International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 803683 | 43 pages |

Asymptotics of Negative Exponential Moments for Annealed Brownian Motion in a Renormalized Poisson Potential

Academic Editor: Nikolai Leonenko
Received24 Dec 2010
Accepted06 Apr 2011
Published28 Jun 2011


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)2𝑑𝑦liminf𝑁sup𝑔𝑁𝑑𝑇𝑁𝑑𝜐