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 [36] for specific topics and for recent development on this subject.

Let 𝜔(𝑑𝑥) be a Poisson field in 𝑑 with intensity measure 𝜈𝑑𝑥, and let 𝐵 be an independent Brownian motion in 𝑑. Throughout, and 𝔼 denote the probability law and the expectation, respectively, generated by the Poisson field 𝜔(𝑑𝑥), while 𝑥 and 𝔼𝑥 denote the probability law and the expectation, respectively, generated by the Brownian motion 𝐵 with 𝐵0=𝑥. For a properly chosen (say, continuous and compactly supported) nonnegative function 𝐾 on 𝑑 (known as a shape function), define the respective random function (known as a Poisson potential) 𝑉(𝑥)=𝑑𝐾(𝑦𝑥)𝜔(𝑑𝑦),(1.1) which heuristically represents the net force at 𝑥𝑑 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 𝜇𝑡,𝜔 defined by 𝑑𝜇𝑡,𝜔𝑑0=1𝑍𝑡,𝜔exp𝑡0𝑉𝐵𝜅𝑠𝑑𝑠,𝑍𝑡,𝜔=𝔼0exp𝑡0𝑉𝐵𝜅𝑠.𝑑𝑠(1.2) Here, 𝜅 is a positive parameter, responsible for the time scaling 𝑠𝜅𝑠, 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 𝜇𝑡 is defined by 𝑑𝜇𝑡𝑑0=1𝑍𝑡exp𝑡0𝑉𝐵𝜅𝑠𝑑𝑠,𝑍𝑡=𝔼𝔼0exp𝑡0𝑉𝐵𝜅𝑠.𝑑𝑠(1.3) Heuristically, the integral 𝑡0𝑉𝐵𝜅𝑠𝑑𝑠(1.4) measures the total net attraction to which the Brownian particle is subject up to the time 𝑡, and henceforth, under the law 𝜇𝑡,𝜔 or 𝜇𝑡, 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 𝐾 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 𝐾(𝑥)=𝜃|𝑥|𝑝,𝑥𝑑,(1.5) which clearly is both locally unbounded and supported by whole 𝑑. 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 𝑥𝑑 when 𝑝𝑑.

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 𝑉(𝑥)=𝑑[].𝐾(𝑦𝑥)𝜔(𝑑𝑦)𝜈𝑑𝑦(1.6) Assume for a while that 𝐾 is locally bounded and compactly supported. Then, 𝑉(𝑥)=𝑑[]=𝐾(𝑦𝑥)𝜔(𝑑𝑦)𝜈𝑑𝑦𝑑𝐾(𝑦𝑥)𝜔(𝑑𝑦)𝜈𝑑𝐾(𝑦𝑥)𝑑𝑦=𝑉(𝑥)𝜈𝑑𝐾(𝑦)𝑑𝑦,(1.7)

that is, 𝑉𝑉=const. Consequently, replacing 𝑉 by 𝑉 in (1.2) and (1.3) does not change the measures 𝜇𝑡,𝜔 and 𝜇𝑡, because both the exponents therein and the normalizers 𝑍𝑡,𝜔 and 𝑍𝑡 are multiplied by the same constant 𝑒𝑡𝔼𝑉(0) (this is where the word “renormalization” comes from). On the other hand, for unbounded and not locally supported 𝐾, 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 𝑑/2<𝑝<𝑑. In that case, 𝑉 is well defined as well as the Gibbs measures 𝑑𝜇𝑡,𝜔𝑑0=1𝑍𝑡,𝜔exp𝑡0𝑉𝐵𝜅𝑠,𝑑𝑠𝑍𝑡,𝜔=𝔼0exp𝑡0𝑉𝐵𝜅𝑠,𝑑𝑑𝑠(1.8)𝜇𝑡𝑑0=1𝑍𝑡exp𝑡0𝑉𝐵𝜅𝑠,𝑑𝑠𝑍𝑡=𝔼𝔼0exp𝑡0𝑉𝐵𝜅𝑠,𝑑𝑠(1.9) see [7, Corollary 1.3]. We use separate notation 𝜇𝑡,𝜔, 𝜇𝑡 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 𝑉, and the renormalized potential 𝑉 are well defined under the mutually excluding assumptions 𝑝>𝑑 and 𝑑/2<𝑝<𝑑, 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 𝑉, partially renormalized at the level ; see [7, Chapter 6]. By definition, 𝑉(𝑥)=𝑑(𝐾(𝑦𝑥))+𝜔(𝑑𝑦)+𝑑([],𝐾(𝑦𝑥))𝜔(𝑑𝑦)𝜈𝑑𝑦(1.10) where [0,] is a renormalization level. Clearly, 𝑉0=𝑉,𝑉=𝑉. It is known (see [7, Chapter 6]) that 𝑉 is well defined for every (0,+) as soon as 𝑉 is well defined for some [0,+], and in that case, there exists a constant 𝐶𝐾,, such that 𝑉𝑉=𝜈𝐶𝐾,,. This makes it possible to define the respective Gibbs measures in a uniform way, replacing 𝑉 in (1.2), (1.3) by 𝑉 with (any) (0,+). In addition, such a definition extends the class of shape functions: for 𝐾 given by (1.5), 𝑉 with (0,+) is well defined for 𝑝>𝑑/2.

The main objective of this paper is to study the asymptotic behavior, as 𝑡+, of the annealed exponential moments 𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠.𝑑𝑠(1.11) This problem is clearly relevant with the model discussed above: in the particular case 𝜅=1, 𝛼𝑡1, this is just the natural question about the limit behavior of the normalizer 𝑍𝑡 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 𝑉 replaced by either a Poisson potential 𝑉 or a partially renormalized potential 𝑉 with (0,+).

The function 𝛼𝑡 in (1.11) appears, on one hand, because of our further intent to study in further publications the a.s. behavior 𝑡0𝑉𝐵𝜅𝑠𝑑𝑠,𝑡.(1.12) 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 𝛼𝑡 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 𝜕𝑡𝑢𝑢(𝑡,𝑥)=𝜅Δ𝑢(𝑡,𝑥)±𝑄(𝑥)𝑢(𝑡,𝑥),(0,𝑥)=1,𝑥𝑑.(1.13) 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 𝑄 is usually chosen as the Poisson potential 𝑉, with the shape function 𝐾 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 {𝑄(𝑥);𝑥𝑑} 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 𝑉 instead of 𝑄. Note that in that case, the field 𝑄 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 𝑢(𝑡,𝑥)=𝔼𝑥±exp𝑡0𝑄𝐵2𝜅𝑠,𝑑𝑠(1.14) when 𝑄 is Hölder continuous and satisfies proper growth bounds. When 𝑄=𝑉 with 𝐾 from (1.5), local unboundedness of 𝐾 induces local irregularity of 𝑄 (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 𝐾 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 𝑑/2<𝑝<𝑑 (Theorem 1.1 in [7]). For the sign “+”, the random field (1.14) is not integrable for any 𝑝. On the other hand, the random field (1.14) is well defined for 𝑑/2<𝑝<min(2,𝑑) (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 [1020] 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 𝑁 of a random Schrödinger operator of the type 𝜅𝐻=2Δ+𝑄.(1.15) This function, written IDS in the sequel, is a deterministic spectral mean-field characteristic of 𝐻. Under quite general assumptions on the random potential 𝑄, it is well defined as 𝑁(𝜆)=lim𝑈𝑑1||𝑈||𝑘1𝜆𝑘,𝑈𝜆,(1.16) where {𝜆𝑘,𝑈} is the set of eigenvalues for the operator 𝐻 in a cube 𝑈 with the Dirichlet boundary conditions, |𝑈| denotes the Lebesgue measure of 𝑈 in 𝑑, and the limit pass is made w.r.t. a sequence of cubes which has same center and extends to the whole 𝑑. 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) 𝜆0 of the spectrum of 𝐻 mainly describes the low-temperature dynamics for a system defined by the Hamiltonian (1.15). This motivates the problem of asymptotic behavior of log𝑁(𝜆), 𝜆𝜆0, 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, 2544] as a partial list of references on the subject.

Connection between the Lifshitz tails asymptotics for the IDS function 𝑁, and the problem discussed above is provided by the representation for the Laplace transform of 𝑁𝑒𝜆𝑡𝑑𝑁(𝜆)=(2𝜋𝜅𝑡)𝑑/2𝔼𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠𝑑𝑠,𝑡0.(1.17) Here, 𝔼𝜅𝑡0,0 denotes the distribution of the Brownian bridge, that is, the Brownian motion conditioned by 𝐵𝜅𝑡=0. Our estimates for (1.11) appear to be process insensitive to some extent and remain true with 𝔼0 in (1.11) replaced by 𝔼𝜅𝑡0,0. This, via appropriate Tauberian theorem, provides information on Lifshitz tail asymptotics for the respective IDS function 𝑁. Note that in this case, the asymptotic behavior of the log𝑁(𝜆) as 𝜆 should be studied, because the bottom of the spectrum is equal 𝜆0=, unlike the (usual) Poisson case, where 𝜆0=0. 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 𝑝 in this definition, we separate three cases 𝛼𝑡𝑡=𝑜(𝑑+2𝑝)/(𝑑+2)𝑡,𝑡,(1.18)(𝑑+2𝑝)/(𝑑+2)𝛼=𝑜𝑡𝛼,𝑡,(1.19)𝑡𝛼𝑡(𝑑+2𝑝)/(𝑑+2)withsome𝛼>0,𝑡,(1.20) 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 𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝔼0𝜈exp𝑑𝜓1𝛼𝑡,𝜉(𝑡,𝑥)𝑑𝑥(1.21) with 𝜓(𝑢)=𝑒𝑢1+𝑢,(1.22)𝜉(𝑡,𝑥)=𝑡0𝐾𝐵𝜅𝑠𝑥𝑑𝑠,(1.23) 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 𝐵 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 𝜈𝑑𝜓1𝛼𝑡𝑡0𝐾(𝑥)𝑑𝑠𝑑𝑥=𝜈𝑑𝜓𝑡𝛼𝑡𝐾(𝑥)𝑑𝑥,(1.24) 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 𝑉 and a partially renormalized Poisson potential 𝑉 (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 𝑉(𝐵𝜅𝑡)𝑉(0) 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, 𝜔𝑑 denotes the volume of the 𝑑-dimensional unit ball. We denote 𝑑=𝑔𝑊12𝑑𝑑𝑔2(𝑥)𝑑𝑥=1,(2.1) where 𝑊12(𝑑) is used for the Sobolev space of functions that belong to 2 together with their first order derivatives. We also denote 𝜑(𝑢)=1𝑒𝑢,Ξ(𝑢,𝑣)=𝜓(𝑢)𝑒𝑢𝜑(𝑣)=𝑒𝑢𝑣1+𝑢,𝑢,𝑣,(2.2) (𝜓 is introduced in (1.22)). Clearly, the functions 𝜓, 𝜑, and Ξ 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 𝑝(𝑑/2,𝑑). (i)In the “light-scale” case, lim𝑡𝛼𝑡𝑡𝑑/𝑝log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝜈𝑑𝜓(𝜃|𝑥|𝑝)𝑑𝑥=𝜈𝜔𝑑𝜃𝑑/𝑝𝑝Γ𝑑𝑝2𝑝𝑑𝑝=𝜈𝜔𝑑𝜃𝑑/𝑝Γ𝑝𝑑𝑝.(2.3)(ii)In the “critical” case, lim𝑡𝑡𝑑/(𝑑+2)log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=sup𝑔𝑑𝜈𝑑𝜓𝜃𝛼𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝜅𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(2.4)(iii)In the “heavy-scale” case, under additional assumption 𝑝<(𝑑+2)/2, lim𝑡𝛼𝑡4/(𝑑+22𝑝)𝑡(𝑑+42𝑝)/(𝑑+22𝑝)log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=sup𝑔𝑑𝜈𝜃22𝑑𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝑑𝑦2𝜅𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(2.5)

Remark 2.2. The additional assumption 𝑝<(𝑑+2)/2 in Statement (iii) is exactly the condition for 𝜉(𝑡,𝑥) 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 𝐾 be bounded and satisfy 𝐾(𝑥)𝜃|𝑥|𝑝,|𝑥|,(2.6) with 𝑝>𝑑. (i)(see [41]) If 𝑝(𝑑,𝑑+2), lim𝑡𝑡𝑑/𝑝log𝔼𝔼0exp𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝜈𝜔𝑑𝜃𝑑/𝑝Γ𝑝𝑑𝑝.(2.7)(ii)(see [40]) If 𝑝=𝑑+2, lim𝑡𝑡𝑑/(𝑑+2)log𝔼𝔼0exp𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=inf𝑔𝑑𝜈𝑑𝜑𝜃𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝜅𝑑𝑦𝑑𝑥+2𝑑||||𝑔(𝑦)2.𝑑𝑦(2.8)(iii)(see [46]) If 𝑝>𝑑+2, lim𝑡𝑡𝑑/(𝑑+2)log𝔼𝔼0expt0𝑉𝐵𝜅𝑠𝑑𝑠=inf𝑔𝑑𝜈𝑑1𝑔(𝑥)>0𝜅𝑑𝑥+2𝑑||||𝑔(𝑦)2.𝑑𝑦(2.9)

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 𝑝>𝑑+2 and 𝑝(𝑑,𝑑+2), 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 𝜅, that is, the “intensity” of the Brownian motion. On the other hand, (2.9) depends on 𝜅 but not on the shape function 𝐾. Since 𝐾 and 𝜅, heuristically, are related to “regular” and “chaotic” parts of the dynamics, an alternative terminology “classic regime” (𝑝>𝑑+2) and “quantum regime” (𝑝(𝑑,𝑑+2)) 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 𝐾 to be either heavy or light tailed replaced by conditions on the scale 𝛼𝑡 to be, respectively, light or heavy. Note that for 𝛼𝑡1, (1.18) and (1.19) transform exactly to 𝑝<𝑑+2 and 𝑝>𝑑+2, respectively. In the classic regime, an analogy between a Poisson potential and a renormalized Poisson potential is very close: for 𝛼𝑡1, (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 𝛼𝑡) and shape dependent (i.e., involves 𝑝).

It is a natural question whether Theorem 2.1 can be extended to other types of potentials, like a Poisson potential 𝑉 or a partially renormalized Poisson potential 𝑉. 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 𝜐 which are convex but are not increasing (like 𝜑 and Ξ). 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 𝐾 can be made very mild: instead of (1.5), we assume (2.6) with 𝑝>𝑑/2 and, when 𝑝<𝑑, 𝑑𝜓(𝐾(𝑥))𝑑𝑥<+,(2.10) which is just the assumption for 𝑉 to be well defined.

Theorem 2.4. Let the shape function 𝐾 satisfy (2.6) and scale function 𝛼𝑡 satisfy (1.18). (i)Statement (i) of Theorem 2.1 holds true assuming 𝐾 satisfies (2.10).(ii)For 𝑝>𝑑, lim𝑡𝛼𝑡𝑡𝑑/𝑝log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝜈𝑑𝜑(𝜃|𝑥|𝑝)𝑑𝑥=𝜈𝜔𝑑𝜃𝑑/𝑝Γ𝑝𝑑𝑝.(2.11)(iii)For 𝑝=𝑑 and >0, log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝜈𝑑𝛼min(𝐾(𝑦),)𝑡𝑡+𝑑𝑦+𝜔𝑑𝑡𝜃Eu𝛼𝑡𝑡+𝑜𝛼𝑡,𝑡,(2.12) where Eu=Γ(1)=0,57721 is the Euler constant. In particular, when 𝐾 has the form (1.5), log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝜈𝜔𝑑𝜃𝑡log𝛼𝑡𝑡+log+Eu𝛼𝑡𝑡+o𝛼𝑡,𝑡.(2.13)

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 𝔼0 replaced by 𝔼𝜅𝑡0,0, 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 𝑄(𝑥), 𝑥𝑑 and a cube 𝑈𝑑, denote by 𝐻𝑄𝑈 the random Schrödinger operator in 𝑈 with the potential 𝑄 and the Dirichlet boundary conditions 𝐻𝑄𝑈𝜅𝑓=2Δ𝑓+𝑄𝑓,𝑓|𝜕𝑈=0.(2.14) When the field 𝑄 is assumed to have locally bounded realizations, the operator 𝐻𝑄𝑈 is a.s. well defined as an operator on 𝐿2(𝑈,𝑑𝑥) and is self-adjoint. In addition, respective semigroup 𝑅𝑄𝑡,𝑈=𝑒𝑡𝐻𝑄𝑈,𝑡0 has a Feynman-Kac representation ([1, page 13]) 𝑅𝑄𝑡,𝑈𝑓(𝑥)=𝔼𝑡𝑥exp𝑡0𝑄𝐵𝜅𝑠𝜒𝑑𝑠𝑈,𝑡𝐵𝑓𝐵𝑡,𝑥𝑈,𝑡0,(2.15) where 𝜒𝑈,𝑡𝐵=l𝐵𝜅𝑠𝑈,𝑠[0,𝑡].(2.16) For general 𝑄, we define 𝐻𝑄𝑈 by the following limit procedure. Consider truncations 𝑄𝑁=(|𝑄|𝑁)sgn𝑄. Under appropriate assumptions on 𝑄, for almost every realization of this field, operators 𝑅𝑄𝑁𝑡,𝑈 converge strongly for every 𝑡0 as 𝑁. In that case, 𝐻𝑄𝑈 is defined as the generator of the limit semigroup 𝑅𝑄𝑡,𝑈, 𝑡0. Assuming the spectrum of 𝐻𝑄𝑈 to be discrete (we verify this assumption below), we denote this spectrum {𝜆𝑄𝑘,𝑈} and define the function 𝑁𝑄𝑈1(𝜆)=||𝑈||𝑘l𝜆𝑄𝑘,𝑈𝜆,𝜆.(2.17)

Proposition 2.6. Let the shape function 𝐾 be such that for some 𝑔>0, the following conditions hold: (i)𝐾𝑔(𝑥)=(𝐾(𝑥)𝑔)+ is compactly supported,(ii)𝐾𝑔=min(𝐾(𝑥),𝑔) is Lipschitz continuous and belongs to the Sobolev space 𝑊12(𝑑).Consider either a partially renormalized potential 𝑄=𝑉 with (0,), or a renormalized potential 𝑄=𝑉, in the latter case assuming additionally (2.10).
Then, (a)for a.s. realization of the potential 𝑄 and every cube U, the described above procedure well defines both the random Schrödinger operator 𝐻𝑄𝑈 and respective function 𝑁𝑄𝑈,(b)there exists an integrated density of states 𝑁𝑄, that is, a deterministic monotonous function such that 𝑁𝑄(𝜆)=lim𝑈𝑑𝑁𝑄𝑈(𝜆),(2.18)
a.s. for every point of continuity of 𝑁𝑄. Respective Laplace transform has the representation 𝑒𝜆𝑡𝑑𝑁𝑄(𝜆)=(2𝜋𝜅𝑡)(𝑑/2)𝔼𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠𝑑𝑠,𝑡0.(2.19)

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 𝑄 (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 𝑉 and 𝑉.

Theorem 2.7. Let 𝐾 satisfy (2.6). (i)For 𝑝(𝑑/2,𝑑), assuming additionally (2.10), one has in limit 𝜆log𝑁𝑉(𝜆)=𝜈𝜔𝑑Γ2𝑝𝑑𝑝𝑝/(𝑑𝑝)𝜃(𝑑𝑝)𝑑𝑑/(𝑑𝑝)(𝜆)𝑑/(𝑑𝑝)(1+𝑜(1)).(2.20)(ii)For 𝑝=𝑑 and (0,), one has in limit𝜆log𝑁𝑉(𝜆)=𝜈𝜔𝑑𝜆𝜃exp𝜈𝜔𝑑𝜃logEu1(1+𝑜(1))=𝜈𝜔𝑑𝜃𝜆exp𝜈𝜔𝑑𝜃Eu1(1+𝑜(1)).(2.21)

Theorem 2.7 involves the asymptotic results for exponential moments (Theorem 2.5) only in a partial form, for the trivial scale function 𝛼𝑡1. 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 𝐻𝛾𝜅=2Δ+𝛾𝑄,𝛾>0.(2.22) Assuming every potential 𝑄𝛾=𝛾𝑄 being such that respective IDS function 𝑁𝑄𝛾 is well defined, denote 𝑁𝑄(𝜆,𝛾)=𝑁𝑄𝛾(𝜆). We call the family 𝑁𝑄(𝜆,𝛾),𝜆,𝛾>0,(2.23)

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 (𝑑+2𝑝)/2>(𝑑+42𝑝)/4. Hence, conditions, that (𝜆)(𝑑+42𝑝)/4/𝛾 and (𝜆)(𝑑+2𝑝)/2/𝛾 is bounded, yield 𝜆0. Therefore, the quantum regime for the integrated density of states field requires that 𝜆 and 𝛾 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 𝜆 (in that case 𝛾 may tend to ), 𝜆0, or 𝜆 to stay bounded away both from 0 and (in these two cases 𝛾0+ necessarily). This is the reason that two conditions (𝜆)𝑝/𝑑/𝛾 and (𝜆)(𝑑+2𝑝)/2/𝛾 are imposed in this case: when 𝜆, the first one includes the second one, but when 𝜆0, the inclusion is opposite.

Theorem 2.8. Let 𝐾 be of the form (1.5) with 𝑝(𝑑/2,𝑑). (i)When (𝜆)𝑝/𝑑/𝛾 and (𝜆)(𝑑+2𝑝)/2/𝛾, log𝑁𝑉(𝜆,𝛾)=𝜈𝜔𝑑Γ2𝑝𝑑𝑝𝑝/(𝑑𝑝)𝜃(𝑑𝑝)𝑑𝑑/(𝑑𝑝)×(𝜆)𝑑/(𝑑𝑝)𝛾𝑑/(𝑑𝑝)(1+𝑜(1)).(2.24)(ii)When (𝜆)𝑑+42𝑝/4/𝛾 and (𝜆)(𝑑+2𝑝)/2/𝛾0, under additional assumption 𝑝<(𝑑+2)/2, log𝑁𝑉(𝜆,𝛾)=2C2𝑑+22𝑝(𝑑+22𝑝)/22𝑑+42𝑝(𝑑+42𝑝)/2(𝜆)(𝑑+42𝑝)/2𝛾2(1+𝑜(1)),(2.25) where C2 denotes the constant in the r.h.s of (2.5).(iii)When 𝜆0 and (𝜆)(𝑑+2𝑝)/2/𝛾 is bounded away both from 0 and from , log𝑁𝑉(𝜆,𝛾)=(𝑑𝑝)C𝜓𝑝𝑝/(𝑑𝑝)(𝑑𝑝)𝑑𝑑/(𝑑𝑝)(𝜆)𝑑/(𝑑𝑝)𝛾𝑝/(𝑑𝑝)(1+𝑜(1)),(2.26) where C𝜓 denotes the constant in the right hand side of (2.4) with 𝛼=1.

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 𝜆0. This in general reflects the fact that for 𝛾0+ 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 >0, denote 𝜉(𝑡,𝑥)=𝑡0(𝐾(𝑦𝑥))+𝑑𝑠,𝜉(𝑡,𝑥)=𝑡0(𝐾(𝑦𝑥))𝑑𝑠.(3.1)

Similarly to (1.21), we have 𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝔼0exp𝑑𝜑1𝛼𝑡,𝜉(𝑡,𝑥)𝑑𝑥(3.2)𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=𝔼0exp𝑑Ξ1𝛼𝑡𝜉(1𝑡,𝑥),𝛼𝑡𝜉(.𝑡,𝑥)𝑑𝑥(3.3) 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 𝑉, we consider a partially renormalized Poisson potential with the properly chosen renormalization level, dependent on 𝑡. Let 𝑔>0 and 𝑡=𝑔𝛼𝑡/𝑡. Then, assuming 𝑝=𝑑, (2.6) and (1.18), we will prove that lim𝑡𝛼𝑡𝑡log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝑡𝐵𝜅𝑠𝑑𝑠=𝜈𝑑Ξ𝜃|𝑥|𝑑𝑔,𝜃|𝑥|𝑑𝑔+𝑑𝑥=𝜈𝜔𝑑𝜃.log𝑔+Eu(3.4) Note that by Proposition  6.1 in [7], 𝑉(𝑥)𝑉(𝑥)=𝜈𝑑(min(𝐾(𝑦),))+𝑑𝑦,(3.5) for any such that 𝑉, 𝑉 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 𝜚, by the Jensen inequality, we have 𝜚1𝛼𝑡𝜉(𝑡,𝑥)=𝜚𝑡01𝛼𝑡𝐾𝐵𝜅𝑠1𝑥𝑑𝑠𝑡𝑡0𝜚𝑡𝛼𝑡𝐾𝐵𝜅𝑠𝑥𝑑𝑠.(3.6) Denote 𝜆𝑡=(𝑡/𝛼𝑡)1/𝑝, 𝐾(𝑥,𝜆)=𝜆𝑝𝐾(𝜆𝑥). By the inequality above, one has the following estimate with nonrandom right hand side: 𝑑𝜚1𝛼𝑡𝜉1(𝑡,𝑥)𝑑𝑥𝑡𝑑t0𝜚𝑡𝛼𝑡𝐾𝐵𝜅𝑠=𝑥𝑑𝑠𝑑𝑥𝑑𝜚𝑡𝛼𝑡𝑡𝐾(𝑥)𝑑𝑥=𝛼𝑡𝑑/𝑝𝑑𝜚𝐾𝑥,𝜆𝑡𝑑𝑥.(3.7) Assumption (1.18) yields 𝜆𝑡. 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 𝜓 or 𝜑 and then prove, respectively, 𝑑𝜓(𝐾(𝑥,𝜆))𝑑𝑥𝑑𝜓(𝐾(𝑥))𝑑𝑥or𝑑𝜑(𝐾(𝑥,𝜆))𝑑𝑥𝑑𝜑(𝐾(𝑥))𝑑𝑥,(3.8)𝜆. By assumption (2.6), for every 𝜀>0, there exists 𝜆𝜀 such that 𝐾(𝑥,𝜆)|𝑥|𝑝[],𝜃𝜀,𝜃+𝜀|𝑥|>𝜀,𝜆>𝜆𝜀.(3.9)

When 𝑝>𝑑, this easily provides lim𝜆|𝑥|>𝜀𝜑(𝐾(𝑥,𝜆))𝑑𝑥=|𝑥|>𝜀𝜑(𝐾(𝑥))𝑑𝑥,𝜀>0.(3.10) Since 𝜑 is bounded on +, (3.10) provides the second relation in (3.8).

When 𝑝(𝑑/2,𝑑), similar argument leads to the relation analogous to (3.10) with 𝜑 replaced by 𝜓. Consequently, with condition (2.10) in mind, it remains to prove that lim𝜀0limsup𝜆|𝑥|𝜀𝜓(𝐾(𝑥,𝜆))𝑑𝑥=0.(3.11) To that end, we choose 𝑟1, 𝜃1 such that 𝐾(𝑥)𝜃1|𝑥|𝑝, |𝑥|>𝑟 and write for 𝜆 large enough |𝑥|𝜀𝜓(𝐾(𝑥,𝜆))𝑑𝑥=|𝑥|𝑟1/𝜆+𝑟1/𝜆<|𝑥|𝜀𝜓(𝐾(𝑥,𝜆))𝑑𝑥𝜆𝑑|𝑥|𝑟1𝜓(𝜆𝑝𝐾(𝑥))𝑑𝑥+|𝑥|𝜀𝜓𝜃1|𝑥|𝑝𝑑𝑥.(3.12) Recall that 𝑝<𝑑, 𝜓(𝑢) is dominated by 𝑢, and 𝐾 is locally integrable under condition (2.10). Then, the first term in the above sum is negligible when 𝜆. This proves (3.10) and completes the proof.

Similarly, for 𝑝=𝑑 from the Jensen’s inequality for the convex function Ξ2, we have Ξ1𝛼𝑡𝜉(1𝑡,𝑥),𝛼𝑡𝜉(1𝑡,𝑥)𝑡𝑡0Ξ𝑡𝛼𝑡𝐾𝐵𝜅𝑠,𝑡𝑥𝛼𝑡𝐾𝐵𝜅𝑠𝑥+𝑑𝑠,(3.13)

and consequently, for 𝑡=𝑔𝛼𝑡/𝑡𝑑Ξ1𝛼𝑡𝜉𝑡1(𝑡,𝑥),𝛼𝑡𝜉𝑡(𝑡,𝑥)𝑑𝑥𝑑Ξ𝑡𝛼𝑡𝐾(𝑥)𝑡,𝑡𝛼𝑡𝐾(𝑥)𝑡+=𝑡𝑑𝑥𝛼𝑡𝑑/𝑝𝑑Ξ𝐾𝑥,𝜆𝑡𝐾𝑔,𝑥,𝜆𝑡𝑔+𝑑𝑥.(3.14) Similarly to (3.8), one can prove 𝑑Ξ𝐾(𝑥,𝜆)𝑔,(𝐾(𝑥,𝜆)𝑔)+𝑑𝑥𝑑Ξ𝐾(𝑥,𝜆)𝑔,(𝐾(𝑥,𝜆)𝑔)+𝑑𝑥,𝜆,(3.15) which provides the upper bound in (3.4).

3.2. Proof of the Lower Bound

For a fixed 𝜀>0, take 𝑅 fixed but large enough so that (𝜃𝜀)|𝑥|𝑝𝐾(𝑥)(𝜃+𝜀)|𝑥|𝑝,|𝑥|𝑅.(3.16) Take 𝛽>0 and consider the set 𝐴𝑡,𝛽=sup𝑠𝑡||𝐵𝜅𝑠||𝛽𝜆𝑡,(3.17) keeping the notation 𝜆𝑡=(𝑡/𝛼𝑡)1/𝑝. By the scaling property and the well-known small balls probability asymptotics for the Brownian motion, we have, for 𝑡 large enough, log0𝐴𝑡,𝛽𝑐𝑡𝛽𝜆𝑡2(3.18) with some constant 𝑐>0. Therefore, condition 𝛼𝑡=𝑜(𝑡(𝑑+2𝑝)/(𝑑+2)) yields 𝛼𝑡𝑡𝑑/𝑝log0𝐴𝑡,𝛽0,𝑡+.(3.19) Take 𝛾>2𝛽. On the set 𝐴𝑡,𝛽, one has ||𝐵𝜅𝑠||𝑡𝑥𝛽𝛼𝑡1/𝑝[],𝑠0,𝑡,|𝑥|𝛾𝜆𝑡.(3.20) Then, for 𝑡 large enough to provide 𝛽𝜆𝑡>𝑅, we have ||𝐵(𝜃𝜀)𝜅𝑠||𝑥𝑝𝐵𝐾𝜅𝑠||𝐵𝑥(𝜃+𝜀)𝜅𝑠||𝑥𝑝[],,𝑠0,𝑡|𝑥|𝛾𝜆𝑡.(3.21) Therefore, a two-sided estimate 𝛽(𝜃𝜀)1+𝛾𝑝|𝑥|𝑝𝐵𝐾𝜅𝑠𝛽𝑥(𝜃+𝜀)1𝛾𝑝|𝑥|𝑝[],𝑠0,𝑡(3.22) is valid on the set 𝐴𝑡,𝛽 for every 𝑥 with |𝑥|>𝛾𝜆𝑡. Observe that (3.22) is a pointwise estimate for a Brownian trajectory from a “small ball” 𝐴𝑡,𝛽 and for a point 𝑥 outside a “large ball” {𝑦|𝑦|𝛾𝜆𝑡}. On the other hand, (3.19) shows the “small Brownian ball” 𝐴𝑡,𝛽 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 𝜓 is nonnegative and nondecreasing, (3.22) yields 𝑑𝜓1𝛼𝑡𝜉(𝑡,𝑥)𝑑𝑥|𝑥|>𝛾𝜆𝑡𝜓𝑡𝛼𝑡𝛽(𝜃𝜀)1+𝛾𝑝|𝑥|𝑝𝑡𝑑𝑥=𝛼𝑡𝑑/𝑝𝐼𝜓𝜀,𝛽,𝛾,(3.23) on 𝐴𝑡,𝛽 with 𝐼𝜓𝜀,𝛽,𝛾=|𝑥|>𝛾𝜓𝛽(𝜃𝜀)1+𝛾𝑝|𝑥|𝑝𝑑𝑥.(3.24) Together with (1.21) and (3.19), this inequality provides liminf𝑡+𝛼𝑡𝑡𝑑/𝑝log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠𝐼𝜓𝜀,𝛽,𝛾,(3.25) for every 𝜀>0, 𝛽>0, 𝛾>0. Since lim𝜀0lim𝛾0lim𝛽0𝐼𝜓𝜀,𝛽,𝛾=𝑑𝜓(𝜃|𝑥|𝑝)𝑑𝑥,(3.26) this completes the proof of the lower bound in (2.3).

Since (𝜑) is nonincreasing and satisfies 𝜑1, (3.22) yields 𝑑𝜑1𝛼𝑡𝜉(𝑡,𝑥)𝑑𝑥|𝑥|𝛾𝜆𝑡𝑑𝑥|𝑥|>𝛾𝜆𝑡𝜑𝑡𝛼𝑡𝛽(𝜃+𝜀)1𝛾𝑝|𝑥|𝑝𝑡𝑑𝑥=𝛼𝑡𝑑/𝑝𝐼𝜑𝜀,𝛽,𝛾,(3.27) on 𝐴𝑡,𝛽 with 𝐼𝜑𝜀,𝛽,𝛾=|𝑥|𝛾𝑑𝑥|𝑥|>𝛾𝜑𝛽(𝜃+𝜀)1𝛾𝑝|𝑥|𝑝𝑑𝑥.(3.28)

Since lim𝜀0lim𝛾0lim𝛽0𝐼𝜑𝜀,𝛽,𝛾=𝑑𝜑(𝜃|𝑥|𝑝)𝑑𝑥,(3.29) this provides the lower bound in (3.5).

Finally, Ξ is nondecreasing in first coordinate and nonincreasing in second coordinate. In addition, Ξ1, and hence (3.22) yields in the case 𝑑=𝑝𝑑Ξ1𝛼𝑡𝜉𝑡1(𝑡,𝑥),𝛼𝑡𝜉𝑡(𝑡,𝑥)𝑑𝑥|𝑥|>𝛾𝜆𝑡Ξ𝑡𝛼𝑡𝛽(𝜃𝜀)1+𝛾𝑑|𝑥|𝑑𝑔𝛼𝑡𝑡,𝑡𝛼𝑡𝛽(𝜃+𝜀)1𝛾𝑑|𝑥|𝑑𝑡𝑔𝛼𝑡𝑡+𝑡𝑑𝑥=𝛼𝑡𝐼Ξ𝜀,𝛽,𝛾,(3.30) on 𝐴𝑡,𝛽 with 𝐼Ξ𝜀,𝛽,𝛾=|𝑥|𝛾𝑑𝑥+|𝑥|>𝛾Ξ𝛽(𝜃𝜀)1+𝛾𝑑|𝑥|𝑑𝛽𝑔,(𝜃+𝜀)1𝛾𝑑|𝑥|𝑑𝑔+𝑑𝑥.(3.31) Together with (3.2) and (3.19), this inequality provides liminf𝑡+𝛼𝑡𝑡log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠𝐼Ξ𝜀,𝛽,𝛾,(3.32) for every 𝜀>0, 𝛽>0, 𝛾>0. Since lim𝜀0lim𝛾0lim𝛽0𝐼Ξ𝜀,𝛽,𝛾=𝑑Ξ𝜃|𝑥|𝑑𝑔,𝜃|𝑥|𝑑𝑔+𝑑𝑥,(3.33)

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, 𝑑Ξ𝜃|𝑥|𝑑𝑔,𝜃|𝑥|𝑑𝑔+𝑑𝑥=𝜔𝑑0Ξ𝜃𝑟𝜃𝑔,𝑟𝑔+𝑑𝑥=𝜔𝑑0𝑒𝜃/𝑟𝜃1𝑟𝑔𝑑𝑟=𝜃𝜔𝑑0𝑒𝑠1𝑠𝑔𝑠2𝑑𝑠=𝜔𝑑𝜃0𝑒𝑠1𝑠1𝑠2,𝑑𝑠+log𝑔(3.34) in the last identity we have used an elementary relation 0𝑠𝑔𝑠1𝑠2𝑑𝑠=log𝑔.(3.35) Integration by parts and n. 538 in [47] gives 0𝑒𝑠1𝑠1𝑠2𝑑𝑠=𝑡01𝑒𝑠𝑠𝑑𝑠0𝑒𝑠𝑠𝑑𝑠=Eu,(3.36) which completes calculation of the integral in (3.4).

Finally, let 𝐾 has the form (1.5). Take 𝑡=𝛼𝑡/𝑡, then 𝑡< for 𝑡 large enough, and 𝑑𝛼min(𝐾(𝑦),)𝑡𝑡+𝑑𝑦=𝑑𝜃|𝑥|𝑑𝑡+𝑑𝑥=𝜔𝑑𝜃0𝑠𝑠𝑡𝑠2𝑑𝑠=𝜔𝑑𝜃loglog𝑡=𝜔𝑑𝜃𝑡log+log𝛼𝑡.(3.37) 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 𝐿𝑑+ and denote 𝜂(𝑡,𝑥)=𝑡0𝐿𝐵𝜅𝑠𝑥𝑑𝑥.(4.1)

Theorem 4.1. Let, for some sequence 𝐿𝑛,𝑛1 of nonnegative continuous compactly supported functions, 𝐿(𝑥)=sup𝑛1𝐿𝑛(𝑥),(4.2) for a.a. 𝑥𝑑. Let 𝜐+ be an increasing convex function with 𝜐(0)=0, and 𝑑𝜐(𝐿(𝑥))𝑑𝑥<+.(4.3) Then, lim𝑡1𝑡log𝔼0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥=sup𝑔𝑑𝑑𝜐𝑑𝑔2𝜅(𝑦)𝐿(𝑥𝑦)𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(4.4) Proof of Theorem 4.1: the lower bound. By Jensen’s inequality,

𝜐1𝑡1𝜂(𝑡,𝑥)=𝜐𝑡𝑡0𝐿𝐵𝜅𝑠1𝑥𝑑𝑠𝑡𝑡0𝜐𝐿𝐵𝜅𝑠𝑥𝑑𝑠,(4.5) and therefore, 𝑑𝜐1𝑡1𝜂(𝑡,𝑥)𝑑𝑥𝑡𝑡0𝑑𝜐𝐿𝐵𝜅𝑠𝑥𝑑𝑥𝑑𝑠=𝑑𝜐(𝐿(𝑥))𝑑𝑥<+.(4.6) For every 𝑅>0, we write 𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥[𝑅,𝑅]𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥,(4.7)

and note that (4.6) provides (1/𝑡)𝜂(𝑡,)1([𝑅,𝑅]𝑑), because 𝜐 has at least linear growth at +.

For a fixed 𝑅, denote +1,𝑅=1[]𝑅,𝑅𝑑,,0(4.8) and consider a convex function Υ𝑅+1,𝑅[0,+]Υ𝑅()=[𝑅,𝑅]𝑑𝜐((𝑥))𝑑𝑥.(4.9) Denote by 𝐵𝑅 the class of bounded measurable functions 𝑓[𝑅,𝑅]𝑑, and put 𝐶Υ,𝑓,𝑅=sup𝐶𝐶+[𝑅,𝑅]𝑑𝑓(𝑥)(𝑥)𝑑𝑥Υ𝑅(),+1,𝑅,𝑓𝐵𝑅.(4.10)

Lemma 4.2. For every +1,𝑅 with Υ𝑅()<+, Υ𝑅()=sup𝑓𝐵𝑅𝐶Υ,𝑓,𝑅+[𝑅,𝑅]𝑑.𝑓(𝑥)(𝑥)𝑑𝑥(4.11)

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, Υ𝑅 is an improper function.

Proof. Consider the set epiΥ𝑅=(,𝑡)+1,𝑅,𝑡Υ𝑅(),(4.12) clearly, epiΥ𝑅 is a convex subset of the Banach space 1([𝑅,𝑅]𝑑)×. In addition, this subset is closed by the Fatou lemma. Therefore, the separation theorem (Theorem 9.2 in [49], Chapter II) provides that epiΥ𝑅 is the intersection of all the closed half-spaces containing epiΥ𝑅. Note that every continuous linear functional on the space 1([𝑅,𝑅]𝑑)× has the form (𝑓,𝑎) with 𝑓𝐵𝑅, 𝑎 and (,𝑡),(𝑓,𝑎)=[𝑅,𝑅]𝑑(𝑥)𝑓(𝑥)𝑑𝑥+𝑎𝑡.(4.13) Take +𝜐 with Υ𝑅()<+, and 𝑡<Υ𝑅(). Then (,𝑡)epiΥ𝑅, and therefore, there exists (𝑓,𝑎) and 𝑐 such that ,𝑡,(𝑓,𝑎)<𝑐,(,𝑡),(𝑓,𝑎)𝑐,(,𝑡)epiΥ𝑅.(4.14) By the definition of epiΥ𝑅, if (,𝑡)epiΥ𝑅, then (,𝑡)epiΥ𝑅 for every 𝑡>𝑡, hence (4.14) is impossible if either 𝑎=0 or 𝑎<0. Divide (4.14) by 𝑎 and denote 𝑓𝑎=𝑓/𝑎,𝑐𝑎=𝑐/𝑎. Then, 𝑐𝑎+[𝑅,𝑅]𝑑(𝑥)𝑓𝑎(𝑥)𝑑𝑥>𝑡,𝑐𝑎+[𝑅,𝑅]𝑑(𝑥)𝑓𝑎(𝑥)𝑑𝑥𝑡,(,𝑡)epiΥ𝑅.(4.15) Take 𝑡=Υ𝑅() in the second inequality in (4.15); this yields 𝑐𝑎𝐶Υ,𝑓𝑎,𝑅. Consequently, 𝑡sup𝑓𝐵𝑅𝐶Υ,𝑓,𝑅+[𝑅,𝑅]𝑑𝑓(𝑥),(𝑥)𝑑𝑥(4.16) which means that Υ𝑅sup𝑓𝐵𝑅𝐶Υ,𝑓,𝑅+[𝑅,𝑅]𝑑𝑓(𝑥)(𝑥)𝑑𝑥,(4.17) because 𝑡<Υ𝑅() is arbitrary. The inverse inequality is obvious.

Take 𝑓𝐵𝑅, then 𝔼0𝑡exp[𝑅,𝑅]𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥𝑒𝐶Υ,𝑓,𝑅𝑡𝔼0exp[𝑅,𝑅]𝑑.𝑓(𝑥)𝜂(𝑡,𝑥)𝑑𝑥(4.18) Note that [𝑅,𝑅]𝑑𝑓(𝑥)𝜂(𝑡,𝑥)𝑑𝑥=𝑡0𝑓𝐵𝜅𝑠𝑑𝑠,(4.19) where 𝑓(𝑦)=[𝑅,𝑅]𝑑𝑓(𝑥)𝐿(𝑦𝑥)𝑑𝑥(4.20) is a bounded function. Henceforth, by the large deviations result by Kac [50] (see also Theorem 4.1.6 in [51]), we have liminf𝑡1𝑡log𝔼0𝑡exp[𝑅,𝑅]𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥𝐶Υ,𝑓,𝑅+lim𝑡1𝑡log𝔼0exp𝑡0𝑓𝐵𝜅𝑠𝑑𝑠𝐶Υ,𝑓,𝑅+sup𝑔𝑑𝑑𝑓(𝑥)𝑔21(𝑥)𝑑𝑥2𝑑||||𝑔(𝑥)2.𝑑𝑥(4.21) Note that 𝑑𝑓(𝑥)𝑔2(𝑥)𝑑𝑥=[𝑅,𝑅]𝑑𝑓(𝑥)𝑑𝑔2(𝑦)𝐿(𝑥𝑦)𝑑𝑦𝑑𝑥.(4.22) Summarizing our proof, liminf𝑡1𝑡log𝔼0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥sup𝑔𝑑𝐶Υ,𝑓,𝑅+[]𝑅,𝑅𝑑𝑓(𝑥)𝑑𝑔21(𝑦)𝐿(𝑥𝑦)𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑥)2,𝑑𝑥(4.23) for every 𝑅>0, 𝑓𝐵𝑅. We take supremum over 𝑓𝐵𝑅 and get, by Lemma 4.2, liminf𝑡1𝑡log𝔼0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥sup𝑔𝑑Υ𝑅𝑑𝑔21(𝑦)𝐿(𝑦)𝑑𝑦2𝑑||||𝑔(𝑥)2.𝑑𝑥(4.24) Note that by Jensen’s inequality, for every 𝑔𝑑, Υ𝑅𝑑𝑔2=(𝑦)𝐿(𝑦)𝑑𝑦[]𝑅,𝑅𝑑𝜐𝑑𝑔2(𝑦)𝐿(𝑥𝑦)𝑑𝑦𝑑𝑥[]𝑅,𝑅𝑑𝑑𝑔2(𝑦)𝜐(𝐿(𝑥𝑦))𝑑𝑦𝑑𝑥𝑑𝜐(𝐿(𝑥))𝑑𝑥<+,(4.25) which makes it possible to apply Lemma 4.2. Finally, taking supremum over 𝑅>0, we obtain the lower bound in (4.4).

Proof of Theorem 4.1 (the upper bound). Assume first that 𝐿 is continuous and supported by some cube [𝑀,𝑀]𝑑. 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 𝜐(𝑢+𝑣)𝜐(𝑢)𝜐(𝑣)𝜐(0) because of the convexity, and 𝜐(0)=0. Hence, the function 𝜐 satisfies 𝜐(𝑢+𝑣)𝜐(𝑢)+𝜐(𝑣),𝑢,𝑣0.(4.26) Thus, for any 𝑁>𝑀, 𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥=𝑧𝑑[]𝑁,𝑁𝑑𝜐1𝑡𝜂(𝑡,2𝑁𝑧+𝑥)𝑑𝑥[𝑁,𝑁]𝑑𝜐1𝑡𝑧𝑑𝜂(𝑡,2𝑁𝑧+𝑥)𝑑𝑥=[]𝑁,𝑁𝑑𝜐1𝑡̃𝜂(𝑡,𝑥)𝑑𝑥,(4.27) where ̃𝜂(𝑡,𝑥)=𝑡0𝐿𝐵𝜅𝑠𝑥𝑑𝑠,𝐿(𝑥)=𝑧𝑑𝐿(2𝑁𝑧+𝑥).(4.28) Denote by 𝑇𝑁𝑑 the torus of the size 2𝑁, that is, the cube [𝑁,𝑁]𝑑 with the sides identified. Let us denote by 𝐽𝑁 the projection on this torus: by definition, for 𝑥𝑑 its projection 𝐽𝑁(𝑥) is the unique point ̃𝑥𝑇𝑁𝑑 such that 𝑥̃𝑥2𝑁𝑑. Denote 𝐵𝑁𝑠=𝐽𝑁(𝐵), the Brownian motion on the torus 𝑇𝑁𝑑. With this notation in mind, we rewrite the right hand side term in (4.27) [𝑁,𝑁]𝑑𝜐1𝑡̃𝜂(𝑡,𝑥)𝑑𝑥=𝑇𝑁𝑑𝜐1𝑡𝜂𝑁(𝑡,𝑥)𝑑𝑥,𝜂𝑁(𝑡,𝑥)=𝑡0𝐿𝐽𝑁𝐵𝜅𝑠𝑥𝑑𝑠.(4.29) Consider the empirical measures for the Brownian motion on the torus 𝑇𝑁𝑑𝑄𝑁𝑡(1𝐴)=𝑡𝑡01𝐵𝑁𝜅𝑠𝐴𝑇𝑑𝑠,𝐴𝑁𝑑.(4.30) Note that 1𝑡𝜂𝑁(𝑡,𝑥)=𝑇𝑁𝑑𝐿𝐽𝑁(𝑦𝑥)𝑄𝑁𝑡(𝑑𝑦),(4.31) and the mapping 𝜇𝑇𝑁𝑑𝜐𝑇𝑁𝑑𝐿𝐽𝑁(𝑦𝑥)𝜇(𝑑𝑦)𝑑𝑥(4.32) is continuous and bounded on the space of all probability distributions on 𝑇𝑁𝑑 with the metrics of weak convergence. Hence, combination of (4.27), the large deviation principle for 𝑄𝑁𝑡 (Theorem 3 in [52]), and Varadhan’s lemma (Proposition 3.8 in [53]) yields limsup𝑡1𝑡𝔼0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥sup𝑔𝑁𝑑𝑇𝑁𝑑𝜐𝑇𝑁𝑑𝐿𝐽𝑁(𝑥𝑦)𝑔2𝜅(𝑦)𝑑𝑦𝑑𝑥2𝑇𝑁𝑑||||𝑔(𝑦)2,𝑑𝑦(4.33) where 𝑁𝑑=𝑔𝑊12𝑇𝑁𝑑𝑇𝑁𝑑𝑔2(𝑥)𝑑𝑥=1.(4.34) By smooth truncation, it is easy to verify that sup𝑔𝑑𝑑𝜐𝑑𝐿(𝑥𝑦)𝑔2𝜅(𝑦)𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2𝑑𝑦liminf𝑁sup𝑔𝑁𝑑𝑇𝑁𝑑𝜐𝑇𝑁𝑑𝐿𝐽𝑁(𝑥𝑦)𝑔2𝜅(𝑦)𝑑𝑦𝑑𝑥2𝑇𝑁𝑑||||𝑔(𝑦)2,𝑑𝑦(4.35) which completes the proof.
Finally, we remove the additional regularity assumption on 𝐿. Recall the assumption (4.2) and note that one can assume the sequence 𝐿𝑛,𝑛1 to be pointwise increasing, because otherwise, one can take 𝐿𝑛=max𝑘𝑛𝐿𝑛 instead.
Write Δ𝑛=𝐿𝐿𝑛 and 𝜂(𝑡,𝑥)=𝜂𝑛(𝑡,𝑥)+𝜁𝑛(𝑡,𝑥),𝜁𝑛(𝑡,𝑥)=𝑡0Δ𝑛𝐵𝜅𝑠𝑥𝑑𝑠.(4.36) For every 𝛾(0,1), we have by convexity 𝜐1𝑡1𝜂(𝑡,𝑥)𝛾𝜐𝜂𝛾𝑡𝑛1(𝑡,𝑥)+(1𝛾)𝜐𝜁(1𝛾)𝑡𝑛(𝑡,𝑥)(4.37) The Jensen inequality, analogously to (4.6), provides that 𝑑𝜐1𝜁(1𝛾)𝑡𝑛(𝑡,𝑥)𝑑𝑥𝑑𝜐1Δ(1𝛾)𝑛(𝑥)𝑑𝑥.(4.38) Then, from the upper bound with a regular kernel 𝐿𝑛, we obtain limsup𝑡1𝑡𝔼0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥sup𝑔𝑑𝛾𝑑𝜐1𝛾𝑑𝐿𝑛(𝑥𝑦)𝑔2𝜅(𝑦)𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2+𝑑𝑦𝑑𝜐1Δ(1𝛾)𝑛(𝑥)𝑑𝑥,(4.39) for any 𝑛1 and 𝛾(0,1). Passing to the limit first as 𝑛 and then as 𝛾1 completes the proof.

4.2. Proof of Theorem 2.1: Critical Case

The kernel (1.5) has the following scaling property: 𝐾(𝑥)=𝜏𝑝/2𝐾(𝜏1/2𝑥) for any 𝜏>0. Then, by the scaling property of the Brownian motion, 𝜉(𝑡,𝑥)𝑑=𝑡0𝐾𝜏1/2𝐵𝑠𝜏𝑥=𝜏𝑝/(21)0𝑡𝜏𝐾𝐵𝜅𝑠𝜏1/2𝑥𝑑𝑠=𝜏𝑝/(21)𝜉𝑡𝜏,𝑥𝜏1/2.(4.40) Henceforth, the integral under the exponent in the right-hand side of (1.21), after the variable change 𝜏1/2𝑥𝑥, can be written as 𝜏𝑑/2𝑑𝜓𝜏𝑝/21𝛼𝑡𝜉(𝑡𝜏,𝑥)𝑑𝑥.(4.41) We take 𝜏𝑡=𝑡2/𝑑+2. Under such a choice, 𝜏𝑡𝑑/2=𝑡𝜏𝑡=𝑡𝑑/𝑑+2. Observe that 𝜏𝑝/21𝛼𝑡1𝛼,𝑡,(4.42) because of (1.20). By monotonicity of 𝜓, we can change the variables 𝑡𝑑/𝑑+2𝑡 and, applying Theorem 4.1 with 𝐿=𝐾,𝜐(𝑢)=𝜓(𝑢/(𝛼±𝜀)), obtain limsup𝑡𝑡𝑑/(𝑑+2)log𝔼𝔼01exp𝛼𝑡𝑡𝑜𝑉𝐵𝜅𝑠𝑑𝑠sup𝑔𝑑𝜈𝑑𝜓𝜃(𝛼𝜀)𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝜅𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2,𝑑𝑦liminf𝑡𝑡𝑑/(𝑑+2)log𝔼𝔼01exp𝛼𝑡𝑡𝑜𝑉𝐵𝜅𝑠𝑑𝑠sup𝑔𝑑𝜈𝑑𝜓𝜃(𝛼+𝜀)𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝜅𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(4.43)

Passing to the limit as 𝜀 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 𝜓𝑐(𝑢)=𝑐2𝜓(𝑐𝑢), 𝑐>0 and 𝜓0(𝑢)=𝑢2/2.

It can be verified that 𝜓𝑐(𝑢)𝜓0(𝑢) when 𝑐0. In particular, 𝜓=𝜓1𝜓0, and hence 𝜏𝑑/2𝑑𝜓𝜏𝑝/21𝛼𝑡1𝜉(𝑡𝜏,𝑥)𝑑𝑥2𝜏𝑝2𝑑/2𝛼2𝑡𝑑𝜉2=1(𝑡𝜏,𝑥)𝑑𝑥2𝑡2𝜏𝑝𝑑/2𝛼2𝑡𝑑1𝑡𝜏𝜉(𝑡𝜏,𝑥)2𝑑𝑥.(4.44) Choose 𝜏𝑡 in such a way that 𝑡2𝜏𝑡𝑝𝑑/2𝛼2𝑡=𝑡𝜏𝑡,(4.45) that is, 𝜏𝑡=𝛼𝑡4/(𝑑+22𝑝)𝑡2/(𝑑+22𝑝).(4.46) Under such a choice, 𝑡𝜏𝑡=𝛼𝑡4/(𝑑+22𝑝)𝑡(𝑑+42𝑝)/(𝑑+22𝑝),(4.47) and the upper bound in (2.5) follows from the upper bound in (4.4) with 𝜐(𝑢)=𝜈𝑢2/2. Note that Theorem 4.1 cannot be applied to this function 𝜐, because (4.3) fails. We refer here to Theorem 1.1 in [45], which, together with the Varadhan’s lemma, provides (2.5).

To get the lower bound, take 𝜏𝑡 from (4.46) and write 𝜏𝑡𝑑/2𝑑𝜓𝜏𝑡𝑝/21𝛼𝑡𝜉𝑡𝜏𝑡𝜏,𝑥𝑑𝑥=𝑡𝑝2𝑑/2𝛼2𝑡𝜏𝑑/2𝑑𝜓𝑐𝑡1𝑡𝜏𝑡𝜉𝑡𝜏𝑡,𝑥𝑑𝑥,(4.48) with 𝑐𝑡=𝛼𝑡1𝑡𝜏𝑡𝑝/2=𝛼𝑡2𝑝/(𝑑+22𝑝)1𝑡1𝑝/(𝑑+22𝑝)=𝑡(𝑑+2𝑝)/(𝑑+22𝑝)𝛼𝑡(𝑑+2)/(𝑑+22𝑝).(4.49) Condition (1.19) and assumption 𝑝<(𝑑+2)/2 provide 𝑐𝑡0. Then, for every fixed 𝑐>0, we have 𝑐𝑡<𝑐 for 𝑡 large enough, and therefore, liminf𝑡𝛼𝑡4/(𝑑+22𝑝)𝑡(𝑑+42𝑝)/(𝑑+22𝑝)log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠sup𝑔𝑑𝜈𝑑𝜓𝑐𝜃𝛼𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝜅𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2,𝑑𝑦(4.50)

by the lower bound in (4.4) with 𝜐(𝑢)=𝜈𝜓𝑐(𝑢). Since 𝜓𝑐(𝑥)𝜓0(𝑥)=𝑥2/2 monotonously, this proves the lower bound in (2.5).

4.4. Brownian Motion in a Wiener Sheet Potential

Let us recall briefly the construction of the Brownian motion in a Wiener sheet potential, introduced in [45]. Let 𝑊 be a Wiener sheet on 𝑑, independent on 𝐵. Write, for a given shape function 𝐾, 𝑈(𝑥)=𝑑𝐾(𝑦𝑥)𝑊(𝑑𝑦),𝐔𝑡=𝑡0𝑈𝐵𝜅𝑠𝑑𝑠=𝑑𝜉(𝑡,𝑥)𝑊(𝑑𝑥),(4.51) for the Wiener sheet potential and respective total net attraction, obtained by a Brownian particle from this potential. If 𝐾 is of the form (1.5), the potential 𝑈 is not well defined in mean square sense, because 𝐾2(𝑑) for any 𝑝. On the other hand, for 𝑝(𝑑/2,(𝑑+2)/2), one has 𝔼0𝑑𝜉2(𝑡,𝑥)𝑑𝑥<+,(4.52)

and consequently 𝐔𝑡 is well defined.

It follows from [45, Corollary 1.5] (with 𝛽=2) and the variation relation derived in [54, Theorem 1.5] (with 𝑝 there equal to 1) that under conditions of Statement (iii) of Theorem 2.1, lim𝑡𝛼𝑡4/(𝑑+22𝑝)𝑡(𝑑+42𝑝)/(𝑑+22𝑝)log𝔼𝔼01exp𝛼𝑡𝐔𝑡=sup𝑔𝑑𝜃22𝑑𝑑𝑔2(𝑦)||||𝑥𝑦𝑝𝑑𝑦2𝜅𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(4.53) Comparing (4.53) and Statement (iii) of Theorem 2.1 below, we see that in the quantum regime, asymptotic behavior of the Brownian motion in a renormalized Poisson potential is principally determined by respective asymptotics of the Brownian motion in a Wiener sheet potential. Such a relation between these random potentials is quite natural, according to the following version of the central limit theorem.

Proposition 4.4. Let 𝐾 have the form (1.5) with 𝑝(𝑑/2,(𝑑+2)/2). Then, for every 𝑡>0, the distribution of 𝜈1/2𝑡0𝑉𝐵𝜅𝑠𝑑𝑠,(4.54) w.r.t. 0 weakly converges as 𝜈 to the distribution of 𝐔𝑡 w.r.t. 𝑊0, where 𝑊 denotes the distribution of the Wiener sheet 𝑊.

Proof. Characteristic function of (4.54) is equal 𝔼0𝜈exp𝑑𝑒𝑖𝑧𝜈1/2𝜉(𝑡,𝑥)1𝑖z𝜈1/2𝜉(𝑡,𝑥)𝑑𝑥,(4.55) see Proposition 2.2 in [7]. For every 𝑧,𝑡>0,𝑥𝑑, 𝜈𝑒𝑖𝑧𝜈1/2𝜉(𝑡,𝑥)1𝑖𝑧𝜈1/2𝑧𝜉(𝑡,𝑥)22𝜉2(𝑡,𝑥),𝜈.(4.56) In addition, 𝜈|||𝑒𝑖𝑧𝜈1/2𝜉(𝑡,𝑥)1𝑖𝑧𝜈1/2|||𝜉(𝑡,𝑥)𝐶0𝑧2𝜉2(𝑡,𝑥),(4.57) with some constant 𝐶0. Since for 𝑝(𝑑/2,(𝑑+2)/2)𝔼0𝑒𝐶𝑑𝜉2(𝑡,𝑥)𝑑𝑥<,𝑡>0,𝐶>0,(4.58) (see [45]), we have by the dominated convergence theorem that the characteristic functions (4.55) converge pointwise to 𝔼0𝑧exp22𝑑𝜉2(𝑡,𝑥)𝑑𝑥,(4.59) which is just the characteristic function of 𝐔𝑡.

5. The integrated Density of States

5.1. Proof of Proposition 2.6

Statement A. We proceed in two steps. First, we show that under conditions of the proposition, almost all realizations of 𝑄 are bounded from below on a given cube. We consider the case 𝑄=𝑉; the case of a partially renormalized potential is quite analogous. To simplify notation, we assume in the sequel 𝜈=1.

Write 𝑉(𝑥)=𝑑𝐾𝑔([]+𝑥𝑦)𝜔(𝑑𝑦)𝑑𝑦𝑑𝐾𝑔([]𝑥𝑦)𝜔(𝑑𝑦)𝑑𝑦.(5.1) The function 𝐾𝑔 is supported by some ball {𝑥|𝑥|𝑅}, which brings the lower bound 𝜔𝑑𝑅𝑑 for the first summand. Henceforth, without loss of generality, we can remove this term. In what follows, we consider a renormalized Poisson potential with 𝐾𝑔 instead of 𝐾. To simplify the notation, we just consider 𝑉 assuming that 𝐾 is bounded, Lipschitz, and belongs to 𝑊12(𝑑).

It is a simple observation that for a smooth compactly supported function 𝐿𝑑, every realization of respective renormalized Poisson potential belongs to 𝑊12(𝑑), and 𝑑[]=𝐿(𝑥𝑦)𝜔(𝑑𝑥)𝑑𝑦𝑑[].𝐿(𝑥𝑦)𝜔(𝑑𝑦)𝑑𝑦(5.2) This, by usual approximation argument, provides that almost all realizations of the renormalized Poisson potential with kernel 𝐾 belong to 𝑊12(𝑑), and (5.2) holds true in Sobolev sense for 𝐾=𝐿.

Since 𝐾 is Lipschitz, 𝐾 is bounded; the function 𝐾 is bounded, as well. Then, Proposition 2.7 in [7] provides ||𝔼exp||+||𝑉(0)||𝑉(0)<+.(5.3) By shift invariance, this gives 𝔼𝑈||exp||+||𝑉(𝑥)||𝑉(𝑥)𝑑𝑥<+.(5.4) Therefore, almost all realizations of 𝑉 belong to 𝑊1(𝑈)=𝑝>1𝑊1𝑝(𝑈) and henceforth are continuous by Sobolev’s inclusion theorem (e.g., [55]). In particular, these realizations are bounded.

The second part of the proof is represented by the following deterministic lemma.

Lemma 5.1. Let 𝑄𝑑 be a function bounded from below and 𝑈 a given cube.
Denote 𝑄𝑁=𝑄𝑁,(5.5) and consider the Schrödinger operators 𝐻𝑄𝑁𝑈, 𝑁1 with the Dirichlet boundary conditions, defined by (2.14) with 𝑄 replaced by 𝑄𝑁.
Then, (i)𝑅𝑄𝑁𝑡,𝑈=𝑒𝑡𝐻𝑄𝑁𝑈,𝑡0 converge strongly as 𝑁 to a continuous semigroup 𝑅𝑄𝑡,𝑈,𝑡0 of self-adjoint operators in 𝐿2(𝑈,𝑑𝑥). 𝑅𝑄𝑡,𝑈, 𝑡0 admits the Feynman-Kac representation (2.15),(ii)every operator 𝑅𝑄𝑡,𝑈, 𝑡0 is of trace class,(iii)for the generator 𝐻𝑄𝑈 of the semigroup 𝑅𝑄𝑡,𝑈, 𝑡0, the function (2.17) is well defined, and its Laplace transform admits the representation 𝑒𝜆𝑡𝑑𝑁𝑄𝑈(𝜆)=(2𝜋𝜅𝑡)𝑑/21||𝑈||𝑈𝔼𝜅t0,0exp𝑡0𝑄𝐵𝜅𝑠𝜒+𝑥𝑑𝑠𝑈,𝑡𝐵+𝑥𝑑𝑥.(5.6)

Proof. For every 𝑁, the semigroup 𝑅𝑄𝑁𝑡,𝑈=𝑒𝑡𝐻𝑄𝑁𝑈,𝑡0 admits the Feynman-Kac representation (2.15). An alternative form of this representation ([1, page 13]) is that 𝑅𝑄𝑁t,𝑈 is an integral operator with the kernel 𝑟𝑄𝑁𝑡,𝑈(𝑥,𝑦)=𝑝𝑡(𝑥,𝑦)𝔼𝑡𝑥,𝑦exp𝑡0𝑄𝑁𝐵𝜅𝑠𝜒𝑑𝑠𝑈,𝑡𝐵,(5.7) where 𝑝𝑡(𝑥,𝑦)=(2𝜋𝜅𝑡)𝑑/2𝑒|𝑥𝑦|2/2𝜅𝑡(5.8) is the transition probability density for the process 𝐵𝜅𝑠, 𝑠0, and 𝔼𝑡𝑥,𝑦 denotes the expectation w.r.t. law of the Brownian bridge which takes values 𝑥 and 𝑦 at 𝑠=0 and 𝑠=𝑡, respectively. By Propositions 3.2 and 3.5 in [1], for a given 𝑁, the function 𝑟𝑄𝑁𝑡,𝑈 is continuous and symmetric. Then, by the monotone convergence theorem, there exists a monotonous limit 𝑟𝑄𝑡,𝑈(𝑥,𝑦)=lim𝑁𝑟𝑄𝑁𝑡,𝑈(𝑥,𝑦)=(2𝜋𝜅𝑡)𝑑/2𝑒|𝑥𝑦|2/(2𝜅𝑡)𝔼𝑡𝑥,𝑦exp𝑡0𝑄𝐵𝜅𝑠𝜒𝑑𝑠𝑈,𝑡𝐵,(5.9) which is bounded and symmetric. Integral operators 𝑅𝑄𝑁𝑡,𝑈 converge to the operator 𝑅𝑄𝑡,𝑈 with kernel 𝑟𝑄𝑡,𝑈 in the Hilbert-Schmidt norm, because respective kernels converge in 𝐿2(𝑈×𝑈,𝑑𝑥). This proves Statement (i) immediately. Since 𝑅𝑄𝑡,𝑈=𝑅𝑄𝑡/2,𝑈𝑅𝑄𝑡/2,𝑈,(5.10) and every 𝑅𝑄𝑡/2,𝑈 is a Hilbert-Schmidt operator, the operator 𝑅𝑄𝑡,𝑈 is of trace class. This proves Statement (ii).
We have just proved that 𝑅𝑄𝑡,𝑈 is of trace class, which means that it has a purely point spectrum and Trace𝑅𝑄𝑡,𝑈=𝑘𝜆𝑘,𝑄,𝑡<+,(5.11) where {𝜆𝑘,𝑄,𝑡} denote eigenvalues of 𝑅𝑄𝑡,𝑈, counted with their multiplicities. By the spectral decomposition theorem, this yields that the generator 𝐻𝑄𝑈 of the semigroup 𝑅𝑄𝑡,𝑈, 𝑡0 has a purely point spectrum locally finite on every interval (,𝜆]. In addition, 𝜆𝑘,𝑄,𝑡=𝑒𝜆𝑘,𝑄𝑡, where {𝜆𝑘,𝑄} are respective eigenvalues of 𝐻𝑄𝑈, counted with their multiplicities. Therefore, the function (2.17) is well defined, and its Laplace transform has the form 𝑒𝜆𝑡𝑑𝑁𝑄𝑈(1𝜆)=||𝑈||𝑘𝑒𝑡𝜆𝑘,𝑈=1||𝑈||Trace𝑅𝑄𝑡,𝑈=1||𝑈||𝑈×𝑈𝑟𝑄𝑡/2,𝑈(𝑥,𝑦)𝑟𝑄𝑡/2,𝑈(𝑦,𝑥)𝑑𝑥𝑑𝑦,(5.12) in the last equality, we have used the standard relation (e.g., [56, Chapter III, Section  9]) Trace𝐴𝐴=𝐴2HS,(5.13) with the Hilbert-Shmidt norm of the operator 𝐴 in the right hand side. By the Feynman-Kac representation (2.15) and the Markov property of the Brownian bridge, the last integral can be written as (2𝜋𝜅𝑡)(𝑑/2)1||𝑈||𝑈𝔼𝜅𝑡𝑥,𝑥exp𝑡0𝑄𝐵𝜅𝑠𝜒𝑑𝑠𝑈,𝑡𝐵𝑑𝑥.(5.14) which completes the proof of the lemma.

Statement B (sketch of the proof). In the second part of the proposition, the classic argument which goes back to [21] is applicable. In order to keep the exposition self-sufficient, we give a brief sketch of this argument here.
The random fields 𝑄=𝑉 and 𝑄=𝑉 are ergodic (or metrically transitive) in the sense that the 𝜎-algebra generated by functionals, invariant w.r.t. the transformations 𝑆𝑄()𝑄(+),𝑑(5.15) is degenerate. The argument here is a straightforward modification of the classic one for one-dimensional moving average integrals; see Theorem 1.1 and Example 3 in Chapter XI, [57]. Then, the Birkhoff’s ergodic theorem (its modification for random fields, e.g., Chapter 6.5 in [58]) yields that for any integrable function 𝑓 on the space of realizations of the field 𝑄, lim𝑈𝑑1||𝑈||𝑈𝑓(𝑄(+𝑥))𝑑𝑥=𝔼𝑓(𝑄),(5.16) both almost surely and in mean sense.
By Proposition 2.7 in [7], 𝔼𝑒𝑐𝑄(0)<+ for every 𝑐>0. This, by the Jensen inequality, provides 𝔼𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠𝑑𝑠<+,𝑡0,(5.17) the argument here is the same as at the beginning of Section 3.1. Then, (5.16) applied to the function 𝑓𝑞𝔼𝜅𝑡0,0exp𝑡0𝑞𝐵𝜅𝑠𝑑𝑠(5.18) yields lim𝑈𝑑1||𝑈||𝑈𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠+𝑥𝑑𝑠𝑑𝑥=𝔼𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠𝑑𝑠(5.19) both in mean and in a.s. sense. Straightforward calculation shows that with probability 1, lim𝑈𝑑1||𝑈||𝑈𝜒𝑈,𝑡𝐵+𝑥𝑑𝑥=1.(5.20) Together with mean convergence (5.19), this provides lim𝑈𝑑1||𝑈||𝑈𝔼𝜅𝑡0,0exp𝑡0𝑄𝐵𝜅𝑠+𝑥𝑑𝑠1𝜒𝑈,𝑡𝐵+𝑥𝑑𝑥=0,(5.21) which completes the proof.

5.2. Proof of Theorem 2.5
5.2.1. Classic Regime

Arguments in Section 3.1 are process insensitive. Henceforth, the upper bounds in (2.3), (2.11), and (3.4) hold true with 𝔼0 replaced by 𝔼𝜅𝑡0,0.

On the other hand, Brownian bridge measures enjoy the scaling property similar to the Brownian one ([1, page 140]): its law 𝑡0,0 is the image of measure of 10,0 under the map 𝑤()𝑡𝑤𝑡.(5.22) In addition, the Brownian bridge measure 10,0 has the small balls asymptotics similar to the Brownian one [59] 𝜀2log10,0sup𝑠[0,1]||||1𝐵(𝑠)𝜀2𝑗2(𝑑2)/2,(5.23) where 𝑗(𝑑2)/2 is the smallest positive root of the Bessel function 𝐽(𝑑2)/2. Henceforth, the heavy small ball argument from Section 3.2 can be applied to get the lower bounds in (2.3), (2.11), and (3.4) with 𝔼𝜅𝑡0,0 instead of 𝔼0.

The argument which deduce (2.12) from (3.4) is process insensitive.

5.2.2. Quantum Regime

The upper bound in (4.4) with 𝔼𝜅𝑡0,0 instead of 𝔼0 can be deduced from the same upper bound in its original form. In the proof, we combine the standard trick based on the Markov property of the Brownian bridge (e.g., Lemma 3 in [27]) with the “universal” upper bound provided by the convexity; see the end of Section 4.1.

Write 𝜂(𝑡,𝑥)=𝜂(𝑡1,𝑥)+𝜁(𝑡,𝑥),𝜁(𝑡,𝑥)=𝑡𝑡1𝐿𝐵𝜅𝑠𝑥𝑑𝑠.(5.24) For every 𝛾(0,1), we have by convexity 𝜐1𝑡1𝜂(𝑡,𝑥)𝛾𝜐1𝛾𝑡𝜂(𝑡1,𝑥)+(1𝛾)𝜐(1𝛾)𝑡𝜁(𝑡,𝑥).(5.25) Analogously to (4.6), we have 𝑑𝜐1(1𝛾)𝑡𝜁(𝑡,𝑥)𝑑𝑥𝑡𝑡1𝑑𝜐1𝐿𝐵(1𝛾)𝑡𝜅𝑠=𝑥𝑑𝑥𝑑𝑠𝑑𝜐1(1𝛾)𝑡𝐿(𝑥)𝑑𝑥.(5.26) The last term vanishes when 𝑡. Therefore, for every 𝛾(0,1), limsup𝑡1𝑡𝔼𝜅𝑡0,0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥limsup𝑡1𝑡𝔼𝜅𝑡0,0exp𝛾𝑡𝑑𝜐1.𝛾𝑡𝜂(𝑡1,𝑥)𝑑𝑥(5.27) On the other hand, applying the Markov property at time 𝑡1, we arrive at 𝔼𝜅𝑡0,0exp𝛾𝑡𝑑𝜐1𝛾𝑡𝜂(𝑡1,𝑥)𝑑𝑥=𝔼0exp𝛾𝑡𝑑𝜐1𝑝𝛾𝑡𝜂(𝑡1,𝑥)𝑑𝑥1𝐵𝜅(𝑡1),0𝔼0exp𝛾𝑡𝑑𝜐1𝑝𝛾𝑡𝜂(𝑡,𝑥)𝑑𝑥1𝐵𝜅(𝑡1).,0(5.28) Because 𝑝1(𝑥,𝑦)(2𝜋𝜅)𝑑/2,𝑥,𝑦𝑑,(5.29)

we get from the upper bound in (4.4) that for every 𝛾(0,1), limsup𝑡1𝑡𝔼𝜅𝑡0,0𝑡exp𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥sup𝑔𝑑𝛾𝑑𝜐1𝛾𝑑𝐿(𝑥𝑦)𝑔2𝜅(𝑦)𝑑𝑦𝑑𝑥2𝑑||||𝑔(𝑦)2.𝑑𝑦(5.30) Passing to the limit 𝛾1 completes the proof of the upper bound.

The lower bound in (4.4) with 𝔼𝜅𝑡0,0 instead of 𝔼0 can be obtained by almost the same argument, as it was used to prove lower bound in (4.4) in its initial form. Only minor changes of the argument are required; let us discuss these changes.

Consider the large deviation result by Kac, which was the basic point in the proof of the lower bound lim𝑡1𝑡log𝔼0expt0𝑓𝐵𝜅𝑠𝑑𝑠𝐶Υ,𝑓,𝑅+sup𝑔𝑑𝑑𝑓(𝑥)𝑔21(𝑥)𝑑𝑥2𝑑||||𝑔(𝑥)2.𝑑𝑥(5.31) Note that under 0, 𝐵𝜅𝑠𝑠𝐵𝜅𝑡𝜅𝑡(5.32) is a Gaussian process with the covariance 𝜅(𝑠𝑠)𝜅(𝑠𝑠/𝑡) and, therefore, has the distribution 𝑡0,0 ([1, page 140]). Then, for any Lipschiz continuous bounded function 𝑓, (5.31) holds true with 𝔼𝜅𝑡0,0 instead of 𝔼0.

Note that the statement of the Lemma 4.2 still holds true when 𝐵𝑅 is replaced with any class of functions 𝐾𝐵𝑅 separating points in 1([𝑅,𝑅]𝑑); in particular, one can take 𝐾=𝐵𝐿𝑅, the class of Lipschitz continuous bounded functions. Clearly, for 𝑓𝐵𝐿𝑅 the function 𝑓(𝑦)=[𝑅,𝑅]𝑑𝑓(𝑥)𝐿(𝑦𝑥)𝑑𝑥(5.33) is Lipschitz continuous and bounded. Applying the modified (5.31) and proceeding literally as in the proof of the lower bound in Theorem 4.1, we get the required lower bound.

Once we have proved the modified large deviation asymptotics (4.4), we can repeat the arguments from Sections 4.2 and 4.3 (with the the scaling property of the Brownian bridge used instead of the same property of the Brownian motion) and deduce (2.4), (2.5) with 𝔼0 replaced by 𝔼𝜅𝑡0,0.

5.3. Proof of Theorem 2.7

Theorem 9.7 in [42], Chapter IV gives (2.20) as a straightforward corollary of (2.3). Because 𝑡𝑡log𝑡 is a regularly varying function of the order 1, this theorem is not applicable when (2.21) is considered.

From (2.13) (with 𝔼𝜅𝑡0,0 instead of 𝔼0), we have log𝑒(𝜆+𝜈𝜔𝑑𝜃(log+Eu))𝑡𝑑𝑁𝑉(𝜆)=𝜈𝜔𝑑𝜃𝑡log𝑡+𝑜(𝑡),𝑡+.(5.34) Henceforth, the proof of (2.21) by elementary transformations can be reduced to the proof of the following lemma.

Lemma 5.2. Let, for a nonnegative and nondecreasing function 𝑁(𝜆), 𝜆, log𝑒𝜆𝑡𝑑𝑁(𝜆)=𝑎𝑡log𝑡+𝑜(𝑡),𝑡+,(5.35) with some 𝑎>0.
Then, 𝜆log𝑁(𝜆)=𝑎exp𝑎1(1+𝑜(1)),𝜆.(5.36)

Proof. The upper bound, in a standard way, is provided by the Chebyshev inequality: for every 𝑡>0, 𝜆<0, log𝑁(𝜆)𝜆𝑡+log𝑒𝜒𝑡𝑑𝑁(𝜒).(5.37) Take 𝑡𝜆=exp[𝜆/𝑎1], the solution of the minimization problem 𝑡𝜆=argmin(𝜆𝑡+𝑎𝑡log𝑡).(5.38) Clearly, 𝑡𝜆+, 𝜆. By (5.35) and (5.37), log𝑁(𝜆)𝑎𝑡𝜆𝑡+𝑜𝜆,𝜆,(5.39) which gives the upper bound in (5.36).
Assume that the lower bound in (5.36) fails; that is, there exist 𝑏>𝑎 and a sequence 𝜆𝑛 such that 𝜆log𝑁𝑛𝜆𝑏exp𝑛𝑎1,𝑛1.(5.40) Fix 𝑐<𝑎 and 𝛿>0, which will be specified below. Since the upper bound in (5.36) is already proved, there exists Λ𝑐<0 such that 𝜆log𝑁(𝜆)𝑐exp𝑎1,𝜆<Λ𝑐.(5.41) Let 𝑛 be large enough for 𝜆𝑛<Λ𝑐. Denote 𝑡𝑛=𝑒(𝜆𝑛𝛿)/𝑎1 and write 𝑒𝜆𝑡𝑛𝑑𝑁(𝜆)=Λ𝑐𝑒𝜆𝑡𝑛𝑒𝑁(𝜆)𝑑𝜆+Λ𝑐𝑡𝑛+Λ𝑐𝑒𝜆𝑡𝑛=𝑑𝑁(𝜆)𝜆𝑛2𝛿𝑒𝜆𝑡𝑛𝑁(𝜆)𝑑𝜆+𝜆𝑛𝜆𝑛2𝛿𝑒𝜆𝑡𝑛𝑁(𝜆)𝑑𝜆+Λ𝑐𝜆𝑛𝑒𝜆𝑡𝑛+𝑒𝑁(𝜆)𝑑𝜆Λ𝑐𝑡𝑛+Λ𝑐𝑒𝜆𝑡𝑛𝑑𝑁(𝜆)=𝐼1𝑛+𝐼2𝑛+𝐼3𝑛+𝐼4𝑛.(5.42) Let us estimate 𝐼1𝑛𝐼4𝑛. Clearly, 𝐼4𝑛𝐶𝑒Λ𝑐𝑡𝑛,(5.43) with appropriate constant 𝐶. Assumption (5.40) yields, via monotonicity, 𝐼2𝑛𝜆2𝛿exp𝑛𝑡2𝛿𝑛𝑏𝑒𝜆𝑛/(𝑎1)=2𝛿exp𝛿+𝑎log𝑡𝑛𝑡+𝑎𝑛𝑏𝑒𝛿/𝑎𝑡𝑛=2𝛿exp𝑎𝑡𝑛log𝑡𝑛𝑏𝑒𝛿/𝑎𝑡𝑎𝛿𝑛,(5.44) here we have used the relation 𝜆𝑛=𝑎log𝑡𝑛𝑎+𝛿,(5.45) which comes from the definition of 𝑡𝑛.
Consider the function Θ𝜆𝜆𝑡𝑛𝑐𝑒𝜆/𝑎1. Straightforward computation shows that assuming 𝑐>𝑎𝑒𝛿/𝑎,(5.46) its derivative is increasing on (,𝜆𝑛2𝛿], and Θ𝑛𝜆𝑛=𝑐2𝛿𝑎𝑒𝛿/𝑎𝑡1𝑛>1,(5.47) for 𝑛 large enough. Then, with (5.41) in mind, we get 𝐼1𝑛𝜆𝑛2𝛿𝑒𝜆𝜆𝑛+2𝛿𝑒Θ𝑛(𝜆𝑛2𝛿)𝑑𝜆=𝑒Θ𝑛(𝜆𝑛2𝛿)=exp𝑎𝑡𝑛log𝑡𝑛𝑐𝑒𝛿/𝑎𝑡𝑎𝛿𝑛.(5.48) Similar argument leads to 𝐼3𝑛𝜆+𝑛𝑒𝜆+𝜆𝑛𝑒Θ𝑛(𝜆𝑛)𝑑𝜆=𝑒Θ𝑛(𝜆𝑛)=exp𝑎𝑡𝑛log𝑡𝑛𝛿+𝑐𝑒𝛿/𝑎𝑡𝑎𝑛.(5.49) Now, we can finalize the proof. Take 𝛿>0 such that 𝑏𝑒𝛿/𝑎𝑎𝛿>0. Note that 𝑎𝑒𝛿/𝑎𝑎𝛿>0,𝛿+𝑎𝑒𝛿/𝑎𝑎>0.(5.50) Therefore, 𝑐<𝑎 can be chosen in such a way that (5.46) holds true and 𝑐𝑒𝛿/𝑎𝑎𝛿>0,𝛿+𝑐𝑒𝛿/𝑎𝑎>0.(5.51) Under such a choice of the constants 𝛿 and 𝑐, (5.43), (5.44) (5.48), and (5.49) provide that for 𝑛 large enough, log𝑒𝜆𝑡𝑛𝑑𝑁(𝜆)𝑎𝑡𝑛log𝑡𝑛𝜀𝑡𝑛,(5.52) with some positive 𝜀. This contradicts to (5.35) and proves that assumption (5.40) is impossible.

Remark 5.3. In the proof of the lower bound in Lemma 5.2, we have combined the upper bound from the same lemma with the estimates, typical for the Laplace method (e.g., [60]). Note that such a structure of the proof is similar to the one for the Gärtner-Ellis theorem (see Section 1.1 in [51]) although we cannot deduce the statement of the lemma from the Gärtner-Ellis theorem directly.

5.4. Proof of Theorem 2.8

Our argument is based on the following version of the Gärtner-Ellis theorem.

Lemma 5.4. Consider a sequence 𝑁𝑛, 𝑛1 of nonnegative monotonous functions on , which vanish at , and assume that there exist 𝑎>1, 𝑐>0, and a sequence Υ𝑛+ such that 1Υ𝑛log𝑒𝜇Υ𝑛𝑥𝑑𝑁𝑛(𝑥)𝐼(𝜇)=𝑐𝜇𝑎,𝑛,𝜇>0.(5.53) Then, 1Υ𝑛log𝑁𝑛(𝑥)=𝐼(𝑥)(1+𝑜(1)),𝑛,(5.54) uniformly by 𝑥[𝐴,𝐵] for every [𝐴,𝐵](0,+). Here, 𝐼(𝑥)=sup𝜇>0[]𝜇𝑥𝐼(𝜇)=(𝑐(𝑎1))1/(𝑎1)𝑎1𝑎𝑎/(𝑎1)𝑥𝑎/(𝑎1).(5.55)

The only difference between conditions of Lemma 5.4 and standard assumptions of the Gärtner-Ellis theorem is that functions 𝑁𝑛 are not assumed to be distribution functions and are allowed to define nonprobability measures. One can see easily that this difference is inessential, and Lemma 5.4 can be proved in the same way with the Gärtner-Ellis theorem (or with Lemma 5.2 above).

Corollary 5.5. Let the field {𝑁(𝜆,𝛾),𝜆,𝛾>0} be such that (a)every function 𝑁(,𝛾), 𝛾>0 is nonnegative and nondecreasing,(b)for every 𝑡>0, 𝑁(𝑡,𝛾)=𝑒𝜆𝑡𝑁(𝑑𝜆,𝛾)<,(5.56)(c)for given 𝑎>1, 𝑏, 𝑐>0, and given sequences 𝜆𝑛>0, 𝛾𝑛>0, 𝑛1 with 𝜆𝑎𝑛𝛾𝑛𝑏, 𝑁log𝜇𝜆𝑛1/(𝑎1)𝛾𝑛𝑏/(𝑎1),𝛾𝑛=𝑐𝜇𝑎𝜆𝑛𝑎/(𝑎1)𝛾𝑛𝑏/(𝑎1)(1+𝑜(1)),𝑛,𝜇>0.(5.57) Then, log𝑁𝜆𝑛𝑥,𝛾𝑛=(𝑐(𝑎1))1/(𝑎1)𝑎1𝑎𝑎/(𝑎1)𝜆𝑛𝑎/(𝑎1)𝛾𝑛𝑏/(𝑎1)𝑥𝑎/(𝑎1)(1+𝑜(1)),𝑛,(5.58) uniformly by 𝑥[𝐴,𝐵] for every [𝐴,𝐵](0,+).

Proof. Put Υ𝑛=𝜆𝑛𝑎/(𝑎1)𝛾𝑛𝑏/(𝑎1), 𝑁𝑛𝜆(𝑥)=𝑁𝑛𝑥,𝛾𝑛,𝑥.(5.59) Because 𝑎>1 and 𝜆𝑎𝑛𝛾𝑛𝑏, we have Υ𝑛. Condition (5.57) provides (5.53). Henceforth, (5.58) follows by Lemma 5.4.

Now, we can finalize the proof of Theorem 2.8.

Statement  (i)
Take 𝑑𝑎=𝑏=𝑝,𝑐=𝜈𝜔𝑑𝜃𝑑/𝑝𝑝Γ𝑑𝑝2𝑝𝑑𝑝.(5.60) For given sequences 𝜆𝑛<0, 𝛾𝑛>0, 𝑛1 and arbitrary 𝜇>0 denote 𝑡𝑛=𝜇𝜆𝑛1/(𝑎1)𝛾𝑛𝑏/(𝑎1),𝛼𝑡𝑛=1𝛾𝑛.(5.61) Condition (𝜆𝑛)𝑝/𝑑/𝛾𝑛 yields 𝑡𝑛, and condition (𝜆𝑛)(𝑑+2𝑝)/2/𝛾𝑛 yields 𝛼𝑡𝑛=𝑜(𝑡𝑛). Therefore, (5.57) is provided by (2.3). In addition, we have (𝜆𝑛)/𝛾𝑛 because 𝑝/𝑑<1, (𝑑+2𝑝)/2>1, and consequently (𝜆𝑛)𝑎𝛾𝑛𝑏. Applying Corollary 5.5 with 𝑥=1 and 𝜆𝑛 replaced by 𝜆𝑛, we obtain the required statement.

Statement  (ii)
Take 𝑎=𝑑+42𝑝4𝑑+22𝑝,𝑏=𝑑+22𝑝,𝑐=C2,(5.62) and keep the notation (5.58). Condition (𝜆𝑛)(𝑑+42𝑝)/4/𝛾𝑛 yields (𝜆𝑛)𝑎𝛾𝑛𝑏. Condition (𝜆𝑛)(𝑑+2𝑝)/2/𝛾𝑛0 yields 𝑡𝑛(𝑑+2𝑝)/(𝑑+2)=𝑜(𝛼𝑡𝑛). Finally, these two conditions yield 𝜆𝑛0, and consequently (𝜆𝑛)(𝑑+22𝑝)/4/𝛾𝑛 which is equivalent to 𝑡𝑛.
Therefore, (5.57) is provided by (2.5). Applying Corollary 5.5 with 𝑥=1 and 𝜆𝑛 replaced by 𝜆𝑛, we obtain the required statement.

Statement  (iii)
In the critical case, one can transform easily (2.4) to lim𝑡𝛼𝑡𝑡𝑑/𝑝log𝔼𝔼01exp𝛼𝑡𝑡0𝑉𝐵𝜅𝑠𝑑𝑠=C𝜓.(5.63) Using this relation and following the proof of Statement (i) with appropriate modifications, we obtain the required statement.

Acknowledgments

The authors are grateful for the referee for useful comments. Xia Chen Research partially supported by NSF Grant no. DMS-0704024. Alexey Kulik Research partially supported by Ukrainian Nat. Acad. of Sciences, Grant no. 0107U002029.