where λ ∈ {λ j : j ∈ N}. This can be made precise in terms of spectral calculus. For this sake, we determine the eigenfunctions of A. The point spectrum of A (conveniently labelled) is given by for k ∈ N such that γ 2 < 4λ k .
We point out that the set {k ∈ N : γ 2 4λ k } may be empty (subcritical damping).

Hypothesis 4.2.
We assume the non-resonance condition γ 2 = 4λ k for all k ∈ N.
The eigenfunctions of A can be calculated explicitly in terms of the eigenfunctions (e k ) k∈N of −∆, where ∆e k = −λ k e k . That is,
Remark 4.1. Note that formally the limit (4.13) in I) implies a limit of type (4.14). However, in Case I) we obtain the stronger profile cutoff phenomenon, while in Case II) we still obtain the window cutoff phenomenon.
The following theorem shows the cutoff phenomenon for the overdamped and subcritically damped wave equation in the Wasserstein distance on H with additive noise for any p > 0.
Then it holds the following profile cutoff phenomenon with the (abstract) cutoff profile P p,z (̺): ii) Let z ∈ H Λ , z = 0, ω * := γ/2 given in item II) of Proposition 4.1 and Then we have the following window cutoff phenomenon Proof of Theorem 4.1. Let p > 0, z ∈ H Λ , z = 0, t 0.
Proof of case i). For any (s ε ) ε>0 such that s ε → ∞ as ε → 0, limits (2.10) and (2.11) given in Lemma 2.3 yield that and both limits coincide. By (4.16) we observe e −ω * tε = ε and Proof of case ii). For any (s ε ) ε>0 such that s ε → ∞ as ε → 0, limits (2.10) and (2.11) given in Lemma 2.3 yield that The choice of t ε given in (4.18) and triangle inequality imply Conversely, The preceding inequalities and the subadditivity of the map [0, ∞) ∋ r → r 1∧p with the help of Proposition 4.1 item II) imply Sending ̺ → −∞ in both sides of the preceding inequality, (4.24) and (4.27) imply the first limit of (4.19).
In case of overdamping the cutoff profile can be calculated explicitly.
Corollary 4.1 (Wave cutoff profile for p 1, in case of overdamping (γ/2) 2 > λ 1 ). Assume Hypotheses 4.1 and 2.2 for some p 1. We keep the notation of Proposition 4.1 and consider case I). Then for all z ∈ H Λ , z = 0, ω * := ω * (z), v := v(z) and t ε being as in (4.16) we have the following profile cutoff phenomenon with the explicit cutoff profile Proof. The proof is a direct consequence of item (d) in Lemma 2.1 applied to the right-hand side of (4.17) in Theorem 4.1.
Proof. The (standard and inverse) triangle inequality with the help of (2.14) in Corollary 2.1, (3.9), Lemma 2.2 and (4.4) imply Profile and window cutoff both imply the simple cutoff phenomenon.
The proof is virtually identical to the proof of Corollary 3.2 in [8] with e −Qt being replaced by S γ (t). Analogously to Corollary 3.4, we have the following cutoff phenomenon for large initial data. W p (X 1 tε+̺ (z/ε), µ 1 ) = P p,z (̺) for any ̺ ∈ R.
b) Under the specific assumptions of item ii) of Theorem 4.1 for some z ∈ H Λ , z = 0, it follows that
The following two lemmas give the proof of Proposition 4.1.

(4.43)
We observe that |ω j | 2 = λ j for all j ∈ N. By Young's inequality we have (4.44) Since z ∈ H Λ , the right-hand side (4.44) is sumable over all j ∈ N and hence the right-hand side of (4.43) is finite. In addition, lim sup In the sequel, we prove that lim inf t→∞ |e tγ/2 S γ (t)z| 2 > 0. (4.46) Since z = 0, there exists j 0 ∈ N such that |b j0 | 2 + |b −j0 | 2 > 0. By (4.43) we have (4.47) We claim that Indeed, assume by contradiction that there exists a sequence (t k ) k∈N , t k → ∞ as k → ∞ satisfying By the Bolzano-Weierstrass theorem we obtain the existence of a subsequence (t km ) m∈N of (t k ) k∈N such that lim m→∞ e iθj 0 t km = z + and lim m→∞ e −iθj 0 t km = z − .

Profile cutoff for the heat equation with multiplicative noise
This section treats the stochastic heat equation with multiplicative noise of the following type (5.1) dX ε t = AX ε t dt + εX ε t dL t for any t 0, X ε 0 = h ∈ H where A satisfies Hypothesis 3.1. We consider the space L 2 (H, H) equipped with the Hilbert-Schmidt norm · .

5.1.
Multiplicative Q-Brownian motion. Let Hypothesis 3.1 be satisfied for A and (e j ) j∈N0 be the orthonormal basis of eigenvectors of A in L 2 (H, H). We consider the following diagonal structures for the linear operators (5.2) G i = diag(g i j ) j∈N , i = 1, 2 with respect to the basis (e j ) j∈N . We study the solution semiflow of the heat equation with multiplicative noise given by for any t 0, X ε 0 = h ∈ H, where ε ∈ (0, 1), with the independence condition: (2) are mutually independent one dimensional standard Brownian motions. (5.4) We point out that the diagonal structure (5.2) implies . The existence and uniqueness of (5.3) is given in 1.2(a) of [54]. We stress that the commutative relations are natural in the case of the stochastic heat equation, whereas, in the case of the stochastic wave equation, the commutative of the respective operators A and G j is too restrictive in order to cover the natural case of noise acting only in the velocity component. By Theorem 16.5 in [57], p.290, we have the existence of a unique invariant probability measure µ ε for (5.3). It is not hard to see that µ ε = δ 0 . Then . For G 2 = 0, formula (1) p. 106 in [31] states Since h = j∈N0 h, e j e j , the diagonal structure (5.2) implies Following step by step the proof of formula (1) in [31] under (5.2) and (5.4), it is not hard to see that The finite dimensional case of this representation has been studied in formula (4.11) p. 100 in [52]. We stress that by (5.2) (2) t are both diagonal operators with respect to (e j ) j∈N0 and hence commute. Consequently, Recall that µ ε = δ 0 for any ε ∈ (0, 1]. In this setting, we obtain the profile cutoff phenomenon in the following sense. Note that the statement is given in the Wasserstein distance of order 2 due to the particular bilinear structure of (5.5) which allows to use the adjoint operators and express the right-hand side of (5.5) as a square. We keep the notation of (5.8) and define Then for any ̺ ∈ R we have the profile cutoff phenomenon Proof. We start with the computation of the right-hand side of (5.5). Recall the representation (5.7). Then for any h ∈ H, ε ∈ (0, 1) and t 0 we have where in the last equality we have used (5.2) and (5.4). Since G 1 + G * 1 and G 2 + G * 2 are symmetric operators, a standard diagonalization argument yields as ε → 0, where in the preceding limit we have used (5.8) and limit (5.9).
Theorem 5.1 shows profile cutoff for Example 2.1 in [42] in the case of small ε = r in his notation. The following corollary yields the profile cutoff phenomenon for perturbations by a general Q-Brownian motion.
for any t 0, Then for any ̺ ∈ R and (a ε ) ε∈(0,1) being a positive function satisfying (5.9) we have the profile cutoff phenomenon where t ε is defined in (5.10), and λ N h and v h are given in (5.8).
Proof. The proof follows the lines of the proof of Theorem 5.1 due to Analogously to Corollary 3.2 and Corollary 3.3 we have the following statements.
Corollary 5.2 (Profile error estimate). Let the assumptions of Corollary 5.1 be satisfied. Then for all ̺ ∈ R there exists a positive constant K := K(̺, C * , λ * ) such that for all ε small enough with t ε + ρ 0 we have for any h ∈ H, where C * and λ * are given in (2.4), and N h , N * h are defined in (3.2). Corollary 5.3 (Simple cutoff phenomenon). Let the assumptions of Corollary 5.1 be satisfied. For h ∈ H, where t ε is given in (5.10).
The linearity of (5.11) in contrast of the affine structure of (2.2) changes the reparametrization of the large initial value problem as follows.
Corollary 5.4 (Large initial data for the heat equation We then consider a pure jump Lévy process (L t ) t 0 , L t = t 0 0< z <1 z N (dz, dt), where N is the compensated Poisson random measure associated to ν on a given probability space (Ω, F , P) satisfying the usual conditions in the sense of Protter, see [63]. Fix η ∈ (0, 1) and let (X where I is the identity on H. That is, it satisfies P-almost surely for all t 0 s− ) N (dz, ds), (5.17) where S is the semigroup defined in (2.1). Since η > 0, the existence and uniqueness of (5.17) is straightforward by a standard interlacing procedure for compound Poisson processes, see for instance [6].
Note that the evaluation X (η,ε) t (h) of the unique mild solution of (5.17) is well-defined for all h ∈ H. However, a priori we do not have an analogously explicit representation as in (5.7). For h ∈ D(A), we obtain the following explicit exponential representation of (X (η,ε) t (h)) t 0 , which coincides with a strong solution of (5.16), see the proof of Lemma 5.1 in Appendix B. This representation turns out to be crucial for the cutoff result.
The following lemma extends Lemma 5.1 passing to the limit η → 0, which can be carried out by a standard stochastic analysis for Poisson random measures.
As a consequence of Lemma 5.2 we have the profile cutoff phenomenon for infinite intensity case of ν.
Proof. The proof follows the lines of the proof of Theorem 5.2. with small Markovian noise with applications in biophysics, climatology and statistics" of Facultad de Ciencias at Universidad de los Andes.

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