#### Abstract

In (Chen and Kulik, 2009), a method of renormalization was proposed for constructing some more physically realistic random potentials in a Poisson cloud. This paper is devoted to the detailed analysis of the asymptotic behavior of the annealed negative exponential moments for the Brownian motion in a renormalized Poisson potential. The main results of the paper are applied to studying the Lifshitz tails asymptotics of the integrated density of states for random Schrödinger operators with their potential terms represented by renormalized Poisson potentials.

#### 1. Introduction

This paper is motivated by the model of Brownian motion in Poisson potential, which describes how a Brownian particle survives from being trapped by the Poisson obstacles. We recall briefly the general setup of that model, referring the reader to the book by Sznitman [1] for a systematic representation, to [2] for a survey, and to [3–6] for specific topics and for recent development on this subject.

Let be a Poisson field in with intensity measure , and let be an independent Brownian motion in . Throughout, and denote the probability law and the expectation, respectively, generated by the Poisson field , while and denote the probability law and the expectation, respectively, generated by the Brownian motion with . For a properly chosen (say, continuous and compactly supported) nonnegative function on (known as a shape function), define the respective random function (known as a Poisson potential) which heuristically represents the net force at generated by the Poisson obstacles. The model of Brownian motion in a Poisson potential is defined in two different settings. In the quenched setting, the setup is conditioned on the random environment created by the Poisson obstacles, and the model is described in the terms of the Gibbs measure defined by Here, is a positive parameter, responsible for the time scaling , introduced here for further references convenience. In the annealed setting, the model averages on both the Brownian motion and the environment, and respective Gibbs measure is defined by Heuristically, the integral measures the total net attraction to which the Brownian particle is subject up to the time , and henceforth, under the law or , the Brownian paths heavily impacted by the Poisson obstacles are penalized and become less likely.

In the Sznitman’s model of “soft obstacles,” the shape function is assumed to be locally bounded and compactly supported. However, these limitations may appear to be too restrictive in certain cases. Important particular choice of a shape function, physically motivated by the Newton’s law of universal attraction, is which clearly is both locally unbounded and supported by whole . This discrepancy is not just a formal one and brings serious problems. For instance, under the choice (1.5), the integral (1.1) blows up at every when .

To resolve such a discrepancy, in a recent paper [7], it was proposed to consider, apart with a Poisson potential (1.1), a renormalized Poisson potential Assume for a while that is locally bounded and compactly supported. Then,

that is, const. Consequently, replacing by in (1.2) and (1.3) does not change the measures and , because both the exponents therein and the normalizers and are multiplied by the same constant (this is where the word “renormalization” comes from). On the other hand, for unbounded and not locally supported , the renormalized potential (1.6) may be well defined, while the potential (1.1) blows up. The most important example here is the shape function (1.5) under the assumption . In that case, is well defined as well as the Gibbs measures see [7, Corollary 1.3]. We use separate notation , because the Gibbs measures (1.2) and (1.3) are not well defined now.

The above exposition shows that using the notion of the renormalized Poisson potential, one can extend the class of the shape functions significantly. Note that in general, the domain of definition for (1.6) does not include the one for (1.1). For instance, for the shape function (1.5), the potential , and the renormalized potential are well defined under the mutually excluding assumptions and , respectively. This, in particular, does not give one a possibility to define respective Gibbs measures in a uniform way. This inconvenience is resolved in the terms of the Poisson potential , partially renormalized at the level ; see [7, Chapter 6]. By definition, where is a renormalization level. Clearly, . It is known (see [7, Chapter 6]) that is well defined for every as soon as is well defined for some , and in that case, there exists a constant such that . This makes it possible to define the respective Gibbs measures in a uniform way, replacing in (1.2), (1.3) by with (any) . In addition, such a definition extends the class of shape functions: for given by (1.5), with is well defined for .

The main objective of this paper is to study the asymptotic behavior, as , of the annealed exponential moments This problem is clearly relevant with the model discussed above: in the particular case , , this is just the natural question about the limit behavior of the normalizer in the formula (1.3) for the annealed Gibbs measure. In the quenched setting, similar problem was studied in the recent paper ([8]). In some cases, we also consider (1.11) with a renormalized Poisson potential replaced by either a Poisson potential or a partially renormalized potential with .

The function in (1.11) appears, on one hand, because of our further intent to study in further publications the a.s. behavior On the other hand, this function can be naturally included into the initial model. One can think about making penalty (1.4) to be additionally dependent on the length of the time interval by dividing the total net attraction for the Brownian particle by some scaling parameter. Because of this interpretation, further on, we call the function a “scale”.

Let us discuss two other mathematically related problems, studied extensively both in mathematical and in physical literature. The first one is known as the continuous parabolic Anderson model This problem appears in the context of chemical kinetics and population dynamics. Its name goes back to the work by Anderson [9] on entrapment of electrons in crystals with impurities. In the existing literature, the random field is usually chosen as the Poisson potential , with the shape function assumed to be bounded (and often locally supported), so that the potential function (1.1) can be defined. A localized shape is analogous to the usual setup in the discrete parabolic Anderson model, where the potential is an i.i.d. sequence; we refer the reader to the monograph [10] by Carmona and Molchanov for the overview and background of this subject.

On the other hand, there are practical needs for considering the shape functions of the type (1.5), which means that the environment has both a long range dependency and extreme force surges at the locations of the Poison obstacles. To that end, we consider (1.13) with a renormalized Poisson potential instead of . Note that in that case, the field represents fluctuations of the environment along its “mean field value” rather than the environment itself although this “mean field value” may be infinite.

It is well known that (1.13) is solved by the following Feynman-Kac representation when is Hölder continuous and satisfies proper growth bounds. When with from (1.5), local unboundedness of induces local irregularity of (Proposition 2.9 in [7]), which does not allow one to expect that the function (1.14) solves (1.13) in the strong sense. However, it is known (Proposition 1.2 and Proposition 1.6 in [7]) that under appropriate conditions, the function (1.14) solves (1.13) in the mild sense. It is a local unboundedness of again, that brings a serious asymmetry to the model, making essentially different the cases “+” and “−” of the sign in the right hand sides of (1.13) and (1.14). For the sign “−”, the random field (1.14) is well defined and integrable for (Theorem 1.1 in [7]). For the sign “+”, the random field (1.14) is not integrable for any . On the other hand, the random field (1.14) is well defined for (Theorem 1.4 and Theorem 1.5 in [7]).

In view of (1.14), our main problem relates immediately to the asymptotic behavior of the moments of the solution to the parabolic Anderson problem (1.13) with the sign “−”. Here, we cite [10–20] as a partial list of the publications that deal with various asymptotic topics related to the parabolic Anderson model.

Another problem related to our main one is the so called *Lifshitz tails* asymptotic behavior of the *integrated density of states* function of a random Schrödinger operator of the type
This function, written IDS in the sequel, is a deterministic spectral mean-field characteristic of . Under quite general assumptions on the random potential , it is well defined as
where is the set of eigenvalues for the operator in a cube with the Dirichlet boundary conditions, denotes the Lebesgue measure of in , and the limit pass is made w.r.t. a sequence of cubes which has same center and extends to the whole . The classic references for the definition of the IDS function are [21, 22]; see also a brief exposition in Sections 2 and 5.1 below.

Heuristically, the bottom (i.e., the left-hand side) of the spectrum of mainly describes the low-temperature dynamics for a system defined by the Hamiltonian (1.15). This motivates the problem of asymptotic behavior of , , studied extensively in the literature. The name of the problem goes back to the papers by Lifshitz [23, 24]; we also give [1, 21, 22, 25–44] as a partial list of references on the subject.

Connection between the Lifshitz tails asymptotics for the IDS function , and the problem discussed above is provided by the representation for the Laplace transform of Here, denotes the distribution of the Brownian bridge, that is, the Brownian motion conditioned by . Our estimates for (1.11) appear to be process insensitive to some extent and remain true with in (1.11) replaced by . This, via appropriate Tauberian theorem, provides information on Lifshitz tail asymptotics for the respective IDS function . Note that in this case, the asymptotic behavior of the as should be studied, because the bottom of the spectrum is equal , unlike the (usual) Poisson case, where . This difference is caused by the renormalization procedure, which brings the negative part to the potential.

We now outline the rest of the paper. The main results about negative exponential moments for annealed Brownian motion in a renormalized Poisson potential are collected in Theorem 2.1. They are formulated for the shape function defined by (1.5). Depending on in this definition, we separate three cases calling them a “light-scale,” a “heavy-scale,” and a “critical” case, respectively. There is a close analogy between our “light” versus “heavy” scale classification for a renormalized Poisson potential and the well-known “classic” versus “quantum” regime classification for a (usual) Poisson potential; see detailed discussion in Section 2.

In all three cases listed above, our approach relies on the identity with see Proposition 2.7 and Proposition 3.1 in [7].

Further analysis of the Wiener integral in the r.h.s. of (1.21) in the light-scale case is quite straightforward. First, the upper bound follows from Jensen’s inequality and is “universal” in the sense that the Brownian motion therein can be replaced by an arbitrary process. Then, we choose a ball in the Wiener space, which simultaneously is “sufficiently heavy” in probability and “sufficiently small” in size. This smallness allows one to transform the integral in the r.h.s. of (1.21) into which after a straightforward transformation gives a lower bound that coincides with the universal upper bound obtained before.

We call this approach the “small heavy ball method”. It is quite flexible, and by means of this method, we also give a complete description of the light-scale asymptotic behavior for a Poisson potential and a partially renormalized Poisson potential (Theorem 2.4). This method differs from the functional methods, typical in the field, which go back to the paper [41] by Pastur. It gives a new and transparent principle explaining the transition from quantum to classical regime; note that the phenomenology of such a transition is a problem discussed in the literature intensively; see [32, Section 3.5] for a detailed overview. In the context of the small heavy ball method, we can identify the classic regime with the situation where a sufficient amount of Brownian paths stay in a suitable neighborhood. So, the relation donimates in this regime.

In the quantum regime, that is, in the critical and the heavy-scale cases, the contribution of Brownian paths cannot be neglected. In this situation, the key role in our analysis of the Wiener integral in the r.h.s. of (1.21) is played by a large deviations result (Theorem 4.1) formulated and proved in Section 4. In the same section, by means of appropriate rescaling procedure, the asymptotics of the Wiener integral in the r.h.s. of (1.21) in the quantum regime is obtained. In the heavy-scale case, this asymptotics appears to be closely related to the large deviations asymptotics for a Brownian motion in a Wiener sheet potential, studied in ([45]); we discuss this relation in Section 4.4.

Finally, we discuss an application of the main results of the paper to the Lifshitz tails asymptotics of the integrated density of states functions for random Schrödinger operators, with their potential terms represented by either renormalized Poisson potential or partially renormalized Poisson potential.

#### 2. Main Results

Throughout the paper, denotes the volume of the -dimensional unit ball. We denote where is used for the Sobolev space of functions that belong to together with their first order derivatives. We also denote ( is introduced in (1.22)). Clearly, the functions , , and are convex; this simple observation is crucial for the most constructions below.

Our main results about the asymptotics of negative exponential moments for annealed Brownian motion in a renormalized Poisson potential are represented by the following theorem.

Theorem 2.1. *Let . *(i)*In the “light-scale” case,
*(ii)*In the “critical” case,
*(iii)*In the “heavy-scale” case, under additional assumption ,
*

*Remark 2.2. *The additional assumption in Statement (iii) is exactly the condition for to be square integrable (see [45]), and henceforth, for respective central limit theorem to hold true, see Proposition 4.4 and discussion in Section 4.4 below.

Let us discuss this theorem in comparison with the following, well-known in the field, results for annealed Brownian motion in a Poisson potential.

Theorem 2.3. *Let be bounded and satisfy
**
with . *(i)*(see [41]) If ,
*(ii)*(see [40]) If ,
*(iii)*(see [46]) If ,
*

It is an effect, discovered by Pastur in [41], that the asymptotic behavior of the Brownian motion in a Poisson potential is essentially different in the cases and , called frequently “light tailed” and “heavy tailed,” respectively. This difference was discussed intensively in the literature, especially in the connection with the asymptotic behavior of respective IDS function. The main asymptotic term in (2.7) is completely determined by the potential and does not involve , that is, the “intensity” of the Brownian motion. On the other hand, (2.9) depends on but not on the shape function . Since and , heuristically, are related to “regular” and “chaotic” parts of the dynamics, an alternative terminology “classic regime” () and “quantum regime” () is frequently used.

Theorem 2.1 shows that the dichotomy “classic versus quantum regimes” is still in force for the model with a renormalized Poisson potential, with conditions on the shape function to be either heavy or light tailed replaced by conditions on the scale to be, respectively, light or heavy. Note that for , (1.18) and (1.19) transform exactly to and , respectively. In the classic regime, an analogy between a Poisson potential and a renormalized Poisson potential is very close: for , (2.3) and (2.7) coincide completely. However, in the quantum regime, the right hand side in (2.5), although being principally different from (2.3), is both scale dependent (i.e., involves ) and shape dependent (i.e., involves ).

It is a natural question whether Theorem 2.1 can be extended to other types of potentials, like a Poisson potential or a partially renormalized Poisson potential . We strongly believe that such an extension is possible in a whole generality; however, we cannot give such an extension in the quantum regime (i.e., critical and heavy-scaled cases) so far, because we do not have an analogue of Theorem 4.1 for functions which are convex but are not increasing (like and ). Such a generalization is a subject for further research.

In the classic regime (i.e., light scale case), such an extension can be made efficiently. Moreover, in this case, the assumptions on the shape function can be made very mild: instead of (1.5), we assume (2.6) with and, when , which is just the assumption for to be well defined.

Theorem 2.4. *Let the shape function satisfy (2.6) and scale function satisfy (1.18). *(i)*Statement (i) of Theorem 2.1 holds true assuming satisfies (2.10).*(ii)*For ,
*(iii)*For and ,
where is the Euler constant. In particular, when has the form (1.5),
*

The following theorem shows that statements of Theorems 2.1 and 2.4 are process insensitive to some extent.

Theorem 2.5. *Relations (4.4), (2.3)–(2.5), (2.11), (2.12), and (2.13) hold true with replaced by , that is, the expectation w.r.t. the law of the Brownian bridge.*

This theorem makes it possible to investigate the Lifshitz tails asymptotics for the integrated density of states of the random Schrödinger operators with (partially) renormalized Poisson potentials. Let us outline the construction of respective objects.

For a given random field , and a cube , denote by the random Schrödinger operator in with the potential and the Dirichlet boundary conditions When the field is assumed to have locally bounded realizations, the operator is a.s. well defined as an operator on and is self-adjoint. In addition, respective semigroup has a Feynman-Kac representation ([1, page 13]) where For general , we define by the following limit procedure. Consider truncations . Under appropriate assumptions on , for almost every realization of this field, operators converge strongly for every as . In that case, is defined as the generator of the limit semigroup , . Assuming the spectrum of to be discrete (we verify this assumption below), we denote this spectrum and define the function

Proposition 2.6. *Let the shape function be such that for some , the following conditions hold: *(i)* is compactly supported,*(ii)* is Lipschitz continuous and belongs to the Sobolev space .**Consider either a partially renormalized potential with , or a renormalized potential , in the latter case assuming additionally (2.10). **Then, *(a)*for a.s. realization of the potential and every cube , the described above procedure well defines both the random Schrödinger operator and respective function ,*(b)*there exists an integrated density of states , that is, a deterministic monotonous function such that ** a.s. for every point of continuity of . Respective Laplace transform has the representation
*

Note that in the proof of Proposition 2.6 (Section 5.1 below), most difficulties are concerned with the statement (A) because of local irregularity of the potential (Proposition 2.9 in [7]).

As a corollary of Theorem 2.5 and representation (2.19), we deduce the following Lifshitz tails asymptotics for random Schrödinger operators with random potentials and .

Theorem 2.7. *Let satisfy (2.6). *(i)*For , assuming additionally (2.10), one has in limit *(ii)*For and , one has in limit*

Theorem 2.7 involves the asymptotic results for exponential moments (Theorem 2.5) only in a partial form, for the trivial scale function . This observation naturally motivates the following extension of the definition of the IDS function and respective generalization of Theorem 2.7.

Consider the family of random Schrödinger operators Assuming every potential being such that respective IDS function is well defined, denote . We call the family

the *integrated density of states field* of the family of random Schrödinger operators (2.22). In the Theorem 2.8 below, we describe the asymptotic behavior of this field for random Schrödinger operators with a renormalized Poisson potential. Let us anticipate this theorem by a brief discussion.

Three statements of Theorem 2.8 below relate directly to our light-scale, heavy-scale, and critical cases, respectively. This means that the integrated density of states field for random Schrödinger operators with a renormalized Poisson potential may demonstrate asymptotic behavior typical either to the classic or to the quantum regime, while for the integrated density of states function, only, the classic regime is available.

Next, observe that . Hence, conditions, that and is bounded, yield . Therefore, the quantum regime for the integrated density of states field requires that and tend to 0 in an adjusted way (Statement (ii) of Theorem 2.8 below). On the contrary, conditions of the Statement (i) of the same theorem allow (in that case may tend to ), , or to stay bounded away both from 0 and (in these two cases necessarily). This is the reason that two conditions and are imposed in this case: when , the first one includes the second one, but when , the inclusion is opposite.

Theorem 2.8. *Let be of the form (1.5) with . *(i)*When and ,
*(ii)*When and , under additional assumption ,
where C _{2} denotes the constant in the r.h.s of (2.5).*(iii)

*When and is bounded away both from 0 and from , where denotes the constant in the right hand side of (2.4) with .*

Note that under the assumptions of Theorem 2.8, the right hand sides of (2.24), (2.25), and (2.26) tend to . So, Theorem 2.8 controls the exponential decay of the IDS field, similarly to Theorem 2.7. What may look nontypical in this theorem when compared with other references in the field is that some part of the statements are formulated when . This in general reflects the fact that for the negative part of the spectrum becomes negligible. Theorem 2.8, in particular, quantifies such a negligibility.

#### 3. Classic Regime

In this section, we prove Theorem 2.4, which includes Statement (i) of Theorem 2.1 as a partial case. For a given , denote

Similarly to (1.21), we have The first relation is provided by Proposition 2.7 and Proposition 3.1 in [7], the proof for the second one is completely analogous and is omitted.

In what follows, we analyse the Wiener integrals in the r.h. sides of (1.21) and (3.2). However, (3.3) appears not to be well designed for an immediate analysis, which motivates the following auxiliary construction. Instead of , we consider a partially renormalized Poisson potential with the properly chosen renormalization level, dependent on . Let and . Then, assuming , (2.6) and (1.18), we will prove that Note that by Proposition 6.1 in [7], for any such that , is well defined. Henceforth, changing a renormalization level just multiplies respective exponential moment by an explicit constant. Therefore, (2.12) is provided by (3.4).

In Sections 3.1 and 3.2, we prove, respectively, upper and lower bounds in (2.3), (2.11), and (3.4) with the constants represented in an integral form. Calculation of the integrals is postponed to Section 3.3.

##### 3.1. Proof of the Upper Bound

For any convex function , by the Jensen inequality, we have Denote , . By the inequality above, one has the following estimate with nonrandom right hand side: Assumption (1.18) yields . Therefore, in order to prove the upper bound either in (2.3) or in (2.11), it is sufficient to apply (3.7) to either or and then prove, respectively, . By assumption (2.6), for every , there exists such that

When , this easily provides Since is bounded on , (3.10) provides the second relation in (3.8).

When , similar argument leads to the relation analogous to (3.10) with replaced by . Consequently, with condition (2.10) in mind, it remains to prove that To that end, we choose , such that , and write for large enough Recall that , is dominated by , and is locally integrable under condition (2.10). Then, the first term in the above sum is negligible when . This proves (3.10) and completes the proof.

Similarly, for from the Jensen’s inequality for the convex function , we have

and consequently, for Similarly to (3.8), one can prove which provides the upper bound in (3.4).

##### 3.2. Proof of the Lower Bound

For a fixed , take fixed but large enough so that Take and consider the set keeping the notation . By the scaling property and the well-known small balls probability asymptotics for the Brownian motion, we have, for large enough, with some constant . Therefore, condition yields Take . On the set , one has Then, for large enough to provide , we have Therefore, a two-sided estimate is valid on the set for every with . Observe that (3.22) is a pointwise estimate for a Brownian trajectory from a “small ball” and for a point outside a “large ball” . On the other hand, (3.19) shows the “small Brownian ball” is “heavy” in the sense that its probability is sufficiently large, in respective logarithmic scale. These observations provide a straightforward tool for proving lower bounds in (2.3)–(3.4).

Since is nonnegative and nondecreasing, (3.22) yields on with Together with (1.21) and (3.19), this inequality provides for every , , . Since this completes the proof of the lower bound in (2.3).

Since is nonincreasing and satisfies , (3.22) yields on with

Since this provides the lower bound in (3.5).

Finally, is nondecreasing in first coordinate and nonincreasing in second coordinate. In addition, , and hence (3.22) yields in the case on with Together with (3.2) and (3.19), this inequality provides for every , , . Since

this completes the proof of the lower bound in (3.4).

##### 3.3. Calculation of the Integrals

In the above proof, we have obtained (2.3), (2.11), and (3.4) with the constants represented as certain integrals. Explicit calculation of these integrals can be made in easy and standard way, using sphere substitution and integration by parts. For such a calculation of the integral (2.3), we refer to Lemma 7.1 in [8]; calculation of the integral (2.11) is completely analogous and omitted. Here, we calculate the integral in (3.4) and prove (2.13).

By sphere substitution, and change of variables,
in the last identity we have used an elementary relation
Integration by parts and **n.** 538 in [47] gives
which completes calculation of the integral in (3.4).

Finally, let has the form (1.5). Take , then for large enough, and Combined with (2.12), this calculation provides (2.13).

#### 4. Quantum Regime

##### 4.1. Large Deviations

Our analysis of the asymptotic behavior of the Brownian motion in a renormalized Poisson integral in the quantum regime (i.e., in the critical and heavy-scale cases) is based on the following large deviations result. Consider some function and denote

Theorem 4.1. *Let, for some sequence of nonnegative continuous compactly supported functions,
**
for a.a. . Let be an increasing convex function with , and
**
Then,
**
Proof of Theorem 4.1: the lower bound. By Jensen’s inequality, *

and therefore, For every , we write

and note that (4.6) provides , because has at least linear growth at .

For a fixed , denote and consider a convex function Denote by the class of bounded measurable functions , and put

Lemma 4.2. *For every with ,
*

*Remark 4.3. *This statement is a version of the classic theorem in the finite-dimensional convex analysis about representation of the epigraph of a convex function as an intersection of upper half-spaces; see Theorem 12.1 in [48]. The idea of the proof, in our case, is principally the same, but we have to take care about topological aspects and about the fact that in general, is an improper function.

*Proof. *Consider the set
clearly, is a convex subset of the Banach space . In addition, this subset is closed by the Fatou lemma. Therefore, the separation theorem (Theorem 9.2 in [49], Chapter II) provides that is the intersection of all the closed half-spaces containing . Note that every continuous linear functional on the space has the form with , and
Take with , and . Then , and therefore, there exists and such that
By the definition of , if , then for every , hence (4.14) is impossible if either or . Divide (4.14) by and denote . Then,
Take in the second inequality in (4.15); this yields . Consequently,
which means that
because is arbitrary. The inverse inequality is obvious.

Take , then Note that where is a bounded function. Henceforth, by the large deviations result by Kac [50] (see also Theorem 4.1.6 in [51]), we have Note that Summarizing our proof, for every , . We take supremum over and get, by Lemma 4.2, Note that by Jensen’s inequality, for every , which makes it possible to apply Lemma 4.2. Finally, taking supremum over , we obtain the lower bound in (4.4).

*Proof of Theorem 4.1 (the upper bound). *Assume first that is continuous and supported by some cube . In that case, we reduce the proof of the upper bound to application of the large deviation principle for empirical measures of the Brownian motion on a torus. Such a reduction is standard, for example, [46]; the projection on the torus is required in order to make it possible to use Donsker-Varadhan’s large deviation principle for empirical measures of a Markov process with a compact state space, [52].

Note that because of the convexity, and . Hence, the function satisfies
Thus, for any ,
where
Denote by the torus of the size , that is, the cube with the sides identified. Let us denote by the projection on this torus: by definition, for its projection is the unique point such that . Denote , the Brownian motion on the torus . With this notation in mind, we rewrite the right hand side term in (4.27)
Consider the empirical measures for the Brownian motion on the torus
Note that
and the mapping
is continuous and bounded on the space of all probability distributions on with the metrics of weak convergence. Hence, combination of (4.27), the large deviation principle for (Theorem 3 in [52]), and Varadhan’s lemma (Proposition 3.8 in [53]) yields
where
By smooth truncation, it is easy to verify that
which completes the proof.

Finally, we remove the additional regularity assumption on . Recall the assumption (4.2) and note that one can assume the sequence to be pointwise increasing, because otherwise, one can take instead.

Write and
For every , we have by convexity
The Jensen inequality, analogously to (4.6), provides that
Then, from the upper bound with a regular kernel , we obtain
for any and . Passing to the limit first as and then as completes the proof.

##### 4.2. Proof of Theorem 2.1: Critical Case

The kernel (1.5) has the following scaling property: for any . Then, by the scaling property of the Brownian motion, Henceforth, the integral under the exponent in the right-hand side of (1.21), after the variable change , can be written as We take . Under such a choice, . Observe that because of (1.20). By monotonicity of , we can change the variables and, applying Theorem 4.1 with , obtain

Passing to the limit as completes the proof of Statement (ii) of Theorem 2.1.

##### 4.3. Proof of Theorem 2.1: Heavy Scale Case

Let us proceed with further transformations of the expression (4.41) for the integral under the exponent in the right hand side of (1.21). Denote , and .

It can be verified that when . In particular, , and hence