From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion in the Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion in traps distributed according to a random perturbation of the lattice.


Introduction and main results
In this article, we consider a diffusing particle moving in random traps. The motion of the particle is given by a simple random walk or a Brownian motion and it is killed at a certain rate when it stays in a trap. Such a model appears in various models in chemical physics and also has some relations to the quantum physics in disordered media. We refer to the papers by Havlin and Ben-Avraham [9] and den Hollander and Weiss [4] for reviews on this model.
The mathematical discription of the trapping model is given by the sub-Markov process with generator where ∆ is the Laplacian on L 2 (R d ) or l 2 (Z d ) and (V ω , P) a nonnegative, stationary, and ergodic random field. Heuristically, the height of V ω corresponds to the rate of killing. Let us write ({X t } t≥0 , {P x } x∈R d or Z d ) for the Markov process generated by −κ∆. One of the quantity of primary interest concerning this process is the survival probability of the particle up to a fixed time t, which is expressed as From this expression, we can identify the survival probability as the Feynman-Kac representation of a solution of the initial value problem Therefore, it is natural to expect that the long time asymptotics of the survival probability gives some information about the spectrum of H ω around the ground state energy and vice versa. This idea has been made rigorous first by Fukushima [6], Nakao [13], and Pastur [14] (with the analysis of some concrete examples) in the following sense: from the annealed long time asymptotics of the survival probability, one can derive the decay rate of the integrated density of states around the ground state energy. Their arguments are based on the fact that the Laplace transform of the integrated density of states can be expressed as the annealed survival probability for the process conditioned to come back to the starting point at time t. Therefore, the above implication follows by an appropriate Tauberian theorem and, since there is the corresponding Abelian theorem (see e.g. Kasahara [10]), the converse is also true. The aim of this article is to study a relation between the quenched asymptotics of u ω (t, x) and the integrated density of states. Let us start by recalling the notion of the integrated density of states. To define it, we assume the following: Assumption 1. In the continuous setting, V ω belongs to the local Kato class K d,loc . (See [3] or [19] for the definition of K d,loc .)@ This assumption is sufficient to ensure that H ω is measurable in ω as an operator. For the notion of measurability of operators, we refer to a lecture notes by Kirsch [11]. (In fact, this is slightly stronger but we need this to utilize a uniform bound for the semigroup e −tHω in the proof.) Under the above assumption, the integrated density of states of H ω is defined as follows: is the k-th smallest eigenvalue of H ω in (−R, R) d with the Dirichlet (resp. Neumann) boundary condition. The existence of the limit in the right hand side can be proved by superadditivity (resp. subadditivity). Now we state our first result.
Theorem 1. Suppose that Assumptions 1 holds and that there exists a regularly varying function φ with index L > 0 such that the integrated density of states N D associated with the operator H ω in (1) admits the upper bound Then, for any fixed x ∈ R d , where ψ is the asymptotic inverse of φ.
The following assumptions are necessary only for the lower bound.
Assumption 3. (Short range correlation) There exists β > 0 and r 0 > 0 such that for λ > 0 and boxes where Now we are ready to state our second result.
Theorem 2. Suppose that Assumptions 1-3 hold and that there exists a regularly varying function φ with index L > 0 such that the integrated density of states N D associated with the operator H ω in (1) admits the lower bound Then, there exists a constant c 1 > 1 such that for any fixed x ∈ R d , where ψ is the asymptotic inverse of φ.
Remark 1. The exponential behavior (5) and (9) of the integrated density of states is called the "Lifshitz tail effect" (cf. [12]) and is typical for the trapping Hamiltonian H ω . The index L is called "Lifshitz exponent". Using these terminologies, we can summarize our results as follows: if we have the Lifshitz tail effect with exponent L > 0, then log u ω (t, x) behaves like −t/(log t) 1/L+o (1) .
Finally we briefly comment on the relation to early studies on the quenched asymptotics of u ω (t, x). We first give historical remarks. The first result in this direction has been obtained for the Brownian motion in the Poissonian traps by Sznitman [18] (see also [19]): with an explicit constant c > 0. The same asymptotics has also been proved for the discrete counterpart (the simple random walk in Bernoulli traps) by Antal [1]. These results are consistent to ours since in these cases, the Lifshitz exponent is known to be d/2 [13,16]. Later, Biskup and König [2] considered the simple random walk in i.i.d. traps with more general distributions. A representative example in their framework is for some γ ∈ (0, ∞). For such a model, they proved the quenched asymptotics with a constantχ > 0 described by a certain variational problem and a function r(t) = (log t) 2/(d+2γ)+o(1) (t → ∞) which is determined by a certain scaling assumption. It is remarkable that they also discussed the annealed asymptotics and as a consequence, the Lifshitz tail effect with the Lifshitz exponent (d + 2γ)/2 was proved. Hence the relation we mentioned in Remark 1 has already appeared in this special class. Next, we comment on some technical points. The lower bound (Theorem 2) is a slight modification of that of Theorem 4.5.1 in p.196 of [19] and not genuinely new. We include it for the completeness and to use in an application given in Section 4.2. On the other hand, the upper bound (Theorem 1) contains some novelties. Besides the generality of the statement, our proof simplifies the existing arguments. To be more precise, our proof contains no localizing argument which all the proofs of above results rely on, see e.g. Lemma 4.6 in [2]. We will see in Section 4.1 that our result indeed gives a simple proof of the quenched asymptotics for the Brownian motion in the Poissonian obstacles.

Proof of the upper bound
We take κ = 1/2 and x = 0 in the proof. The extension to general κ and x are verbatim. Also, we give the proof only for the continuous setting. The proof of the discrete case follows by the same argument. We begin with the following general upper bounds for u ω (t, x) in terms of the principal eigenvalue.
Proof. Let τ denote the exit time of the process from (−t, t) d . Then, by the reflection principle, we have Now, (14) follows immediately from (3.1.9) in p.93 of [19] under Assumption 1.
Due to this lemma, it suffices for (6) to obtain the almost sure lower bound for the principal eigenvalue λ D ω, 1 (−t, t) d . We use the following inequality for the integrated density of states which holds for any λ > 0 and R > 0. The first inequality is an easy application of the so-called "Dirichlet-Neumann bracketing" and can be found in [3], (VI.15) in p.311. Now, fix ǫ > 0 arbitrarily and let λ = (1 − ǫ)ψ(d log t) −1 and R = t. Then it follows from (16) and (5) that for some δ(ǫ) > 0 when t is sufficiently large. This right-hand side is summable along the sequence t k = e k and therefore Borel-Cantelli's lemma shows except finitely many k, P-almost surely. We can extend this bound for all large t as follows: since ψ(d log t) is slowly varying in t, we have for t k−1 ≤ t ≤ t k when k is sufficiently large. Combined with Lemma 1, this proves the upper bound (6).

Proof of the Lower bound
We take κ = 1/2 and x = 0 again. Also, we only consider the continuous case. As in the proof of the upper bound, the principal eigenvalue plays a key role. Let us write λ N k (U) for the k-th smallest eigenvalue of −1/2∆ in U with the Neumann boundary condition. Then we have another inequality for the integrated density of states which holds for any λ ∈ (0, 1) and R > 0. The first inequality can be found in [3] again, (VI.16) in p. 331, and the third one is a consequence of the classical Weyl asymptotics for the free Laplacian, see e.g. Proposition 2 in Section XIII.15 of [15]. For arbitrary ǫ > 0, let λ = (1 + ǫ)ψ(d log t) −1 . Then, using (20) and (9), we find for some δ(ǫ) > 0 when t is sufficiently large. Now we introduce some notations to proceed the proof. Let us fix a positive number and define Note that min i =j d (C i , C j ) > diam(C i ) and both of them go to infinity as t → ∞. Therefore, by using (21) and Assumption 3 recursively, we obtain for sufficiently large t. Since the right hand side is summable in t ∈ N, Borel-Cantelli's lemma tells us that P-almost surely, for all large t ∈ N. The next lemma translates (26) to an upper bound for the Dirichlet eigenvalue: There exists a constant c 1 > 1 such that P-almost surely, for all large t.
Proof. We choose C i (i ∈ I) for which λ N ω, 1 (C i ) ≤ (1 + ǫ)ψ(d log t) −1 . This is possible for large t ∈ N by (26) and then it also holds for all large t with slightly larger ǫ by regularly varying property of ψ. Let φ N i denote the L 2 -normalized nonnegative eigenfunction corresponding to λ N ω, 1 (C i ) and ∂ ǫ C i (i ∈ I) the set We further take a nonnegative function ρ ∈ C 1 c (C i ) which satisfies Such a function can easily be constructed by a standard argument using mollifier. Substituting ρφ N i to the variational formula for the principal eigenvalue, we obtain To bound the right hand side, we first use the uniform bound on eigenfunctions φ N i ∞ ≤ c 5 λ N ω, 1 (C i ) d/4 (see e.g. (3.1.55) in p.107 of [19]) to see Next, it is clear from (29) and the above uniform bound that Taking ǫ = (2c 6 ) −1 and plugging these bounds into (30), the result follows.
We also need the following almost sure upper bound.
Lemma 3. Under Assumption 2, we have P-almost surely, for sufficiently large t.
Proof. By Chebyshev's inequality, Since the last expression is summable in t ∈ N, the claim follows by Borel-Cantelli's lemma and monotonicity of sup x∈ Now, we can finish the proof of the lower bound. We pick ω for which Lemma 2 and Lemma 3 holds. Then we can find a box C i (i ∈ I) satisfying for sufficiently large t. Let φ D i denote L 2 -normalized nonnegative eigenfunction associated with λ D ω, 1 (C i ). It is easy to see that there exists a box q We also know the following uniform upper bound: from (3.1.55) in [19]. Let us recall that the semigroup generated by H ω has the kernel p ω (s, x, y) under Assumption 1 (see Theorem B.7.1 in [17]). We can bound this kernel from below by using the Dirichlet heat kernel p (−t,t) d (s, x, y) in (−t, t) d as follows: where the second inequality follows by Lemma 3 and a Gaussian lower bound for the Dirichlet heat kernel in [20]. Taking s = t/(log t) M and noting that |q| < 2s, we arrive at inf for sufficiently large t. Plugging (35)-(39) into an obvious inequality, we arrive at where in the third line, we have replaced p ω by the kernel of the semigroup generated by H ω with the Dirichlet boundary condition outside C i . This completes the proof of the lower bound of Theorem 2 since s = t/(log t) M was chosen to be o(t/ψ(log t)).

Examples
We apply our results to two models in this section. The first is the Brownian motion in the Poissonian obstacles, where we see that our result recovers the correct upper bound. The second is the Brownian motion in a perturbed lattice traps introduced in [7], for which the quenched result is new.

Poissonian obstacles
Let us consider the standard Brownian motion (κ = 1/2) killed by the random potential of the form where (ω = i δ ω i , P ν ) is a Poisson point process with intensity ν > 0 and W is a nonnegative, bounded, and compactly supported function. As is mentioned in Section 1, Sznitman proved in [18] the quenched asymptotics for this model: where c(d, ν) = λ d (νω d /d) 2/d with λ d denoting the principal Dirichlet eigenvalue of −1/2∆ in B(0, 1) and ω d = |B(0, 1)|. We can recover the upper bound by using classical Donsker-Varadhan's result [5] and Theorem 1. Indeed, the above potential clearly satisfies Assumption 1 and the asymptotics of the integrated density of states has been derived by Nakao [13] by applying an exponential Tauberian theorem to Donsker-Varadhan's asymptotics Now an easy computation shows that the asymptotic inverse of the right hand side of (43) is and then Theorem 1 proves the upper bound in (42).

Remark 2.
In this case, the lower bound given by Theorem 2 is not sharp as is obvious from the statement. (In the proof, we lose the precision in Lemma 2.) However, the lower bound can be complemented by a rather direct and simple argument in the Poissonian soft obstacles case, see [18]. So our argument simplifies the harder part.

Perturbed lattice traps
In this subsection, we use our results to derive the quenched asymptotics for the model introduced in [7]. We consider the standard Brownian motion (κ = 1/2) killed by the potential of the form where ({ω q } q∈Z d , P θ ) (θ > 0) is a collection of independent and identically distributed random vectors with density and W is a nonnegative, bounded, and compactly supported function. The author has derived the annealed asymptotics for this model in [7] and also proved the following Lifshitz tail effect as a corollary: We can prove the quenched asymptotics from this result.
Theorem 3. For any θ > 0 and x ∈ R d , we have with P θ -probability one.
Proof. The Assumption 1 is clearly satisfied since V ω is locally bounded almost surely. Hence the upper bound readily follows by computing the asymptotic inverse of (49) and using Theorem 1. To use Theorem 2, we have to verify Assumptions 2 and 3. The former is rather easy and can be found in Lemma 11 in [8]. The latter is verified as follows: we first fix r 0 > 0 sufficiently large so that supp W ⊂ B(0, r 0 /4). For r > r 0 and boxes {A k } 1≤k≤n as in Assumption 3, let us define events = for all q ∈ Z d with d (q, A 1 ) ≥ r/2, d (q + ω q , A 1 ) ≥ r/4 .
Then, λ N ω, 1 (A 1 ) and {λ N ω, 1 (A k )} 2≤k≤n are mutually independent on E 1 ∩ E 2 thanks to our choice of r 0 . Therefore, the left hand side of (8) is bounded by P θ (E c 1 ) + P θ (E c 2 ). Let us denote the s-neighborhood of A 1 by N s (A 1 ). The first term is estimated as P θ (E c 1 ) ≤ P θ |ω q | ≥ r/4 for some q ∈ Z d ∩ N r/2 (A 1 ) for large r, where we have used diam(A 1 ) < r in the last line. Next, we bound the second term P θ (E c 2 ). Using the distribution of ω q , we have P θ (E c 2 ) = P θ q + ω q ∈ N r/4 (A 1 ) for some q ∈ Z d \ N r/2 (A 1 ) We can assume by shift invariance that A 1 is centered at the origin. We divide the sum into two parts {|q| ≤ r} and {|q| > r}. The former part of the sum is bounded by exp − d (q, N r/4 (A 1 )) θ ≤ c 10 r d exp −(r/4) θ . (54) For the latter part, we use the fact that N r/4 (A 1 ) ⊂ B(0, 3r/4), which follows from the assumption diam(A 1 ) < r. By using this fact, we find d (q, N r/4 (A 1 )) ≥ |q| − 3r/4 > |q|/4 for |q| > r (55) and therefore q∈Z d \N r/2 (A 1 ), |q|>r exp − d (q, N r/4 (A 1 )) θ ≤ q∈Z d , |q|>r It is not difficult to see that this right hand side is bounded by exp{−(r/8) θ } for sufficiently large r. Combining all the estimates, we arrive at P θ (E c 1 ) + P θ (E c 2 ) ≤ N(d, θ)r d 2 + c 10 r d exp −(r/8) θ (57) for large r, which verifies Assumption 3.