The quenched asymptotics for nonlocal Schr\"odinger operators with Poissonian potentials

We study the quenched long time behaviour of the survival probability up to time $t$, $\mathbf{E}_x\big[e^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big],$ of a symmetric L\'evy process with jumps, under a sufficiently regular Poissonian random potential $V^{\omega}$ on $\mathbb{R}^d$. Such a function is a probabilistic solution to the parabolic eq. involving the nonlocal Schr\"odinger operator based on the generator of $(X_t)_{t \geq 0}$ with potential $V^{\omega}$. For a large class of processes and potentials, we determine rate functions $\eta(t)$ and positive constants $C_1, C_2$ such that \[-C_1 \leq \liminf_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big]}{\eta(t)} \leq \limsup_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big]}{\eta(t)} \leq -C_2, \] almost surely with respect to $\omega$, for every fixed $x \in \mathbb{R}^d$. The functions $\eta(t)$ 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 L\'evy-Khintchine exponents $\psi(\xi) = (|\xi|^2+m^{2/\alpha})^{\alpha/2}-m$, $\alpha \in (0,2)$, $m>0$, for which \[\lim_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s)ds}\big]}{t/(\log t)^{2/d}} = \frac{\alpha}{2} m^{1-\frac{2}{\alpha}} \, \left(\frac{\rho \omega_d}{d}\right)^{\frac{d}{2}} \, \lambda_1^{BM}(B(0,1)), \quad \mbox{for almost all $\omega$,}\] where $\lambda_1^{BM}(B(0,1))$ is the principal eigenvalue of the Brownian motion in the unit ball, $\omega_d$ is the Lebesgue measure of a unit ball and $\rho>0$ corresponds to $V^{\omega}$. We also identify two interesting regime changes ('transitions') in the growth properties of $\eta(t)$

almost surely with respect to ω, for every fixed x ∈ R d . 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 + 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 (Ω, 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·Xt = e −tψ(ξ) , ξ ∈ R d , t > 0, 0 Research supported by the National Science Centre (Poland) internship grant on the basis of the decision No. DEC-2012/04/S/ST1/00093 and by the Foundation for Polish Science. 1 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} [15,16]. 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 parabolic Anderson model based on the Laplacian (both continuous and discrete), with various potentials, has been studied with varying intensity. For an excellent review of the history of the research in this area we refer to the book of König [25].
Under suitable regularity assumptions, the solution to the problem (1.1) can be probabilistically represented by means of the Feynman-Kac formula: 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 (1.4) ψ(x) = ψ (α) (x) + o(|x| α ), |x| → 0, 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 where n is a symmetric finite measure on unit sphere S d−1 when α ∈ (0, 2), or (1.6) ψ (α) (ξ) = ξ · Aξ, 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 [10] (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 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 falls not within the scope of present paper and so we will discuss this case elsewhere.
The key observation used in the quenched case is that when the profile function W is of bounded support, then Q−a.s. there exist large areas with no potential interaction. Typically, with high probability, the process tends to remain in those 'atypical', 'favorable' areas, which affects 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 only) was first observed and rigorously established by Sznitman in [32]. He proves that in that case, for any x ∈ R d , and Q−almost all ω, 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 [11]. 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]).
In this paper, we address the quenched asymptotics for 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 drives the process out of those spots: if the process does not stay there long enough, then the effect of 'no-potential-interaction' is spoiled and as a consequence the quenched rate can be the same as the annealed rate. What is decisive here is the intensity of long jumps of the process: for processes with Lévy measures whose tails decay fast enough at infinity, we see the same phenomenon as that for Brownian motion.
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 stable-type small jumps).
The annealed rate is always governed by the exponent α appearing in (1.4), which 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.
(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 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 examples of intensity of jumps / process parameters rate η(t) lower bound for lim inf t→∞ and lim sup t→∞ for specific isotropic Lévy processes. First six examples are pure jump processes with Lévy-Khintchine exponents as in (1.2) with A ≡ 0 and ν(dx) = ν(|x|)dx, where ν(r) are subsequent profiles given in the first column. Here λ (α) 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.
processes that are of interests in mathematical physics and in technical sciences, including the relativistic α−stable process and some tempered stable processes [6,18].
(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 the two interesting regime changes ('transitions') in the growth properties of the quenched rates occur. 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 consistent with the annealed rate and becomes heavily dependent on the decay of the intensity of large jumps of the process). The second transition occurs as the intensity of large jumps becomes stretched exponential or faster. In this case, the long jumps intensity-driven quenched rate takes the form t/(log t) 2/d , which is the quickest possible one, obtained also for Brownian motion. It is worth to point out that similar large jumps intensity-dependent transition in the ground state fall-off properties of the nonlocal Schrödinger operators has been recently identified in [19] 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 assumption (U) preceding Theorem 3.1. It asserts the existence of the profile function F (r) that dominates the tail P 0 (|X t | > 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 [7] while for those with polynomial and intermediate tails we use the estimates obtained recently in [21] (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 [24].
At the end of the Introduction, let's say a few words about how the general theorems Th. 3.1, Th. 4.1 are obtained. 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 [32] 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 [11] 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. 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.

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]), while (2.1) guarantees that 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 [31,15,16,1].
The generator L of the process (X t ) t≥0 is a nonlocal pseudodifferential operator uniquely determined by its Fourier transform It is a negative-definite self-adjoint operator with a core The corresponding Dirichlet form (E, D(E)) can be defined by It holds that E(f, g) = (−Lf, g), for f ∈ D(L) and g ∈ D(E).
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 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 : (2.5) and the local Kato class K X loc -of functions V such that for every ball The condition defining the Kato class can be reformulated in terms of the kernel p(t, x) restricted to small t and small x: it is shown in [13, Corollary 1.3] that (2.5) is equivalent to Sharp estimates of p(t, x) that are available in the literature (see e.g. [21,20,2,7]) often allow to find more explicit form of (2.6).
Further, let N be a Poisson point process on R d , with intensity ρ dx, ρ > 0, defined on some probability space (Ω, M, Q), and let W : where µ ω is the random counting measure on R d corresponding to the Poisson point process N . 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 [22,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 nonincreasing function of the Euclidean distance, 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 a := inf {r > 0 : W (x) = 0 for Lebesgue-almost all x ∈ B(0, r) c }.

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 [9,5]. 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 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 ℓ ω R are vaguely convergent as R → ∞ to a nonrandom measure ℓ on R + , called the integrated density of states (see e.g. [26, p. 635]). The cumulative distribution function of the measure ℓ 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 for all relevant arguments x (the range will be clear from the context). For an open set U ⊂ R d , C ∞ c (U ) stands for C ∞ -functions with compact support inside U. B(x, R) denotes the open Euclidean ball with center x ans radius R > 0. We also say that a measurable function W : R d → R + is not identically zero, if |{x ∈ R d : W (x) > 0}| > 0 (by |U | we denote the Lebesgue measure of the set U ). Important constants are denoted with upper case letters C, K, Q, possibly with subscripts. Technical constants are numbered within each proof separately as c 1 , c 2 , ....

Preliminary estimates
We start with two preliminary results. First, a lemma, proven for nonrandom potentials. Recall that the constant t 0 comes from the assumption (2.1). Lemma 3.1. Let (X t ) 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 C 1 = C 1 (X, d) such that for any open, nonempty set U ⊂ R d one has Proof. The proof goes along standard arguments. Let U, x, and t be as in the assumptions. We have and for any R > 0 Further, Collecting these estimates we obtain 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 Consequently, for any box D n as above and any λ > 0 one has By using standard min-max formulas for eigenvalues (see, e.g., [35, 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 will be 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 assumption (U) on the process X. In the condition below, the constant t 0 comes from the assumption (2.1).
The next theorem is our main result in this section.
then for every fixed x ∈ R d one has where the function h F,α,κ is defined in (3.6) with F given by (U) and g(t) 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 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 [11, (2.3)-(2.6)] with φ(r) = κ 0 r d/α and the sequence t n = 2 n , from the assumption (3.8), we get that for every ε ∈ (0, 1), Q−almost surely we can find Piecing together (3.10), (3.12), and (3.13) 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 (3.11) one has 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 (3.11) holds with R = h(t) ∨ C 3 t, and moreover R ≥ R ε (ω). Thus we may substitute in (3.14) the value 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 (3.17) follows.
We conclude that Q−almost surely Letting ε → 0 through rational numbers, we get (3.9). The proof is complete.
The next corollary will eventually enable us to obtain, for certain processes, the existence of lim t→∞ . Corollary 3.1. Let the assumptions of Theorem 3.1 above be satisfied. More specific, let (U) hold with a function F . If there exists In particular, when lim r→∞ 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

The lower bound for regularly distributed Lévy measures
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 non-diffusion case. Instead, we estimate u ω (t, x) directly. In this part (similarly as in [32,11]), we require the potential profile W to be bounded and compactly supported. As usual in problems of this kind, we first prove that Q−almost surely there exist sufficiently large regions without potential interaction, then we force the process to go to this region and then stay there for a long enough time. This behaviour will be described analytically.

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.  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, at most 2 d of them. 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

Using an elementary inequality
and the expression in the exponent is equal to (recall r = m) We also quote Lemma 3.2 from [11] (we have α = 1 in present case).

Lemma 4.2.
Suppose that the profile function W is compactly supported and bounded. Then Q−almostsurely, for sufficiently large R one has

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 To begin with, we introduce the following notation: Also, recall that λ (α) 1 (B(0, 1)) is the principal Dirichlet eigenvalue of the symmetric α-stable 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.
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 Bt,V ω (·, ·, ·) = p Bt (·, ·, ·), and consequently λ ψ 1 (B t , V ω ) = λ ψ 1 (B t ) (recall that λ ψ (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 interaction, respectively).
Let φ be the normalized, positive Dirichlet L 2 −eigenfuntion, supported in B t , corresponding to this principal eigenvalue. For sufficiently large t we have the following chain of inequalities: From the translation invariance of the process and assumption (4.5) we see that The chain of inequalities continues as To estimate the infimum of the kernel p V ω (1, 0, y) for y ∈ B t , we take J t = (−2 √ dM (t), 2 √ dM (t)) d . For y ∈ B t one has y ∈ (−M (t), M (t)) d \ (−m(t), 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: Inserting these estimates inside (4.7) and using (4.6) again, we obtain that Q−a.s., for sufficiently large t: 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), .

It follows
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, Letting ε → 0 through rationals gives the statement.
The next corollary gives a direct lower bound for lim inf log u ω (t,x) , similar to that in Corollary 3.1.
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.  (B(0, 1)). Theorem 5.1 above states that (C) and (5.1) are sufficient conditions for the validity of (3.8), 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.
We now provide some reasonable and easy-to-check sufficient conditions under which the basic asymptotic assumption (C) holds true. Proposition 5.2. Let X be a Lévy process determined by the Lévy-Khintchine exponent ψ as in (1.2) with Gaussian coefficient A = (a ij ) 1≤i,j≤d and Lévy measure ν. The following hold.
The first assertion follows from the dominated convergence theorem together with the finiteness of the second moment of ν.
To prove the second assertion, we write 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 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) = R d \{0} (1 − cos(ξ · z))ν(dz). 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).

Proposition 5.3. Let X be a Lévy process determined by the Lévy-Khintchine exponent ψ as in (1.2) with
Gaussian coefficient A = (a ij ) 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 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). ψ given by (1.2) and such that e −tψ(·) ∈ L 1 (R d ), t > 0. Suppose there exist C 5 > 0 and r 0 > 0 such that for every Borel set
Proof. Let r 0 > 0 be as in the assumptions and let R > 4r 0 be fixed. One can check by using the strong Markov property that for any 2r 0 < |y| < R/2 and any t > 0 Set ν(x, ·) := ν(· − x). Using (5.11) and the Ikeda-Watanabe formula [14, Theorem 1], we can write for such y and t 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 where we have denoted Λ = Λ(t ′ , r 0 ) := C 5 with η(t, r) defined by (5.9). The Proposition follows.
In the sequel we will make use of the following symmetrization of the exponent ψ. Denote It follows from a combination of [34,Remark 4.8] and [30,Section 3] that there exist constants C 6 , C 7 > 0, independent of the process (i.e. of A and ν), such that 1 ∧ |y| 2 r 2 ν(dy) (5.14) ( A denotes the operator norm of a square matrix A). A direct proof of this estimate with explicit constants can be found in [12,Lemma 6]. It immediately follows from the definition that H is non-increasing and that the doubling property H(r) ≤ 4H(2r), r > 0, holds. In particular, Ψ(2r) ≤ 4C −1 6 C 7 Ψ(r), for all r > 0. Also, by (5.14) we get that ν B(0, r) c ≤ C 6 −1 Ψ(1/r), r > 0.

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.
Further, let V ω be a Poissonian potential with bounded, compactly supported, nonnegative and nonidentically zero profile W . Then the following hold.
We now illustrate our Theorem 5.2 with several examples.

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 (1 ∧ (r|z|) 2 )ν ∞ (dz) = o(r δ ) as r → 0, (5.19) and consider a symmetric Lévy process with Lévy-Khinchine exponent ψ as in (1.2) with arbitrary diffusion matrix A and Lévy measure ν = ν (δ) + ν ∞ . Then the assumptions of Theorem 5.2 hold with α = δ and Typical examples of measures ν ∞ satisfying (5.19) are as follows.
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 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 Moreover, let V ω be a Poissonian potential with bounded, compactly supported, nonnegative and nonidentically zero profile W . Then, for any fixed x ∈ R d , one has 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, (3.8) 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.
Lemma 5.2. Let X be a symmetric Lévy process with characteristic exponent ψ as in (1.2), with defining parameters A ≡ 0 and ν(dx) = ν(x)dx, such that lim inf |ξ|→∞ and ν(x) ≥ C 13 for |x| ≤ C 14 , then then there exist C 15 , t 1 > 0 such that Proof. The assumption lim inf |ξ|→∞ ψ(ξ) log |ξ| > 0 immediately gives that e −tψ(·) ∈ L 1 (R d ), for sufficiently large t > 0, and also that ν(R d \ {0}) = ∞. Thus, by the Fourier inversion formula, p(t, x) exist and are bounded for all t large enough. To find an upper bound on p(t, x), we use [20, Theorem 1]. Its assumption (1) follows directly from the upper bound on the density ν(x) with the profile function f (r) = r −d−δ and (2) can be directly derived from the monotonicity and the doubling property of such f . Indeed, for every s, r > 0 we may write Since f is nonincreasing, both integrals I 1 and I 2 can be easily estimated by f (s/2) |y|>r ν(y)dy, which is smaller or equal to c 1 f (s)Ψ(1/r), for all s, r > 0. Thus the assumption (2) holds true. It remains to justify the last assumption (3). First note that by the upper estimate of the density ν(x) and Proposition 5.2 (i) one has ψ(ξ) = ξ · Aξ + o(|ξ| 2 ), with A = ( a ij ) 1≤i,j≤d , where a ij = 1 2 R d y i y j ν(y)dy.
Proof. First of all we observe that indeed by Proposition 5.2 (i) one has i.e. the basic asymptotic assumption (C) holds true. By [21, Lemma 5 (a)], we have Ψ ν (x) ≍ ψ ν (|x|), x ∈ R d , where Ψ ν is the symmetrization of ψ ν introduced in in (5.13). From (5.14) we thus have In particular, (5.1) holds true. We now address the upper bound and the lower bound separately.
Thus, by direct asymptotic calculations, we obtain that and consequently, Since κ 0 = ρ(λ (2) ) d/2 , in light of (5.24), this gives 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 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 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. Then there exist C 16 > 0, r 0 > 0 and t 1 > 0 such that p(t, x) ≤ C 16 te −θ log |x| 4 β , |x| > r 0 ∨ t, t ≥ t 1 .
We now pass to the case when the decay of the Lévy density is stretched exponential, exponential, or superexponential.
Theorem 5.5. Let X be a symmetric Lévy process with characteristic exponent ψ as in (1.2) with the Gaussian coefficient A = (a ij ) 1≤i,j≤d such that either A ≡ 0 or inf |ξ|=1 ξ ·Aξ > 0, and a symmetric Lévy measure ν(dx) = ν(x)dx such that there exist θ > 0, β ∈ (0, ∞), γ ≥ 0 and δ ∈ (0, 2) such that either 1 (U ) and λ (2) correspond to the diffusion process with Gaussian matrix A as in (5.20). In particular, if A = a Id for some a ≥ 0 and ν is radial nonincreasing, then Proof. We proceed along the same scheme as in the proof of Theorem 5.4 above. By Proposition 5.2 the basic assumption (C) is satisfied (with A as in (5.20)).