The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials

We study the quenched long time behaviour of the survival probability up to time t, Exe−∫0tVω(Xs)ds,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf {E}_{x}\left [e^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})\mathrm {d}s}\right ],$\end{document} of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential Vω on ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document}. Such a function is a probabilistic solution to the parabolic equation involving the nonlocal Schrödinger operator based on the generator of the process (Xt)t≥ 0 with potential Vω. For a large class of processes and potentials of finite range, we determine rate functions η(t) and compute explicitly the positive constants C1,C2 such that −C1≤lim inft→∞logExe−∫0tVω(Xs)dsη(t)≤lim supt→∞logExe−∫0tVω(Xs)dsη(t)≤−C2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ -C_{1} \leq \liminf \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})\mathrm {d}s}\right ]}{\eta (t)} \leq \limsup \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})\mathrm {d}s}\right ]}{\eta (t)} \leq -C_{2}, $\end{document} almost surely with respect to ω, for every fixed x∈ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x \in \mathbb {R}^{d}$\end{document}. The functions η(t) and the bounds C1,C2 heavily depend on the intensity of large jumps of the process. In particular, if its decay at infinity is ‘sufficiently fast’, then we prove that C1 = C2, i.e. the limit exists. Representative examples in this class are relativistic stable processes with Lévy-Khintchine exponents ψ(ξ) = (|ξ|2 + m2/α)α/2 − m, α ∈ (0,2), m > 0, for which we obtain that limt→∞logExe−∫0tVω(Xs)dst/(logt)2/d=α2m1−2αρωddd2λ1BM(B(0,1)),for almost allω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})ds}\right ]}{t/(\log t)^{2/d}} = \frac {\alpha }{2}m^{1-\frac {2}{\alpha }} \left (\frac {\rho \omega _{d}}{d}\right )^{\frac {d}{2}} \lambda _{1}^{BM}(B(0,1)), \quad \text {for almost all} \omega ,$\end{document} where λ1BM(B(0,1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{1}^{BM}(B(0,1))$\end{document} is the principal eigenvalue of the Brownian motion killed on leaving the unit ball, ωd is the Lebesgue measure of a unit ball and ρ > 0 corresponds to Vω. We also identify two interesting regime changes (‘transitions’) in the growth properties of the rates η(t) as the intensity of large jumps of the processes varies from polynomial to higher order, and eventually to stretched exponential order.

almost surely with respect to ω, for every fixed x ∈ R d . The functions η(t) and the bounds C 1 , C 2 heavily depend on the intensity of large jumps of the process. In particular, if its decay at infinity is 'sufficiently fast', then we prove that C 1 = C 2 , i.e. the limit exists. Representative examples in this class are relativistic stable processes with Lévy-Khintchine exponents ψ(ξ ) = (|ξ | 2 + m 2/α ) α/2 − m, α ∈ (0, 2), m > 0, for which we obtain that

Introduction
This paper is concerned with the large time asymptotic behaviour of the solutions of the spatially continuous parabolic nonlocal Anderson problem with Poissonian interaction, driven by a Lévy process in R d . More precisely, we consider the equation where L is the generator of the underlying process and V ω (x) = R d W (x − y)μ ω (dy) is a random Poissonian potential with sufficiently regular profile function W : R d → R + . By μ ω we denote the Poisson random measure on R d with intensity ρ dx, ρ > 0, over a given probability space ( , Q). Processes considered throughout the paper, X = (X t , P x ) t≥0, x∈R d , are symmetric Lévy processes with jumps, with characteristic functions E 0 e iξ ·X t = e −tψ(ξ) , ξ ∈ R d , t > 0, whose characteristic exponents (symbols) ψ are given by the Lévy-Khintchine formula (1 − cos(ξ · z))ν(dz). (1.2) Here A = (a ij ) 1≤i,j ≤d is a symmetric non-negative definite matrix, and ν is a symmetric Lévy measure, i.e. a Radon measure on R d \ {0} that satisfies R d \{0} (1 ∧ |z| 2 )ν(dz) < ∞ and ν(E) = ν(−E), for every Borel E ⊂ R d \ {0} [14,15]. We always assume that X is strong Feller and e −t 0 ψ(·) ∈ L 1 (R d ), for some t 0 > 0 (for more details see Section 2.1). Since its introduction in the 50's of the past century, the Anderson model based on the Laplacian (both continuous and discrete), with various potentials, has been studied with varying intensity. The notion 'parabolic Anderson model' appeared first in the paper of Gärtner and Molchanov [10]. For an excellent, updated review of the history of the research in this area we refer to the book of König [24].
Under suitable regularity assumptions, the solution to the problem (1.1) can be probabilistically represented by means of the Feynman-Kac formula: u ω (t, x) = E x e − t 0 V ω (X s ) ds . (1.3) One is interested in the long-time behaviour of u ω (t, x), in both the annealed sense (averaged with respect to Q) and the quenched sense (almost sure with respect to Q). In this paper, we will analyse the quenched behaviour of functionals u ω (t, x) for Lévy processes whose exponent ψ can be written as ψ(x) = ψ (α) (x) + o(|x| α ), |x| → 0, (1.4) for some α ∈ (0, 2], and satisfies some mild assumptions concerning its behaviour at infinity. In formula (1.4), ψ (α) is the characteristic exponent of a symmetric (not necessarily isotropic) α−stable process, i.e. a Lévy process with characteristic exponent 1 − cos(ξ · rz) r 1+α n(dz)dr, (1.5) where n is a symmetric finite measure on unit sphere S d−1 when α ∈ (0, 2), or where A = (a ij ) 1≤i,j ≤d is a symmetric nonnegative definite matrix when α = 2. When n is the uniform distribution on S d−1 for α ∈ (0, 2) or A ≡ a Id with some a > 0 for α = 2, then the process is called isotropic α−stable. We assume the nondegeneracy condition inf |ξ |=1 ψ (α) (ξ ) > 0. The annealed asymptotics of u ω (t, x) has been first analyzed by Donsker-Varadhan [8] (for stable processes, including the Brownian motion) and of Okura [27] (for symmetric Lévy processes satisfying (1.4)). When the profile W is of order o(1/|x| d+α ) when |x| → ∞, they prove that (1.7) In this formula, ω d is the volume of the unit ball, and denotes the infimum of principal eigenvalues for the symmetric α−stable process with exponent (1.5) in U with outer Dirichlet conditions on U c . Okura's work covers also the case when ψ(x) = O(ψ (α) (x)), |x| → 0, but only when the potential is heavy-tailed. This does not fall within the scope of present paper and so we will discuss this case elsewhere. A version of (1.7) for subordinate Brownian motions on the Sierpiński gasket was obtained in [20]. The quenched behaviour for the diffusion processes interacting with Poissonian random fields is also quite well understood. The key observation used in this case is that when the profile function W is of bounded support, then Q−a.s. there exist a large area where the potential V ω (x) is equal to zero. Typically, with high probability, the process tends to remain in that 'atypical', 'favorable' areas, which determine the a.s. behaviour of the functional. As a result, the quenched behaviour can differ from the annealed asymptotics.
This phenomenon (for the Brownian motion) was first rigorously established by Sznitman in [34]. He proves that in that case, for any x ∈ R d , and Q−almost all ω, lim t→∞ log u ω (t, x) t/(log t) 2 (B(0, 1)), (1.8) where λ BM 1 (B(0, 1)) is the principal eigenvalue for the Brownian motion killed on exiting B(0, 1). This result was reproven by Fukushima [9]. For the Brownian motion on some irregular spaces such as the Sierpiński gasket, one also sees a similar phenomenon: rates of the annealed and the quenched asymptotics differ (see [28,29]). The quenched behaviour for Lévy processes with jumps in Euclidean spaces has not been studied before (even in the most regular cases) and the identification of the corresponding quenched rate has been an open problem for a long time now.
In this paper, we address the quenched asymptotics for a wide class of symmetric Lévy processes with jumps influenced by potentials with compact-range profiles. Key examples include a vast selection of isotropic unimodal Lévy processes, subordinate Brownian motions, processes with nondegenerate Brownian components and with non-isotropic Lévy densities as well as processes with less regular Lévy measures that have product or discrete large jumps components. While the 'favorable' spots in the Poissonian configuration are still present, the jumping nature of Lévy processes can alter the picture: if the intensity of large jumps decays polymomially, which is the case of stable processes, then the quenched and the annealed rate coincide. What is really decisive here is the intensity of large jumps of the process: for processes with Lévy measures whose tails decay are faster than polynomial at infinity, one should expect the same phenomenon as that for Brownian motion, i.e. the quenched rate being substantially different from the annealed rate.
For more clarity, we have collected the results obtained for particular classes of processes with various types of large jump intensities in Table 1 below (for simplicity we restricted the presentation to the family of isotropic unimodal Lévy processes with stabletype small jumps). The detailed discussion of specific families of processes (also including the selection of less regular cases) is given in Section 5.
The annealed rate t d/(d+α) is always governed by the exponent α appearing in (1.4), which in turn is determined by the behaviour of the exponent of ψ near zero. Formula (1.4) together with some mild assumptions concerning the behaviour of the symbol at infinity permit to obtain the annealed asymptotics of u ω (t, x) and also to identify the constant in (1.7).
The question of the quenched rate is much more delicate. In this case, the formula (1.4) (even if combined with some information on the behaviour of the characteristic exponent at infinity) is generally insufficient. This is particularly evident when α = 2. It occured to us as a surprise that the effective derivation of the quenched rate (and the corresponding bounds) requires deep analysis of the subtle properties of Lévy processes with prescribed Lévy measures, depending on the type of their fall-off at infinity.
As usual, in this paper the upper and the lower bounds of u ω (t, x) are addressed separately. First, in Sections 3 and 4 we prove two general results: Theorem 3.1 concerning the upper bound, and Theorem 4.1 concerning the lower bound.
The rest of the paper (Section 5) is devoted to the application of our general results for specific classes of processes.
(1) For processes satisfying (1.4) with α ∈ (0, 2) (Theorem 5.2 and Examples 5.1-5.2), and also for those with α = 2 but polynomially decaying Lévy measures (Theorem 5.3 and Examples 5.3, 5.5 (2)), the quenched and annealed rates coincide and are both equal to t d/(d+α) (we were not able to show that the lower and the upper constant agree).
A similar phenomenon in a discrete setting has been recently observed in [25], where the annealed and quenched asymptotics of the heavy tailed (stable-type) random walk under the lattice Weibull-type random potential on Z d were considered. More precisely, in that case the annealed and quenched rates agree, but the authors were not able to prove the existence of the limit in the quenched case (the upper and the lower bounds differ).
(2) If the Lévy measure decays stretched exponentially or faster (one must necessarily have α = 2 in this case), then the annealed rate is t d/(d+2) , while the quenched rate is Upper bound for lim sup 1,ν and λ (2) 1,ν denote the principal eigenvalues for the given stable process (in the first line) and diffusions determined by Gaussian matrices as in (5.20) (in the next five lines), killed on leaving the ball B(0, 1). We compare these examples with the case of Brownian motion which is included in the last line. In the last column we indicate for which processes the convergence follows bigger and equal to t/(log t) 2/d . This is the same rate as that for the Brownian motion.
In this case, we not only identify the quenched rate, but also often obtain the limit (Theorem 5.5 and Examples 5.4, 5.5 (1)). This case covers many important examples of processes that are of interests in mathematical physics, including the relativistic α−stable process and some tempered stable processes [5,17]. (3) We also consider a class of processes with Lévy measures that have intermediate decay: slower than stretched exponential, but still faster than polynomial (Theorem 5.4). The annealed rate is perforce equal to t d/(d+2) , but the quenched rate obtained is t (βd)/(2+βd) , β being a parameter specific to the process.
It is seen from this picture that we encountered two interesting regime changes in the growth properties of the quenched rates. The first one can be observed when the intensity of large jumps of the processes varies from polynomial to higher order, in the sense that the quenched rate becomes faster than the annealed rate (i.e. it is no longer the same as the annealed rate and becomes substantially dependent on the decay of the intensity of large jumps of the process). The second transition occurs as the intensity of large jumps becomes at least stretched exponential. In this case, we prove that the limit exists and the quenched rate takes the form t/(log t) 2/d , which is the fastest possible one. It agrees with the rate obtained for Brownian motion. It is worth pointing out that similar large jumps intensitydependent transition in the ground state fall-off properties of the nonlocal Schrödinger operators has been recently identified in [18]. However, interestingly, in our present work even the processes with stretched-exponential decay of the intensity of large jumps have the fastest possible quenched rate. Such a phenomenon seems to be new and has not been observed before.
The verification of the assumptions of our general Theorems 3.1 and 4.1 for various types of Lévy measures (i.e. in each of the situations (1)-(3) above) requires a separate analysis. The applicability of our results essentially depends on the verifiability of the condition (U) preceding Theorem 3.1. It asserts that the tail P 0 (|X t | > r) is dominated by a profile function F (r) for large r. This profile plays a crucial role in determining the quenched rate and therefore, in applications, it is a key initial step to establish it as precisely as possible. It does not come as a surprise that such a profile should be determined by the tail of the corresponding Lévy measure. When (1.4) holds with α ∈ (0, 2), then the corresponding profiles F (r) are derived by using the general estimates for the tails of the supremum functional obtained in [30]. When α = 2, the problem is more complicated and it requires an application of the sharp estimates of the transition probability densities that are available in the literature. For Lévy measures with stretched exponential and lighter tails, we apply directly the results of [6] while for those with polynomial and intermediate tails we use the estimates obtained recently in [22] (Lemmas 5.2-5.3). The case of jump processes with non-degenerate Gaussian components is discussed separately in Proposition 5.3. Another key step in application of our general lower bound was to find a possibly sharpest lower estimate for the Dirichlet heat kernels of the large box which leads to sufficiently precise lower bound of the function G defined in (4.4). For processes with Lévy measures whose tails decay at infinity not faster than exponentially this is established in Proposition 5.4. The cases with lighter tails require an application of more specialized estimates obtained in [23].
At the end of the Introduction, we will explain how the results from general Theorems 3.1 and 4.1 are obtained and how the correct rate is identified. To the best of our knowledge the quenched asymptotics for Lévy processes with jumps has not been studied before. In the literature concerning the Brownian motion, one finds two methods: Sznitman's paper [34] estimates u ω (t, x) directly, using his 'enlargement of obstacles' technique for the more difficult upper bound (similar method was used on the Sierpiński gasket in [28]); Fukushima [9] gives elegant arguments for deriving both the upper and the lower quenched bound from respective upper and lower bounds at zero for the integrated density of states of the corresponding Schrödinger operator (being closely related to the annealed upper and lower bounds) -this is done by means of the Dirichlet-Neumann bracketing for the Laplace operator. In our work, we are able to find a counterpart of Fukushima's method for Lévy processes with jumps to obtain the upper bounds. The main ingredient of the proof is the construction of some auxiliary function f , reflecting the crucial spectral and distributional properties of the underlying process under the Poissonian potential, through which we determine the typical asymptotic profile for the quenched behaviour of u ω (t, x). As the Dirichlet-Neumann bracketing seems not to be available in the nonlocal case, we had to use a different approach for the lower bound. The lower estimate of u ω (t, x) we provide is proven directly, without using any properties of the annealed limits.
More precisely, when the realization of potential is fixed, then the strategy of the process to survive up to time t is similar to that of Brownian motion: first the process moves to the 'obstacle-free' area and then it tends to remain there. Such a region of size r = r(t) can be typically (i.e. Q−a.s.) found at distance e cr d from the origin (Lemma 4.1). Both the cost of getting to this area, and the cost of staying there can affect the quenched rate. The size r(t) depends on the process: the faster the process spreads out, the bigger this size can be. For example, when the intensity of large jumps is polynomial (∼ |x| −d−α , jump stable process), then we can take r(t) ∼ t 1 d+α . Incidentally, the term related to the cost of getting to this region (expressed by the intensity of large jumps of the process) and the term including the principal Dirichlet eigenvalue for this region (responsible for the probability of not exiting the set up to time t) are of the same order, and they both enter the quenched rate. This should be contrasted with the case of Brownian motion and jump processes with light tails, where the correct rate is obtained by choosing a box of logarithmic size. Now the probability of getting to this box is relatively small (so that the corresponding term is asymptotically irrelevant), and, in the limit, the correct quenched behaviour is governed solely by the term including the principal eigenvalue of the potential-free region.
The annealed survival strategy is essentially different: we make the process remain in the ball of radius r centered at the origin which is known to be free of Poisson obstacles with probability e −cr d . The optimal annealed rate is then obtained by choosing a correct timedependent size r(t) for which the survival probability of the process in this region (which is again measured by the principal eigenvalue of the killed process) is balanced by that probability. The main observation is now that despite the fact the both quenched and annealed strategies are different, for Lévy processes with polynomial intensities of large jumps the bounds leading to the identification of the corresponding rates can be of the same order.

Lévy Processes
Recall that X = (X t ) t≥0 is assumed to be a symmetric jump Lévy process in R d , d ≥ 1, with Lévy-Khintchine exponent ψ as in (1.2). We will always assume that X is strong Feller and there exists t 0 > 0 for which e −t 0 ψ(·) ∈ L 1 (R d ). (2.1) Note that the strong Feller property is equivalent to the existence of measurable transition densities p(t, x, y) = p(t, y − x) for the process (see e.g. [31,Th. 27.7] Consequently, X is strong Markov with respect to its natural filtration and has a modification with càdlàg paths. The càdlàg property will be assumed throughout the paper. For more details on Lévy processes we refer to [1,14,15,31]. The generator L of the process (X t ) t≥0 is a nonlocal pseudodifferential operator uniquely determined by its Fourier transform The corresponding Dirichlet form (E, D(E)) can be defined by The transition densities p U (t, x, y) of the process killed upon exiting an open, bounded set U ⊂ R d are given by the Dynkin-Hunt formula (2.4) Here and thereafter, τ U = inf {t ≥ 0 : X t / ∈ U } denotes the exit time of the process from the set U . The L 2 −semigroup of operators with kernel p U (·, ·, ·), also called the Dirichlet semigroup, will be denoted by P U t : t ≥ 0 . Since U is bounded, the operators P U t are trace-class (consequently, compact) and admit a complete set of positive eigenvalues Sometimes, to specify which process we are working with, these eigenvalues will be denoted by λ ψ i (U ), where ψ is the Lévy exponent of (X t ) t≥0 . In the special case of symmetric α-stable processes, α ∈ (0, 2], its corresponding Dirichlet form will be denoted by (E (α) , D(E (α) )), and the eigenvalues of the Dirichlet semigroup -by λ (α) i (U ). For the standard Brownian motion running at twice the speed, we will use the notation (E BM , D(E BM )) and λ BM i (U ), respectively.

Poisson Potentials
The process X will be subject to interaction with a nonnegative, random Poissonian potential V ω . To properly set the assumptions, recall that the Kato class relative to X, K X , consists of those measurable functions V : 5) and the local Kato class K X loc -of functions V such that for every ball Sharp estimates of p(t, x) that are available in the literature (see e.g. [2,6,21,22]) often allow to find more explicit form of (2.6). Further, consider a Poisson point process on R d , with intensity ρ dx, ρ > 0, defined on some probability space ( , M, Q), and let W : where μ ω is the random counting measure on R d corresponding to the Poisson point process. For such profiles W, the potential V ω (·) belongs Q−almost surely to K X loc . This can be directly justified by following the argument in [19, Proposition 2.1], where it has been proven for the subordinate Brownian motions on the Sierpiński gasket. One can check that when the profile W is continuous, or when it is a radial and non-increasing function, then the condition (2.7) is satisfied under the assumption W ∈ L 1 (R d ). Starting from Section 4 we will be interested in the Poissonian potentials with finite-range (compactly supported) profiles W , for which (2.7) holds automatically. By the range of a profile W we mean

Random Semigroups and the Integrated Density of States
Suppose that W : R d → R + is a profile function belonging to K X loc for which (2.7) holds. As indicated above, V ω given by (2.8) belongs to K X loc , Q−almost surely. Therefore we can legitimately define the random Feynman-Kac semigroups P V ω t : t ≥ 0 and P U,V ω t : t ≥ 0 related to the 'free' process and the process killed on exiting an open, bounded and nonempty set U ⊂ R d . They consist of operators and admit the measurable, strictly positive, bounded and symmetric kernels p V ω (t, x, y) and p U,V ω (t, x, y), respectively. It is known that Q−a.s. the semigroup P V ω t : t ≥ 0 coincides with the semigroup generated by the operator −H ω , where H ω = −L + V ω is the random nonlocal Schrödinger operator based on the generator L of the process X, with Poissonian potential V ω [4,7]. The semigroup P U,V ω t : t ≥ 0 corresponds then to the random nonlocal Schrödinger operator H ω U with exterior Dirichlet conditions on U . The operators P U,V ω t are Hilbert-Schmidt, so that the spectrum of the operator H ω U is Q−a.s. discrete: Again, we will single out the case of α−stable processes and denote the respective eigenvalues by λ (α) i (U, V ω ). Similarly, P V t and P U,V t will denote operators relative to nonrandom potentials 0 ≤ V ∈ K X loc .
Consider now the process killed on exiting the boxes U = U R = (−R, R) d , and the random empirical measures on R + , based on the spectra the generators of such processes, normalized by the volume: From the maximal ergodic theorem it follows that Q−a.s. the measures ω R are vaguely convergent as R → ∞ to a nonrandom measure on R + , called the integrated density of states (see e.g. [26, p. 635]). The cumulative distribution function of the measure will be denoted by N D (λ). The superscript D indicates that we are dealing with the Dirichlet exterior conditions (as opposed to the Neumann conditions, which are not pursued in this paper).

Notation
We say that the function g is asymptotically equivalent to the function f at infinity, which is denoted by g ≈ f , when lim x→∞ f (x)/g(x) = 1. Likewise, when we say f g, we mean that there exists a constant C ∈ [1, ∞), such that 1 denotes the open Euclidean ball with center x ans radius R > 0. We also say that a measurable function Constants whose values are relevant in further arguments are denoted with upper case letters C, K, Q, possibly with subscripts. Constants whose importance is limited to the place of their occurrence are numbered within each proof separately as c 1 , c 2 , ....

Preliminary Estimates
We start with two preliminary results. First we give a lemma about nonrandom potentials. Recall that the constant t 0 comes from the assumption (2.1). t≥0 be a symmetric strong Feller Lévy process with Lévy-Khintchine exponent ψ as in (1.2) and (2.1), and let 0 ≤ V ∈ K X loc . Then there exists a constant Proof The proof goes along standard arguments (cf. [35, (3.1.13)]). Let U, x, and t be as in the assumptions. When |U | = ∞, then the statement is obvious, so assume |U | < ∞. We have Consequently, we obtain which is the desired bound (3.1).
In the random setting, we will need the following lemma on the mean number of eigenvalues not exceeding a given level λ > 0. Lemma 3.2 Let X be a symmetric strong Feller Lévy process with characteristic exponent ψ satisfying (1.2) and (2.1), and let V ω be a Poissonian potential defined in (2.8). For n ∈ Z + let D n = (−2 n , 2 n ) d . Then for every λ > 0 we have By using standard min-max formulas for eigenvalues (see, e.g., [33, Section 12.1]), one can check that Moreover, the space homogeneity of the process together with the stationarity of the potential V ω give Taking the expected value E Q on both sides of (3.4), we immediately get (3.2).

A General Upper Bound
We first introduce an auxiliary function through which we determine the typical asymptotic profile for the quenched asymptotics of the function u ω (t, x).
For every α ∈ (0, 2], κ > 0, and a nonincreasing function F : One can directly see that for any fixed α ∈ (0, 2], κ > 0, and a given function F (r) we have In particular, the inverse function is well defined. It is strictly increasing and satisfies When the parameters α and κ are fixed, they will be dropped. Observe that the function h F,α,κ (t) satisfies: The function h F,α,κ (t) will play a central role in determining the rate of decay of the functionals considered.
In what follows we will work under the following regularity condition (U) on the process X. In the condition below, the constant t 0 comes from the assumption (2.1).

9)
then for every fixed x ∈ R d one has
Proof Fix x ∈ R d and let r 0 , t 1 , γ , α, κ 0 and F be as in the assumptions. Specifically, we may and do assume that r 0 ≥ 1 is so large that F (r) ≤ 1 for r ≥ r 0 . We will write h for h F,α,κ 0 . By Lemma 3.1, for every t ≥ t 0 /2 and every open set U x, we have (3.11) In particular, we can choose Now, since for this choice of U we have B(x, R) ⊂ U, from the Lévy inequality and assumption (U) we obtain: We now estimate λ 1 (U 2R , V ω ) for large R. Inequality (3.3) from Lemma 3.2 holds for dyadic boxes D n and reads: Running the argument from [9, (2.3)-(2.6)] with φ(r) = κ 0 r d/α and the sequence t n = 2 n , from the assumption (3.9), we get that for every ε ∈ (0, 1), Q−almost surely we can find (3.14) Piecing together (3.11), (3.13), and (3.14) we get that for every ε ∈ (0, 1), Q−a.s. there exists R ε > 1 such that for all t and R satisfying R ≥ R ε and (3.12) one has and further, with an absolute constant c 1 ≥ 0. Let now h(t) be given by (3.6). As h(t) → ∞ when t → ∞, Q−a.s. there exists t 2 ≥ t 1 large enough so that for every t ≥ t 2 the condition (3.12) holds with R = h(t) ∨ C 3 t, and moreover R ≥ R ε (ω). Thus we may substitute into (3.15) the value (3.16) Next, from the definition of h(t), (3.7), and the monotonicity of f F,α,κ 0 , we see that with equality when h(t) ≥ C 3 t. We finally obtain that for all t > t 2 we have with absolute constants c 1 , c 2 ≥ 0, for Q-almost all ω.
To complete the proof, it remains to show that where g(t) := t/(log h(t)) α/d . This is obvious when h(t) ≤ C 3 t, and when h(t) ≥ C 3 t, then from (3.7) we have for some c 3 > 0 and (3.18) follows. We conclude that Q−almost surely Letting ε → 0 through rational numbers, we get (3.10). The proof is complete.
The next corollary will eventually enable us to obtain, for certain processes, the existence of lim t→∞ log u ω (t,x) g(t) .  (3.19) and, consequently, for every fixed x ∈ R d , one has In particular, when lim r→∞

21)
Proof The assumptions give that for any 0 < Q 1 < Q 1 there exists r 0 such that for r > r 0 which is equivalent to saying that for sufficiently large r Letting Q 1 Q 1 we get (3.19). Statements (3.20) and (3.21) follow directly from (3.10).

The Lower Bound
As indicated in the Introduction, the argument deriving the quenched asymptotic lower bound directly from the lower asymptotics of the IDS seems to be not obvious in the nondiffusion case. Instead, we estimate u ω (t, x) directly. In this part (similarly as in [9,34]), we require the potential profile W to be bounded and compactly supported. We first prove that Q−almost surely there exist sufficiently large regions where the potential is equal to zero, then we force the process to go to this region and then stay there for a long enough time. This phenomenon will be expressed as the behaviour of the semigroup of the process.

Typical Potential Configuration
Let ε > 0 be given. For a given number r > 0, let M ε (r) be defined by where ω d denotes the volume of the unit ball in R d . We have a lemma.  (m + a), 'concentric' with those boxes are disjoint. As the realizations of the cloud over disjoint sets are independent random variables, the probability that each such (small) ball contains at least one Poisson point (denote this event by A ) equals to We would like to produce a ball with radius (m + a) that is both: free of Poisson points and separated from zero, so that we exclude from our considerations the boxes whose closure might contain zero. There are at most 2 d of such balls. Let A m be the event that 'every small ball from (−M ε/2 (m), M ε/2 (m)) d \ (−m − a, m + a) d , arising as above, contains a Poisson point', then (1)) , for m → ∞, and the expression in the exponent is equal to (recall r = m) We also quote Lemma 3.2 from [9] concerning the behaviour of the potential (the moment condition [9, (1.10)] is satisfied with α = 1 in the Poissonian case).

A General Lower Bound
Let R > R 0 > 0 be given and let p U R (t, x, y) be the Dirichlet kernel of our process (X t ) t≥0 in the box U R := (−R, R) d . To begin with, we introduce the following notation: (1, 0, y). (4.4) Also, recall that λ (α) 1 (B(0, 1)) is the principal Dirichlet eigenvalue of the symmetric αstable process defined by (1.5) in the unit ball B(0, 1) and ω d is the volume of this ball.
We now present our main theorem in this section.

5)
and that V ω is a Poissonian potential defined in (2.8) with bounded profile W of finite range. Then for any x ∈ R d , κ, R 0 > 0 and any nonincreasing function

α,κ (t) is well-defined for large t's).
Proof For simplicity, we run the proof for x = 0 only; for a general x ∈ R d the proof is identical. Let κ > 0 and R 0 > 0 be given. As in the proof of Theorem 3.1, we will write h for h F,α,κ . Let ε > 0 be given and let a be the range of the potential profile W , then for t > 0 let m(t) and M(t)(= M ε (m(t))) be related by (4.1). The potential range a is fixed so it does not enter the notation. For the time being we require only that m(t) → ∞ when t → ∞. Eventually, the number m(t) will be chosen of order h(t) from (3.6), but in such a manner that M ε (m) will bear no ε−dependence.
Pick ω outside the exceptional sets from Lemmas 4.1 and 4.2. Let B t be the open ball of radius m(t) whose a−neighbourhood contains no Poisson points, obtained from the statement of Lemma 4.1. As there is no interaction with the potential inside this ball, we have that p B t ,V ω (·, ·, ·) = p B t (·, ·, ·), and consequently λ ψ 1 (B t , V ω ) = λ ψ 1 (B t ) (recall that λ ψ 1 (U, V ω ) and λ ψ 1 (U ) denote the principal Dirichlet eigenvalue of the process in U under the influence of the potential V ω , or without potential, respectively).
For sufficiently large t we have the following chain of inequalities: Since the potential V ω is zero on B t , the double integral above is just equal to B t B t p B t (t − 1, x, y)dxdy, which is the heat content of the operator P B t t−1 . We would rather have an estimate with the trace. To this goal, we use the assumption (2.1). For any x, y ∈ R d we have, by the Fourier inversion theorem, p B t (t 0 , x, y) ≤ p(t 0 , x, y) ≤ 1 (2π) d R d e −t 0 ψ(z) dz =: A < ∞. We continue the main estimate as: From the translation invariance of the process and assumption (4.5) we see that To estimate the infimum of the kernel , m(t)) d so that for sufficiently large t one has R 0 ≤ y ≤ √ dM(t). Using (4.3) and (4.4) we can write: Substituting the estimates (4.7) and (4.8) into (4.6), we obtain that Q−a.s., for sufficiently large t: (1)) .

At this point we declare the scale m(t).
Recall that all this reasoning is performed for a fixed number ε > 0. Set m(t) = m ε (t) to be the solution of the equation (unique for large t) where h(t) was given by (3.6). Consequently, using (4.1), (4.10) Further, from the relation (3.7) defining h, we see that Consequently, for sufficiently large t we get These properties give that, Q−almost surely, (B(0, 1)).
The next corollary gives a direct lower bound for lim inf log u ω (t,x) , similar to that in Corollary 3.1.

Corollary 4.1 Let the assumptions of the above theorem be satisfied. In particular, let F and G be the monotone functions appearing in its statement. If there exist
(4.14) In particular, when the assumptions (4.11) and (4.12) hold with Q 1 = ∞ and Q 2 = 0, then (B(0, 1)).

(4.15)
Proof The first bound in (4.13) follows from (4.11) exactly by the same argument as in Corollary 3.1. To prove the second bound in (4.13) we write The desired bound immediately follows from (3.7) once we recall that h(t) → ∞ when t → ∞.

Discussion of Specific Cases
We will apply the general results of previous sections to some particular processes, for which the assumptions of Theorems 3.1 and 4.1 hold true. Throughout this section we will work under the assumption that the Lévy-Khinchine exponent ψ is close to the characteristic exponent of a symmetric α−stable process near the origin. More precisely, we assume the following condition.
Under these assumptions, in the paper [27] the annealed asymptotics of u ω (t, x) was proven, and also in [26] the behaviour of the integrated density of states, N D (λ), was established. We have the following.
The constant λ (α) is given by the variational formula where the infimum is taken over all open sets G ⊂ R d of unit Lebesgue measure.
Theorem 6.2 in [26] has been proven for continuous profiles W , but its proof also applies to the local Kato-class case.
Moreover, it follows from the Faber-Krahn isoperimetric inequality (see, e.g. [8,Lemma 3.13] and [3,Theorem 3.5]) that when the process X (α) is isotropic, then the infimum in  (B(0, 1)). Theorem 5.1 above states that (C) and (5.1) are sufficient conditions for the validity of (3.9), which is the main assumption of Theorem 3.1. We now show that when (C) holds, then also the quasi-scaling of principal eigenvalues needed in Theorem 4.1 holds true. The following proposition takes care of that.
Then, from (5.4) it follows that for every R > 1 (the first inequality follows by the standard variational formula for the principal eigenvalue). Finally, by substituting R = (1 + δ)R, we get The statement follows now from (5.5) and (5.6).
We now provide some reasonable and easy-to-check sufficient conditions under which the basic asymptotic assumption (C) holds true. (1.2) with Gaussian coefficient A = (a ij ) 1≤i,j ≤d and Lévy measure ν. The following hold.

Proposition 5.2 Let X be a Lévy process determined by the Lévy-Khintchine exponent ψ as in
where A = ( a ij ) 1≤i,j ≤d with a ij = a ij + 1 2 R d y i y j ν(dy).
Proof Knowing that ψ(ξ ) = ξ · Aξ + R d (1 − cos(ξ · y)) ν(dy), and writing down the Taylor expansion of the function cos s at 0 we get The first assertion follows from the dominated convergence theorem together with the finiteness of the second moment of ν.
Since 0 ≤ ξ · Aξ ≤ A |ξ | 2 , we only need to show that the last member above is of order o(|ξ | α ). We have and the statement follows from the assumption.
In what follows we will often use the following notation. If X = (X t ) t≥0 is a symmetric Lévy process with characteristic exponent ψ as in (1.2), then we write t≥0 is the Gaussian part determined by the Lévy-Khintchine exponent ψ A (ξ ) = ξ · Aξ , and X ν = (X ν t ) t≥0 is the jump part with the exponent ψ ν (ξ ) The following fact on the tails of jump Lévy processes with nondegenerate Gaussian component will also be needed below. It states that one can add a sufficiently regular diffusion process to a purely jump Lévy process without spoiling the assumption (U). 1≤i,j ≤d and Lévy measure ν. Moreover, suppose that inf |ξ |=1 ξ · Aξ > 0. If the process X ν satisfies the assumption (U) with γ > 0, profile F and constants C 2 , r 0 , t 1 , then the entire process X also satisfies a version of (U). More precisely, there are constants C 2 ≥ C 2 and C 4 ∈ (0, 1] such that

Proposition 5.3 Let X be a Lévy process determined by the Lévy-Khintchine exponent ψ as in (1.2) with Gaussian coefficient
In particular, if F (C 4 r) ≥ e −C 4 r for r ≥ 2r 0 , then X satisfies the assumption (U) with C 2 , the same γ and the profile F (r) = F (C 4 r). If F (C 4 r) < e −C 4 r for r ≥ 2r 0 , then the same is true with C 2 , γ and F (r) = e −C 4 r .
Proof For t > t 1 and r ≥ 4t, we may write The first part is estimated using (U). To take care of the Gaussian part, note that under the assumption inf |ξ |=1 ξ · Aξ > 0 the transition densities p A (t, x, y) of the corresponding diffusion process X A exist and enjoy the Gaussian upper estimates: In particular, taking into account that r > 4t ≥ 4t 1 , for some positive constants c 1 − c 5 . The proof is complete.
In the sequel, we will also need the following general lower estimate for the function G defined in (4.4). Recall that for every R > 0 we have denoted U R = (−R, R) d . Below we will also write B(R) for B(0, R).
To complete the proof, we need to estimate the kernel p B(2r 0 ) (t , 0, x) for every t ∈ (0, t] and |x| ≤ r 0 . Let first r 0 /2 < |x| ≤ r 0 . By following through with the argument above and using (5.7), we have In the case when |x| ≤ r 0 /2, use first the Chapman-Kolmogorov identity and then (5.12) to get Here we have used the fact that the set B(r 0 ) \ B(r 0 /2) ∩ B(x, 3r 0 /4) \ B(x, 3r 0 /8) always contains a ball of radius r 0 /8 (in the last line we first restricted the integration to this ball and then we took the infimum). Observe that the last infimum can be estimated exactly in the same way as in (5.12). We thus have as long as 3r 0 8 < |z| ≤ 3r 0 4 . In consequence, for 0 < s < t we have inf with η(t, r) defined by (5.9). The Proposition follows.

Processes with Polynomially Decaying Lévy Measures
In this subsection we show how our general results translate to the case when the Lévy measure is polynomially decaying at infinity. We now give versions of Theorems 3.1 and 4.1 specialized to this case. Recall that for a symmetric α−stable process with Lévy-Khinchine exponent ψ (α) , by λ 1 over all open sets of unit measure. We first consider the class of Lévy processes that are close to non-Gaussian symmetric stable processes in the sense of the condition (C). As we will see later (Lemma 5.1), when the Lévy measure of such process has a density comparable with a nonincreasing function, then its decay at infinity is necessarily stable-like. (2) Mixture of isotropic stable processes (possibly with Brownian component). Let ψ(ξ ) = a 0 |ξ | 2 + n i=1 a i |ξ | α i , n ∈ N, a 0 ≥ 0, and a i > 0, α i ∈ (0, 2), for i = 1, ..., n. Then the assumptions of Theorem 5.2 are satisfied with α = α i min := min i α i and ψ (α) (ξ ) = a i min |ξ | α i min .
Example 5.2 (More general perturbations of symmetric stable processes) Let δ ∈ (0, 2) and let n be a symmetric finite measure on the unit sphere S d−1 such that for some constant c 0 > 0. Denote the corresponding stable Lévy measure by ν (δ) (drdθ) = n(dθ)r −1−δ dr. Note that we do not impose similar growth condition on n from above, which means that ν (δ) is not necessarily absolutely continuous with respect to the Lebesgue measure. Furthermore, let ν ∞ be a (non-necessarily infinite) measure on R d \ {0} such that (1) Other stable Lévy measures: and n is a symmetric finite measure on S d−1 . with θ ∈ (0, 2q) and β > 2q. Then one can take We now turn to the class of processes with Lévy measures that have second moment finite. In this case, for a given Gaussian matrix A = (a ij ) 1≤i,j ≤d , the coefficients A = ( a ij ) 1≤i,j ≤d , with a ij = a ij + 1 2 R d y i y j ν(y)dy, (5.20) are well defined (see Proposition 5.2 (i)). In what follows, by λ (2) 1 (U ) we will always denote the principal eigenvalue of the diffusion process with characteristic exponent ψ (2) (ξ ) = ξ · Aξ , killed on leaving an open bounded set U ⊂ R d . Also, λ (2) denotes the infimum of λ (2) We now discuss the case of Lévy measures with polynomial tails whose decay at infinity is faster than stable. To avoid some technical difficulties and for more clarity, in the theorem below we restrict our attention to the absolutely continuous case (cf. Example 5.5 (2)).

Theorem 5.3
Let X be a symmetric Lévy process with characteristic exponent ψ as in (1.2), with defining parameters A = (a ij ) 1≤i,j ≤d and ν(dx) = ν(x)dx. Assume that there exist C 10 ≥ C 11 > 0 and δ 1 ≥ δ 2 > 2 such that Proof By Proposition 5.2 (i) not only the coefficients a ij given by (5.20) are finite, but also one has i.e. the basic asymptotic assumption (C) holds true with α = 2. Also, (5.1) is satisfied in both cases of (i).
As usual, to establish the upper bound we apply our general Theorem 3.1. When A ≡ 0, then from Lemma 5.2 below we get p(t, x) ≤ c 1 t δ 2 /2 |x| −d−δ 2 , x ∈ R d \{0}, t ≥ t 1 , for some t 1 > 0, so that (U) holds with F (r) = r −δ 2 and γ = δ 2 /2. When inf |ξ |=1 ξ ·Aξ > 0, then the same is true by Proposition 5.3. Also, by Theorem 5.1, (3.9) holds true with κ 0 = ρ(λ (2) ) d/2 and the proof of the upper bound can completed by following the argument in the proof of the upper bound in Theorem 5.2 above, with the function (for large r) To get the lower bound, it is enough to observe that by Proposition 5.4 and the bound on the density ν(x) we have G(2, R) ≥ c 2 R −d−δ 1 for large R. Indeed, the rest of the proof follows the lines of the second part of the justification of the lower bound in Theorem 5.2 with profile function F (r) = r −δ 1 .
To complete the proof of the above theorem we need to prove the following lemma.
Since ν(x) is separated from zero around the origin, this together with lim inf |ξ |→∞

The Upper Bound
Similarly as before, we use our general Theorem 3.1. First we need to check that (U) holds true. When A ≡ 0, then it follows from Lemma 5.3 below that there exist c 2 > 0, c 3 ∈ (0, 1/4], r 0 , t 1 > 0 such that which implies (U) with γ = 1 and F (r) = e −θ(log(c 3 r)) β r d (log c 3 r) −(β−1) (clearly, such profile F is strictly increasing for r > r 0 with sufficiently large r 0 ≥ r 0 ). When inf |ξ |=1 ξ · Aξ > 0, then we get from Proposition 5.3 that the same is true with the profile function F (r) = e −θ(log( c 3 r)) β r d (log c 3 r) −(β−1) for some c 3 ∈ (0, c 3 ) and same γ . Thanks to (5.22) and (C), by Theorem 5.1 we also have lim λ→0 λ d/2 log N D (λ) = −ρ(λ (2) ) d/2 , where λ (2) is determined by the variational formula (5.3) with λ (2) 1 (G) corresponding to the diffusion process with exponent ψ (2) We are now in a position to derive the claimed upper bound. Indeed, with the preparation above, by our general Theorem 3.1 and Corollary 3.1, we get lim sup t→∞ log u ω (t, x) g(t) ≤ − ρ d 2/d λ (2) , Q − a.s., (5.24) with g(t) = t/(log h F,α,κ 0 (t)) 2/d , where h F,α,κ 0 is the inverse function to f F,α,κ 0 given by (3.5) with α = 2, κ 0 = ρ(λ (2) ) d/2 and F (r) = e −θ(log(c 3 r)) β r d (log c 3 r) −(β−1) (as we will see below, here the concrete value of c 3 is irrelevant Since κ 0 = ρ(λ (2) ) d/2 , in light of (5.24), this gives lim sup The Lower Bound First recall that at the beginning of the proof we verified the basic asymptotic assumption (C). In view of Proposition 5.1 it gives that the assumptions of our general Theorem 4.1 (and Corollary 4.1 as well) are satisfied with any K > 1. To match the asymptotic profile from the upper bound, it is enough to take F (r) = e −θ(log r) β . Similarly as before, we first proceed with an arbitrary fixed κ > 0, and in the concluding part of the proof we will choose a suitable κ. Condition (5.7) of Proposition 5.4 is satisfied, so that we also have that there exists c 4 > 0 such that G(1, R) ≥ c 4 e −θ(log(R/2)) β for sufficiently large R.
We now verify the assumptions of Corollary 4.1. First observe that one has Q 1 = ∞ in (4.11). Moreover, since lim sup we also have Q 2 = 1 in (4.12). By (4.14), this yields that for any fixed κ > 0 and x ∈ R d lim inf Recall that here λ (2) 1 (B(0, 1)) corresponds to the diffusion process determined by ψ (2) (ξ ) = ξ · Aξ with A as in (5.20). In light of (5.25), passing to the limit K ↓ 1 through rationals, we finally get We now justify the upper bound (5.23), which is one of the key steps in the proof of the above theorem.