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

Abstract

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

1. Introduction

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

Let 𝜔(𝑑𝑥) be a Poisson field in ℝ^𝑑 with intensity measure 𝜈𝑑𝑥, and let 𝐵 be an independent Brownian motion in ℝ^𝑑. Throughout, ℙ and 𝔼 denote the probability law and the expectation, respectively, generated by the Poisson field 𝜔(𝑑𝑥), while ℙ_𝑥 and 𝔼_𝑥 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 ℝ^𝑑 (known as a shape function), define the respective random function (known as a Poisson potential) 𝑉(𝑥)=_ℝ𝑑𝐾(𝑦−𝑥)𝜔(𝑑𝑦),(1.1) which heuristically represents the net force at 𝑥∈ℝ^𝑑 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 𝜇_𝑡,𝜔 defined by 𝑑𝜇_𝑡,𝜔𝑑ℙ₀=1𝑍_𝑡,𝜔exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,𝑍_𝑡,𝜔=𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠.(1.2) Here, 𝜅 is a positive parameter, responsible for the time scaling 𝑠↦𝜅𝑠, 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 𝜇_𝑡 is defined by 𝑑𝜇_𝑡𝑑ℙ⊗ℙ₀=𝑍1^𝑡exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,𝑍_𝑡=𝔼⊗𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠.(1.3) Heuristically, the integral ^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠(1.4) measures the total net attraction to which the Brownian particle is subject up to the time 𝑡, and henceforth, under the law 𝜇_𝑡,𝜔 or 𝜇_𝑡, 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,𝑥∈ℝ^𝑑,(1.5) which clearly is both locally unbounded and supported by whole ℝ^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 𝑥∈ℝ^𝑑 when 𝑝≤𝑑.

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)=_ℝ𝑑𝐾(𝑦−𝑥)[𝜔(𝑑𝑦)−𝜈𝑑𝑦].(1.6) Assume for a while that 𝐾 is locally bounded and compactly supported. Then, 𝑉(𝑥)==_ℝ𝑑𝐾(𝑦−𝑥)[𝜔(𝑑𝑦)−𝜈𝑑𝑦_ℝ𝑑]=𝑉(𝑥𝐾)−(𝑦𝜈−𝑥_ℝ𝑑)𝜔(𝑑𝑦)−𝜈_ℝ𝑑𝐾(𝑦−𝑥)𝑑𝑦𝐾(𝑦)𝑑𝑦,(1.7)

that is, 𝑉−𝑉=const. Consequently, replacing 𝑉 by 𝑉 in (1.2) and (1.3) does not change the measures 𝜇_𝑡,𝜔 and 𝜇_𝑡, because both the exponents therein and the normalizers 𝑍_𝑡,𝜔 and 𝑍_𝑡 are multiplied by the same constant 𝑒^{𝑡𝔼𝑉(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 𝑑𝜇_𝑡,𝜔𝑑ℙ₀=1𝑍_𝑡,𝜔𝑑exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,𝑍_𝑡,𝜔=𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,(1.8)𝜇_𝑡𝑑ℙ⊗ℙ₀=𝑍1_𝑡exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,𝑍_𝑡=𝔼⊗𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,(1.9) see [7, Corollary 1.3]. We use separate notation 𝜇_𝑡,𝜔, 𝜇_𝑡 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^ℎ, partially renormalized at the level ℎ; see [7, Chapter 6]. By definition, 𝑉^ℎ(𝑥)=_ℝ𝑑(𝐾(𝑦−𝑥)−ℎ)₊𝜔(𝑑𝑦)+_ℝ𝑑(𝐾(𝑦−𝑥)∧ℎ)[𝜔(𝑑𝑦)−𝜈𝑑𝑦],(1.10) where ℎ∈[0,∞] is a renormalization level. Clearly, 𝑉⁰=𝑉,𝑉^∞=𝑉. It is known (see [7, Chapter 6]) that 𝑉^ℎ is well defined for every ℎ∈(0,+∞) as soon as 𝑉^ℎ′ is well defined for some ℎ^∈[0,+∞], and in that case, there exists a constant 𝐶_𝐾,ℎ,ℎ^′ such that 𝑉^ℎ−𝑉^ℎ′=𝜈𝐶_𝐾,ℎ,ℎ^′. This makes it possible to define the respective Gibbs measures in a uniform way, replacing 𝑉 in (1.2), (1.3) by 𝑉^ℎ with (any) ℎ∈(0,+∞). In addition, such a definition extends the class of shape functions: for 𝐾 given by (1.5), 𝑉^ℎ with ℎ∈(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 𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠.(1.11) This problem is clearly relevant with the model discussed above: in the particular case 𝜅=1, 𝛼_𝑡≡1, this is just the natural question about the limit behavior of the normalizer 𝑍_𝑡 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^ℎ with ℎ∈(0,+∞).

The function 𝛼_𝑡 in (1.11) appears, on one hand, because of our further intent to study in further publications the a.s. behavior ^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠,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 𝛼_𝑡 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 𝜕_𝑡𝑢(𝑡,𝑥𝑢)=(0,𝜅𝑥Δ)𝑢=(𝑡1,,𝑥)±𝑥∈𝑄ℝ(𝑥_𝑑)𝑢(𝑡,𝑥),.(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);𝑥∈ℤ^𝑑} 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)=𝔼_𝑥exp±^𝑡0𝑄𝐵_2𝜅𝑠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 [10–20] as a partial list of the publications that deal with various asymptotic topics related to the parabolic Anderson model.

Another problem related to our main one is the so called Lifshitz tails asymptotic behavior of the integrated density of states function 𝑁 of a random Schrödinger operator of the type 𝐻=−𝜅2Δ+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)=lim_{𝑈↑ℝ𝑑}1||𝑈||_𝑘1_{𝜆𝑘,𝑈≤𝜆},(1.16) where {𝜆_𝑘,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 ℝ^𝑑, and the limit pass is made w.r.t. a sequence of cubes which has same center and extends to the whole ℝ^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), 𝜆↘𝜆₀, studied extensively in the literature. The name of the problem goes back to the papers by Lifshitz [23, 24]; we also give [1, 21, 22, 25–44] as a partial list of references on the subject.

Connection between the Lifshitz tails asymptotics for the IDS function 𝑁, and the problem discussed above is provided by the representation for the Laplace transform of 𝑁_ℝ𝑒^−𝜆𝑡𝑑𝑁(𝜆)=(2𝜋𝜅𝑡)^−𝑑/2𝔼⊗𝔼^𝜅𝑡0,0exp−^𝑡0𝑄𝐵_𝜅𝑠𝑑𝑠,𝑡≥0.(1.17) Here, 𝔼^𝜅𝑡0,0 denotes the distribution of the Brownian bridge, that is, the Brownian motion conditioned by 𝐵_𝜅𝑡=0. Our estimates for (1.11) appear to be process insensitive to some extent and remain true with 𝔼₀ in (1.11) replaced by 𝔼^𝜅𝑡0,0. 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 𝑡𝛼_𝑡=𝑜𝑡^{(𝑑+2−𝑝)/(𝑑+2)},𝑡⟶∞,(1.18)_{(𝑑+2−𝑝)/(𝑑+2)}𝛼=𝑜𝛼_𝑡,𝑡⟶∞,(1.19)_𝑡∼𝛼𝑡^{(𝑑+2−𝑝)/(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 𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠=𝔼₀exp𝜈_ℝ𝑑𝜓1𝛼_𝑡𝜉(𝑡,𝑥)𝑑𝑥,(1.21) with 𝜉(𝑡,𝜓𝑥)(=𝑢)=_𝑡0𝑒^−𝑢−1+𝑢,(1.22)𝐾𝐵_𝜅𝑠−𝑥𝑑𝑠,(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 𝜈_ℝ𝑑𝜓1𝛼_𝑡^𝑡0𝐾(−𝑥)𝑑𝑠𝑑𝑥=𝜈_ℝ𝑑𝜓𝑡𝛼_𝑡𝐾(−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^ℎ (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(𝐵_𝜅𝑡)≈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, 𝜔_𝑑 denotes the volume of the 𝑑-dimensional unit ball. We denote ℱ_𝑑=𝑔∈𝑊¹²ℝ^𝑑∶_ℝ𝑑𝑔²(𝑥)𝑑𝑥=1,(2.1) where 𝑊¹²(ℝ^𝑑) is used for the Sobolev space of functions that belong to ℒ₂ together with their first order derivatives. We also denote 𝜑(𝑢)=1−𝑒^−𝑢,Ξ(𝑢,𝑣)=𝜓(𝑢)−𝑒^−𝑢𝜑(𝑣)=𝑒^{−𝑢−𝑣}−1+𝑢,𝑢,𝑣∈ℝ,(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, lim_𝑡→∞𝛼^𝑡𝑡^𝑑/𝑝log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=𝜈_ℝ𝑑𝑉𝐵_𝜅𝑠𝑑𝑠=𝜈𝜔_𝑑𝜓(𝜃|𝑥|^−𝑝)𝑑𝑥𝜃^𝑑/𝑝𝑑−𝑝𝑝Γ2𝑝𝑝−𝑑=−𝜈𝜔^𝑑𝜃^𝑑/𝑝Γ𝑝−𝑑𝑝.(2.3)(ii)In the “critical” case, lim_𝑡→∞𝑡^{−𝑑/(𝑑+2)}log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=sup_{𝑔∈ℱ𝑑}𝑉𝐵_𝜅𝑠𝑑𝑠𝜈_ℝ^𝑑𝜓𝜃𝛼_ℝ^𝑑𝑔²(𝑦)||𝑥−𝑦||^𝑝𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦.(2.4)(iii)In the “heavy-scale” case, under additional assumption 𝑝<(𝑑+2)/2, lim_𝑡→∞𝛼^{𝑡4/(𝑑+2−2𝑝)}𝑡^{−(𝑑+4−2𝑝)/(𝑑+2−2𝑝)}log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=sup_{𝑔∈ℱ𝑑}𝑉𝐵_𝜅𝑠𝑑𝑠⎧⎪⎨⎪⎩𝜈𝜃²2_ℝ^𝑑_ℝ^𝑑𝑔²(𝑦)||𝑥−𝑦||^𝑝𝑑𝑦²𝑑𝑥−𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦⎫⎪⎬⎪⎭.(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), lim_𝑡→∞𝑡^−𝑑/𝑝log𝔼⊗𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠=−𝜈𝜔_𝑑𝜃^𝑑/𝑝Γ𝑝−𝑑𝑝.(2.7)(ii)(see [40]) If 𝑝=𝑑+2, lim_𝑡→∞𝑡=−inf^{−𝑑/(𝑑+2)}l_{𝑔∈ℱ𝑑}og𝔼⊗𝔼₀exp−^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠𝜈_ℝ^𝑑𝜑𝜃_ℝ𝑑𝑔²(𝑦)||𝑥−𝑦||^𝑝𝑑𝑦𝑑𝑥+𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦.(2.8)(iii)(see [46]) If 𝑝>𝑑+2, lim_𝑡→∞𝑡=−inf^{−𝑑/(𝑑+2)}l_{𝑔∈ℱ𝑑}og𝔼⊗𝔼₀exp−^t₀𝑉𝐵_𝜅𝑠𝑑𝑠𝜈_ℝ𝑑1_𝑔(𝑥)>0𝑑𝑥+𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦.(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 𝛼_𝑡 to be, respectively, light or heavy. Note that for 𝛼_𝑡≡1, (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 𝛼_𝑡≡1, (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 𝛼_𝑡) 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^ℎ. 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, _ℝ𝑑𝜓(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 𝛼_𝑡 satisfy (1.18). (i)Statement (i) of Theorem 2.1 holds true assuming 𝐾 satisfies (2.10).(ii)For 𝑝>𝑑, lim_𝑡→∞𝛼^𝑡𝑡^𝑑/𝑝log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=−𝜈_ℝ𝑑𝑉𝐵_𝜅𝑠𝑑𝑠𝜑(𝜃|𝑥|^−𝑝)𝑑𝑥=−𝜈𝜔_𝑑𝜃^𝑑/𝑝Γ𝑝−𝑑𝑝.(2.11)(iii)For 𝑝=𝑑 and ℎ>0, log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=𝜈_ℝ𝑑𝑉^ℎ𝐵_𝜅𝑠𝑑𝑠min(𝐾(𝑦),ℎ)−𝛼𝑡^𝑡₊𝑑𝑦+𝜔_𝑑𝜃Eu𝛼𝑡_𝑡+𝑜𝛼𝑡_𝑡,𝑡⟶∞,(2.12) where Eu=−Γ^(1)=0,57721⋯ is the Euler constant. In particular, when 𝐾 has the form (1.5), log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0=𝜈𝜔_𝑑𝑉^ℎ𝐵_𝜅𝑠𝑑𝑠𝜃log𝑡𝛼_𝑡+logℎ+Eu𝛼𝑡_𝑡+o𝛼𝑡_𝑡,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 𝔼^𝜅𝑡0,0, 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), 𝑥∈ℝ^𝑑 and a cube 𝑈⊂ℝ^𝑑, denote by 𝐻^𝑄𝑈 the random Schrödinger operator in 𝑈 with the potential 𝑄 and the Dirichlet boundary conditions 𝐻^𝑄𝑈𝑓=−𝜅2Δ𝑓+𝑄𝑓,𝑓|^𝜕𝑈=0.(2.14) When the field 𝑄 is assumed to have locally bounded realizations, the operator 𝐻^𝑄𝑈 is a.s. well defined as an operator on 𝐿₂(𝑈,𝑑𝑥) and is self-adjoint. In addition, respective semigroup 𝑅^{𝑄𝑡,𝑈}=𝑒^{−𝑡𝐻𝑄𝑈},𝑡≥0 has a Feynman-Kac representation ([1, page 13]) 𝑅^{𝑄𝑡,𝑈}𝑓(𝑥)=𝔼^𝑡𝑥exp−^𝑡0𝑄𝐵_𝜅𝑠𝑑𝑠𝜒_𝑈,𝑡𝐵_⋅𝑓𝐵_𝑡,𝑥∈𝑈,𝑡≥0,(2.15) where 𝜒_𝑈,𝑡𝐵_⋅=l_{𝐵𝜅𝑠∈𝑈,𝑠∈[0,𝑡]}.(2.16) For general 𝑄, we define 𝐻^𝑄𝑈 by the following limit procedure. Consider truncations 𝑄_𝑁=(|𝑄|∧𝑁)sgn𝑄. Under appropriate assumptions on 𝑄, for almost every realization of this field, operators 𝑅^𝑄𝑁_𝑡,𝑈 converge strongly for every 𝑡≥0 as 𝑁→∞. In that case, 𝐻^𝑄𝑈 is defined as the generator of the limit semigroup 𝑅^{𝑄𝑡,𝑈}, 𝑡≥0. Assuming the spectrum of 𝐻^𝑄𝑈 to be discrete (we verify this assumption below), we denote this spectrum {𝜆^{𝑄𝑘,𝑈}} and define the function 𝑁^𝑄𝑈(𝜆)=1||𝑈||_𝑘l_𝜆^{𝑄𝑘,𝑈}_≤𝜆,𝜆∈ℝ.(2.17)

Proposition 2.6. Let the shape function 𝐾 be such that for some 𝑔>0, the following conditions hold: (i)𝐾^𝑔(𝑥)=(𝐾(𝑥)−𝑔)₊ is compactly supported,(ii)𝐾_𝑔=min(𝐾(𝑥),𝑔) is Lipschitz continuous and belongs to the Sobolev space 𝑊¹²(ℝ^𝑑).Consider either a partially renormalized potential 𝑄=𝑉^ℎ with ℎ∈(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 𝐻^𝑄𝑈 and respective function 𝑁^𝑄𝑈,(b)there exists an integrated density of states 𝑁^𝑄, that is, a deterministic monotonous function such that 𝑁^𝑄(𝜆)=lim_{𝑈↑ℝ𝑑}𝑁^𝑄𝑈(𝜆),(2.18)
a.s. for every point of continuity of 𝑁^𝑄. Respective Laplace transform has the representation _ℝ𝑒^−𝜆𝑡𝑑𝑁^𝑄(𝜆)=(2𝜋𝜅𝑡)^−(𝑑/2)𝔼⊗𝔼^𝜅𝑡0,0exp−^𝑡0𝑄𝐵_𝜅𝑠𝑑𝑠,𝑡≥0.(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^ℎ.

Theorem 2.7. Let 𝐾 satisfy (2.6). (i)For 𝑝∈(𝑑/2,𝑑), assuming additionally (2.10), one has in limit 𝜆→−∞log𝑁^𝑉(𝜆)=−𝜈𝜔_𝑑Γ2𝑝−𝑑𝑝^{−𝑝/(𝑑−𝑝)}𝜃(𝑑𝑑−𝑝)^{𝑑/(𝑑−𝑝)}(−𝜆)^{𝑑/(𝑑−𝑝)}(1+𝑜(1)).(2.20)(ii)For 𝑝=𝑑 and ℎ∈(0,∞), one has in limit𝜆→−∞log𝑁^𝑉ℎ(𝜆)=−𝜈𝜔_𝑑𝜃exp−𝜆=−𝜈𝜔_𝑑𝜈𝜔_𝑑𝜃−logℎ−Eu−1(1+𝑜(1))𝜃ℎexp−𝜈𝜔𝜆_𝑑𝜃−Eu−1(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 𝛼_𝑡≡1. 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 𝐻_𝛾=−𝜅2Δ+𝛾𝑄,𝛾>0.(2.22) Assuming every potential 𝑄_𝛾=𝛾𝑄 being such that respective IDS function 𝑁^𝑄𝛾 is well defined, denote 𝑁^𝑄(𝜆,𝛾)=𝑁^𝑄𝛾(𝜆). We call the family 𝑁^𝑄(𝜆,𝛾),𝜆∈ℝ,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+2−𝑝)/2>(𝑑+4−2𝑝)/4. Hence, conditions, that (−𝜆)^{(𝑑+4−2𝑝)/4}/𝛾→∞ and (−𝜆)^{(𝑑+2−𝑝)/2}/𝛾 is bounded, yield 𝜆→0−. 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 ∞), 𝜆→0−, or 𝜆 to stay bounded away both from 0 and −∞ (in these two cases 𝛾→0+ necessarily). This is the reason that two conditions (−𝜆)^𝑝/𝑑/𝛾→∞ and (−𝜆)^{(𝑑+2−𝑝)/2}/𝛾→∞ are imposed in this case: when 𝜆→−∞, the first one includes the second one, but when 𝜆→0−, 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+2−𝑝)/2}/𝛾→∞, log𝑁^𝑉(𝜆,𝛾)=−𝜈𝜔_𝑑Γ2𝑝−𝑑𝑝^{−𝑝/(𝑑−𝑝)}𝜃(𝑑𝑑−𝑝)^{𝑑/(𝑑−𝑝)}×(−𝜆)^{𝑑/(𝑑−𝑝)}𝛾^{−𝑑/(𝑑−𝑝)}(1+𝑜(1)).(2.24)(ii)When (−𝜆)^{𝑑+4−2𝑝/4}/𝛾→∞ and (−𝜆)^{(𝑑+2−𝑝)/2}/𝛾→0, under additional assumption 𝑝<(𝑑+2)/2, log𝑁^𝑉(𝜆,𝛾)=−2C²𝑑+2−2𝑝^{−(𝑑+2−2𝑝)/2}𝑑+42−2𝑝^{(𝑑+4−2𝑝)/2}(−𝜆)^{(𝑑+4−2𝑝)/2}𝛾⁻²(1+𝑜(1)),(2.25) where C₂ denotes the constant in the r.h.s of (2.5).(iii)When 𝜆→0− and (−𝜆)^{(𝑑+2−𝑝)/2}/𝛾 is bounded away both from 0 and from ∞, log𝑁^𝑉(𝜆,𝛾)=−(𝑑−𝑝)C^𝜓𝑝^{−𝑝/(𝑑−𝑝)}(𝑑−𝑑𝑝)^{𝑑/(𝑑−𝑝)}(−𝜆)^{𝑑/(𝑑−𝑝)}𝛾^{−𝑝/(𝑑−𝑝)}(1+𝑜(1)),(2.26) where C_𝜓 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 𝜆→0−. This in general reflects the fact that for 𝛾→0+ 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 ℎ>0, denote 𝜉^ℎ(𝑡,𝑥)=^𝑡0(𝐾(𝑦−𝑥)−ℎ)₊𝑑𝑠,𝜉_ℎ(𝑡,𝑥)=^𝑡0(𝐾(𝑦−𝑥)∧ℎ)𝑑𝑠.(3.1)

Similarly to (1.21), we have 𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠=𝔼₀exp−_ℝ𝑑𝜑1𝛼_𝑡𝔼⊗𝔼₀𝜉(𝑡,𝑥)𝑑𝑥,(3.2)exp−1𝛼_𝑡^𝑡0𝑉^ℎ𝐵_𝜅𝑠𝑑𝑠=𝔼₀exp_ℝ𝑑Ξ1𝛼_𝑡𝜉_ℎ(𝑡,𝑥),1𝛼_𝑡𝜉^ℎ(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^ℎ, we consider a partially renormalized Poisson potential with the properly chosen renormalization level, dependent on 𝑡. Let 𝑔>0 and ℎ_𝑡=𝑔𝛼_𝑡/𝑡. Then, assuming 𝑝=𝑑, (2.6) and (1.18), we will prove that lim_𝑡→∞𝛼^𝑡𝑡log𝔼⊗𝔼⁰exp−1𝛼_𝑡^𝑡0=𝜈_ℝ𝑑𝑉^ℎ𝑡𝐵_𝜅𝑠𝑑𝑠Ξ𝜃|𝑥|^−𝑑∧𝑔,𝜃|𝑥|^−𝑑−𝑔₊𝑑𝑥=𝜈𝜔_𝑑𝜃log𝑔+Eu.(3.4) Note that by Proposition  6.1 in [7], 𝑉^ℎ′(𝑥)−𝑉^ℎ(𝑥)=𝜈_ℝ𝑑(min(𝐾(𝑦),ℎ)−ℎ^)₊𝑑𝑦,(3.5) for any ℎ≥ℎ^ such that 𝑉^ℎ, 𝑉^ℎ′ 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 𝜚1𝛼_𝑡𝜉(𝑡,𝑥)=𝜚^𝑡0𝛼1_𝑡𝐾𝐵_𝜅𝑠−𝑥𝑑𝑠≤1𝑡^𝑡0𝜚𝑡𝛼_𝑡𝐾𝐵_𝜅𝑠−𝑥𝑑𝑠.(3.6) Denote 𝜆_𝑡=(𝑡/𝛼_𝑡)^1/𝑝, 𝐾(𝑥,𝜆)=𝜆^𝑝𝐾(𝜆𝑥). By the inequality above, one has the following estimate with nonrandom right hand side: _ℝ𝑑𝜚1𝛼_𝑡𝜉(𝑡,𝑥)𝑑𝑥≤1𝑡_ℝ^𝑑^t₀𝜚𝑡𝛼_𝑡=𝐾𝐵_𝜅𝑠−𝑥𝑑𝑠𝑑𝑥_ℝ𝑑𝜚𝑡𝛼_𝑡𝐾(−𝑥)𝑑𝑥=𝑡𝛼_𝑡^𝑑/𝑝_ℝ𝑑𝜚𝐾𝑥,𝜆_𝑡𝑑𝑥.(3.7) Assumption (1.18) yields 𝜆_𝑡→∞. 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, _ℝ𝑑𝜓(𝐾(𝑥,𝜆))𝑑𝑥⟶_ℝ𝑑𝜓(𝐾(𝑥))𝑑𝑥or_ℝ𝑑𝜑(𝐾(𝑥,𝜆))𝑑𝑥⟶_ℝ𝑑𝜑(𝐾(𝑥))𝑑𝑥,(3.8)𝜆→∞. By assumption (2.6), for every 𝜀>0, there exists 𝜆^𝜀 such that 𝐾(𝑥,𝜆)|𝑥|^𝑝∈[𝜃−𝜀,𝜃+𝜀],|𝑥|>𝜀,𝜆>𝜆^𝜀.(3.9)

When 𝑝>𝑑, this easily provides lim_𝜆→∞_|𝑥|>𝜀𝜑(𝐾(𝑥,𝜆))𝑑𝑥=_|𝑥|>𝜀𝜑(𝐾(𝑥))𝑑𝑥,𝜀>0.(3.10) Since 𝜑 is bounded on ℝ⁺, (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 lim_𝜀→0limsup_𝜆→∞_{|𝑥|≤𝜀}𝜓(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|≤𝜀}𝜓(𝐾(𝑥,𝜆))𝑑𝑥=_{|𝑥|≤𝑟1/𝜆}+_{𝑟1/𝜆<|𝑥|≤𝜀}≤𝜆_−𝑑𝜓(𝐾(𝑥,𝜆))𝑑𝑥_{|𝑥|≤𝑟1}𝜓(𝜆^𝑝𝐾(𝑥))𝑑𝑥+_{|𝑥|≤𝜀}𝜓𝜃₁|𝑥|^−𝑝𝑑𝑥.(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 Ξ∶ℝ²→ℝ, we have Ξ1𝛼_𝑡𝜉_ℎ(𝑡,𝑥),1𝛼_𝑡𝜉^ℎ(𝑡,𝑥)≤1𝑡^𝑡0Ξ𝑡𝛼_𝑡𝐾𝐵_𝜅𝑠−𝑥∧ℎ,𝑡𝛼_𝑡𝐾𝐵_𝜅𝑠−𝑥−ℎ₊𝑑𝑠,(3.13)

and consequently, for ℎ_𝑡=𝑔𝛼_𝑡/𝑡_ℝ𝑑Ξ1𝛼_𝑡𝜉_ℎ𝑡(𝑡,𝑥),1𝛼_𝑡𝜉^ℎ𝑡(𝑡,𝑥)𝑑𝑥≤_ℝ𝑑Ξ𝑡=𝑡𝛼_𝑡𝐾(−𝑥)∧ℎ_𝑡,𝛼𝑡_𝑡𝐾(−𝑥)−ℎ_𝑡₊𝑑𝑥𝛼_𝑡^𝑑/𝑝_ℝ𝑑Ξ𝐾𝑥,𝜆_𝑡∧𝑔,𝐾𝑥,𝜆_𝑡−𝑔₊𝑑𝑥.(3.14) Similarly to (3.8), one can prove _ℝ𝑑Ξ𝐾(𝑥,𝜆)∧𝑔,(𝐾(𝑥,𝜆)−𝑔)₊𝑑𝑥⟶_ℝ𝑑Ξ𝐾(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 (𝜃−𝜀)|𝑥|^−𝑝≤𝐾(𝑥)≤(𝜃+𝜀)|𝑥|^−𝑝,|𝑥|≥𝑅.(3.16) Take 𝛽>0 and consider the set 𝐴_𝑡,𝛽=sup_𝑠≤𝑡||𝐵_𝜅𝑠||≤𝛽𝜆_𝑡,(3.17) keeping the notation 𝜆_𝑡=(𝑡/𝛼_𝑡)^1/𝑝. By the scaling property and the well-known small balls probability asymptotics for the Brownian motion, we have, for 𝑡 large enough, logℙ₀𝐴_𝑡,𝛽≥−𝑐𝑡𝛽𝜆_𝑡⁻²(3.18) with some constant 𝑐>0. Therefore, condition 𝛼_𝑡=𝑜(𝑡^{(𝑑+2−𝑝)/(𝑑+2)}) yields 𝛼𝑡^𝑡^𝑑/𝑝logℙ₀𝐴_𝑡,𝛽⟶0,𝑡⟶+∞.(3.19) Take 𝛾>2𝛽. On the set 𝐴_𝑡,𝛽, one has ||𝐵_𝜅𝑠−𝑥||≥𝛽𝑡𝛼_𝑡^1/𝑝,𝑠∈[0,𝑡],|𝑥|≥𝛾𝜆_𝑡.(3.20) Then, for 𝑡 large enough to provide 𝛽𝜆_𝑡>𝑅, we have (𝜃−𝜀)||𝐵_𝜅𝑠−𝑥||^−𝑝≤𝐾𝐵_𝜅𝑠−𝑥≤(𝜃+𝜀)||𝐵_𝜅𝑠−𝑥||^−𝑝,𝑠∈[0,𝑡],|𝑥|≥𝛾𝜆_𝑡.(3.21) Therefore, a two-sided estimate (𝜃−𝜀)1+𝛽𝛾^−𝑝|𝑥|^−𝑝≤𝐾𝐵_𝜅𝑠−𝑥≤(𝜃+𝜀)1−𝛽𝛾^−𝑝|𝑥|^−𝑝,𝑠∈[0,𝑡](3.22) is valid on the set 𝐴_𝑡,𝛽 for every 𝑥 with |𝑥|>𝛾𝜆_𝑡. Observe that (3.22) is a pointwise estimate for a Brownian trajectory from a “small ball” 𝐴_𝑡,𝛽 and for a point 𝑥 outside a “large ball” {𝑦∶|𝑦|≤𝛾𝜆_𝑡}. On the other hand, (3.19) shows the “small Brownian ball” 𝐴_𝑡,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 _ℝ𝑑𝜓1𝛼_𝑡𝜉(𝑡,𝑥)𝑑𝑥≥_{|𝑥|>𝛾𝜆𝑡}𝜓𝑡𝛼_𝑡(𝜃−𝜀)1+𝛽𝛾^−𝑝|𝑥|^−𝑝𝑑𝑥=𝛼𝑡_𝑡^𝑑/𝑝𝐼^{𝜓𝜀,𝛽,𝛾},(3.23) on 𝐴_𝑡,𝛽 with 𝐼^{𝜓𝜀,𝛽,𝛾}=_|𝑥|>𝛾𝜓(𝜃−𝜀)1+𝛽𝛾^−𝑝|𝑥|^−𝑝𝑑𝑥.(3.24) Together with (1.21) and (3.19), this inequality provides liminf_𝑡→+∞𝛼^𝑡𝑡^𝑑/𝑝log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠≥𝐼^{𝜓𝜀,𝛽,𝛾},(3.25) for every 𝜀>0, 𝛽>0, 𝛾>0. Since lim_𝜀→0lim_𝛾→0lim_𝛽→0𝐼^{𝜓𝜀,𝛽,𝛾}=_ℝ𝑑𝜓(𝜃|𝑥|^−𝑝)𝑑𝑥,(3.26) this completes the proof of the lower bound in (2.3).

Since (−𝜑) is nonincreasing and satisfies −𝜑≥−1, (3.22) yields −_ℝ𝑑𝜑1𝛼_𝑡𝜉(𝑡,𝑥)𝑑𝑥≥−_{|𝑥|≤𝛾𝜆𝑡}𝑑𝑥−_{|𝑥|>𝛾𝜆𝑡}𝜑𝑡𝛼_𝑡(𝜃+𝜀)1−𝛽𝛾^−𝑝|𝑥|^−𝑝𝑑𝑥=𝛼𝑡_𝑡^𝑑/𝑝𝐼^{𝜑𝜀,𝛽,𝛾},(3.27) on 𝐴_𝑡,𝛽 with 𝐼^{𝜑𝜀,𝛽,𝛾}=−_{|𝑥|≤𝛾}𝑑𝑥−_|𝑥|>𝛾𝜑(𝜃+𝜀)1−𝛽𝛾^−𝑝|𝑥|^−𝑝𝑑𝑥.(3.28)

Since lim_𝜀→0lim_𝛾→0lim_𝛽→0𝐼^{𝜑𝜀,𝛽,𝛾}=−_ℝ𝑑𝜑(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, Ξ≥−1, and hence (3.22) yields in the case 𝑑=𝑝_ℝ𝑑Ξ1𝛼_𝑡𝜉_ℎ𝑡(𝑡,𝑥),1𝛼_𝑡≥𝜉^ℎ𝑡(𝑡,𝑥)𝑑𝑥_{|𝑥|>𝛾𝜆𝑡}Ξ𝑡𝛼_𝑡(𝜃−𝜀)1+𝛽𝛾^−𝑑|𝑥|^−𝑑𝑡∧𝑔𝑡𝛼^𝑡,𝛼_𝑡(𝜃+𝜀)1−𝛽𝛾^−𝑑|𝑥|^−𝑑−𝑡𝑔𝑡𝛼^𝑡₊𝑑𝑥=𝛼𝑡_𝑡𝐼^{Ξ𝜀,𝛽,𝛾},(3.30) on 𝐴_𝑡,𝛽 with 𝐼^{Ξ𝜀,𝛽,𝛾}=−_{|𝑥|≤𝛾}𝑑𝑥+_|𝑥|>𝛾Ξ(𝜃−𝜀)1+𝛽𝛾^−𝑑|𝑥|^−𝑑∧𝑔,(𝜃+𝜀)1−𝛽𝛾^−𝑑|𝑥|^−𝑑−𝑔₊𝑑𝑥.(3.31) Together with (3.2) and (3.19), this inequality provides liminf_𝑡→+∞𝛼^𝑡𝑡log𝔼⊗𝔼⁰exp−1𝛼_𝑡^𝑡0𝑉𝐵_𝜅𝑠𝑑𝑠≥𝐼^{Ξ𝜀,𝛽,𝛾},(3.32) for every 𝜀>0, 𝛽>0, 𝛾>0. Since lim_𝜀→0lim_𝛾→0lim_𝛽→0𝐼^{Ξ𝜀,𝛽,𝛾}=_ℝ𝑑Ξ𝜃|𝑥|^−𝑑∧𝑔,𝜃|𝑥|^−𝑑−𝑔₊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, _ℝ𝑑Ξ𝜃|𝑥|^−𝑑∧𝑔,𝜃|𝑥|^−𝑑−𝑔₊𝑑𝑥=𝜔_𝑑^∞0Ξ𝜃=𝜔_𝑑𝑟∧𝑔,𝜃𝑟−𝑔₊𝑑𝑥^∞0𝑒^−𝜃/𝑟−1−𝜃=𝜃𝜔_𝑑𝑟∧𝑔𝑑𝑟^∞0𝑒^−𝑠−1−𝑠∧𝑔𝑠²=𝜔_𝑑𝑑𝑠𝜃^∞0𝑒^−𝑠−1−𝑠∧1𝑠²𝑑𝑠+log𝑔,(3.34) in the last identity we have used an elementary relation ^∞0𝑠∧𝑔−𝑠∧1𝑠²𝑑𝑠=log𝑔.(3.35) Integration by parts and n. 538 in [47] gives ^∞0𝑒^−𝑠−1−𝑠∧1𝑠²𝑑𝑠=^𝑡01−𝑒^−𝑠𝑠𝑑𝑠−^∞0𝑒^−𝑠𝑠𝑑𝑠=Eu,(3.36) which completes calculation of the integral in (3.4).

Finally, let 𝐾 has the form (1.5). Take ℎ_𝑡=𝛼_𝑡/𝑡, then ℎ_𝑡<ℎ for 𝑡 large enough, and _ℝ𝑑min(𝐾(𝑦),ℎ)−𝛼𝑡^𝑡₊𝑑𝑦==𝜔_ℝ𝑑𝑑𝜃|𝑥|^−𝑑∧ℎ−ℎ_𝑡₊𝑑𝑥𝜃^∞0𝑠∧ℎ−𝑠∧ℎ_𝑡𝑠²=𝜔_𝑑𝑑𝑠𝜃logℎ−logℎ_𝑡=𝜔_𝑑𝜃logℎ+log𝑡𝛼_𝑡.(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 𝐿∶ℝ^𝑑→ℝ⁺ and denote 𝜂(𝑡,𝑥)=^𝑡0𝐿𝐵_𝜅𝑠−𝑥𝑑𝑥.(4.1)

Theorem 4.1. Let, for some sequence 𝐿_𝑛,𝑛≥1 of nonnegative continuous compactly supported functions, 𝐿(𝑥)=sup_𝑛≥1𝐿_𝑛(𝑥),(4.2) for a.a. 𝑥∈ℝ^𝑑. Let 𝜐∶ℝ⁺→ℝ be an increasing convex function with 𝜐(0)=0, and _ℝ𝑑𝜐(𝐿(𝑥))𝑑𝑥<+∞.(4.3) Then, lim_𝑡→∞1𝑡log𝔼⁰exp𝑡_ℝ𝑑𝜐1=sup_{𝑔∈ℱ𝑑}𝑡𝜂(𝑡,𝑥)𝑑𝑥_ℝ𝑑𝜐_ℝ𝑑𝑔²(𝑦)𝐿(𝑥−𝑦)𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦.(4.4) Proof of Theorem 4.1: the lower bound. By Jensen’s inequality,

𝜐1𝑡𝜂(𝑡,𝑥)=𝜐1𝑡^𝑡0𝐿𝐵_𝜅𝑠−𝑥𝑑𝑠≤1𝑡^𝑡0𝜐𝐿𝐵_𝜅𝑠−𝑥𝑑𝑠,(4.5) and therefore, _ℝ𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥≤1𝑡^𝑡0_ℝ𝑑𝜐𝐿𝐵_𝜅𝑠−𝑥𝑑𝑥𝑑𝑠=_ℝ𝑑𝜐(𝐿(𝑥))𝑑𝑥<+∞.(4.6) For every 𝑅>0, we write _ℝ𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥≥_{[−𝑅,𝑅]}^𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥,(4.7)

and note that (4.6) provides (1/𝑡)𝜂(𝑡,⋅)∈ℒ₁([−𝑅,𝑅]^𝑑), because 𝜐 has at least linear growth at +∞.

For a fixed 𝑅, denote ℒ^+1,𝑅=ℎ∈ℒ₁[−𝑅,𝑅]^𝑑,ℎ≥0,(4.8) and consider a convex function Υ_𝑅∶ℒ^+1,𝑅→[0,+∞]Υ_𝑅(ℎ)=_{[−𝑅,𝑅]𝑑}𝜐(ℎ(𝑥))𝑑𝑥.(4.9) Denote by 𝐵_𝑅 the class of bounded measurable functions 𝑓∶[−𝑅,𝑅]^𝑑→ℝ, and put 𝐶_Υ,𝑓,𝑅=sup𝐶∶𝐶+_{[−𝑅,𝑅]𝑑}𝑓(𝑥)ℎ(𝑥)𝑑𝑥≤Υ_𝑅(ℎ),ℎ∈ℒ^+1,𝑅,𝑓∈𝐵^𝑅.(4.10)

Lemma 4.2. For every ℎ∈ℒ^+1,𝑅 with Υ_𝑅(ℎ)<+∞, Υ_𝑅(ℎ)=sup_{𝑓∈𝐵𝑅}𝐶^Υ,𝑓,𝑅+_{[−𝑅,𝑅]𝑑}𝑓(𝑥)ℎ(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, Υ_𝑅 is an improper function.

Proof. Consider the set epiΥ_𝑅=(ℎ,𝑡)∶ℎ∈ℒ^+1,𝑅,𝑡≥Υ_𝑅(ℎ),(4.12) clearly, epiΥ_𝑅 is a convex subset of the Banach space ℒ₁([−𝑅,𝑅]^𝑑)×ℝ. In addition, this subset is closed by the Fatou lemma. Therefore, the separation theorem (Theorem 9.2 in [49], Chapter II) provides that epiΥ_𝑅 is the intersection of all the closed half-spaces containing epiΥ_𝑅. Note that every continuous linear functional on the space ℒ₁([−𝑅,𝑅]^𝑑)×ℝ has the form (𝑓,𝑎) with 𝑓∈𝐵_𝑅, 𝑎∈ℝ and ⟨(ℎ,𝑡),(𝑓,𝑎)⟩=_{[−𝑅,𝑅]𝑑}ℎ(𝑥)𝑓(𝑥)𝑑𝑥+𝑎𝑡.(4.13) Take ℎ_∗∈ℒ^+𝜐 with Υ_𝑅(ℎ_∗)<+∞, and 𝑡_∗<Υ_𝑅(ℎ_∗). Then (ℎ_∗,𝑡_∗)∉epiΥ_𝑅, and therefore, there exists (𝑓,𝑎) and 𝑐∈ℝ such that ℎ_∗,𝑡_∗,(𝑓,𝑎)<𝑐,⟨(ℎ,𝑡),(𝑓,𝑎)⟩≥𝑐,(ℎ,𝑡)∈epiΥ_𝑅.(4.14) By the definition of epiΥ_𝑅, if (ℎ,𝑡)∈epiΥ_𝑅, then (ℎ,𝑡^)∈epiΥ_𝑅 for every 𝑡′>𝑡, hence (4.14) is impossible if either 𝑎=0 or 𝑎<0. Divide (4.14) by 𝑎 and denote 𝑓_𝑎=−𝑓/𝑎,𝑐_𝑎=𝑐/𝑎. Then, 𝑐_𝑎+_{[−𝑅,𝑅]𝑑}ℎ^∗(𝑥)𝑓_𝑎(𝑥)𝑑𝑥>𝑡_∗,𝑐_𝑎+_{[−𝑅,𝑅]𝑑}ℎ(𝑥)𝑓_𝑎(𝑥)𝑑𝑥≤𝑡,(ℎ,𝑡)∈epiΥ_𝑅.(4.15) Take 𝑡=Υ_𝑅(ℎ) in the second inequality in (4.15); this yields 𝑐_𝑎≤𝐶_{Υ,𝑓𝑎,𝑅}. Consequently, 𝑡_∗≤sup_{𝑓∈𝐵𝑅}𝐶^Υ,𝑓,𝑅+_{[−𝑅,𝑅]𝑑}𝑓(𝑥)ℎ_∗(𝑥)𝑑𝑥,(4.16) which means that Υ_𝑅ℎ_∗≤sup_{𝑓∈𝐵𝑅}𝐶^Υ,𝑓,𝑅+_{[−𝑅,𝑅]𝑑}𝑓(𝑥)ℎ_∗(𝑥)𝑑𝑥,(4.17) because 𝑡_∗<Υ_𝑅(ℎ_∗) is arbitrary. The inverse inequality is obvious.

Take 𝑓∈𝐵_𝑅, then 𝔼₀exp𝑡_{[−𝑅,𝑅]𝑑}𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥≥𝑒^{𝐶Υ,𝑓,𝑅𝑡}𝔼₀exp_{[−𝑅,𝑅]𝑑}𝑓(𝑥)𝜂(𝑡,𝑥)𝑑𝑥.(4.18) Note that _{[−𝑅,𝑅]𝑑}𝑓(𝑥)𝜂(𝑡,𝑥)𝑑𝑥=^𝑡0𝑓𝐵_𝜅𝑠𝑑𝑠,(4.19) where 𝑓(𝑦)=_{[−𝑅,𝑅]𝑑}𝑓(𝑥)𝐿(𝑦−𝑥)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 liminf_𝑡→∞1𝑡log𝔼⁰exp𝑡_{[−𝑅,𝑅]𝑑}𝜐1≥𝐶_Υ,𝑓,𝑅𝑡𝜂(𝑡,𝑥)𝑑𝑥+lim_𝑡→∞1𝑡log𝔼⁰≥𝐶_Υ,𝑓,𝑅exp^𝑡0𝑓𝐵_𝜅𝑠𝑑𝑠+sup_{𝑔∈ℱ𝑑}_ℝ𝑑𝑓(𝑥)𝑔²(𝑥)𝑑𝑥−12_ℝ^𝑑||∇𝑔(𝑥)||²𝑑𝑥.(4.21) Note that _ℝ𝑑𝑓(𝑥)𝑔²(𝑥)𝑑𝑥=_{[−𝑅,𝑅]𝑑}𝑓(𝑥)_ℝ𝑑𝑔²(𝑦)𝐿(𝑥−𝑦)𝑑𝑦𝑑𝑥.(4.22) Summarizing our proof, liminf_𝑡→∞1𝑡log𝔼⁰exp𝑡_ℝ𝑑𝜐1≥sup_{𝑔∈ℱ𝑑}𝑡𝜂(𝑡,𝑥)𝑑𝑥𝐶^Υ,𝑓,𝑅+_{[]−𝑅,𝑅𝑑}𝑓(𝑥)_ℝ𝑑𝑔²(𝑦)𝐿(𝑥−𝑦)𝑑𝑦𝑑𝑥−12_ℝ^𝑑||∇𝑔(𝑥)||²𝑑𝑥,(4.23) for every 𝑅>0, 𝑓∈𝐵_𝑅. We take supremum over 𝑓∈𝐵_𝑅 and get, by Lemma 4.2, liminf_𝑡→∞1𝑡log𝔼⁰exp𝑡_ℝ𝑑𝜐1≥sup_{𝑔∈ℱ𝑑}𝑡𝜂(𝑡,𝑥)𝑑𝑥Υ^𝑅_ℝ𝑑𝑔²(𝑦)𝐿(⋅−𝑦)𝑑𝑦−12_ℝ^𝑑||∇𝑔(𝑥)||²𝑑𝑥.(4.24) Note that by Jensen’s inequality, for every 𝑔∈ℱ_𝑑, Υ_𝑅_ℝ𝑑𝑔²(𝑦)𝐿(⋅−𝑦)𝑑≤𝑦=_{[]−𝑅,𝑅𝑑}𝜐_ℝ𝑑𝑔²(𝑦)𝐿(𝑥−𝑦)𝑑𝑦𝑑𝑥_{[]−𝑅,𝑅𝑑}_ℝ𝑑𝑔²(𝑦)𝜐(𝐿(𝑥−𝑦))𝑑𝑦𝑑𝑥≤_ℝ𝑑𝜐(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,𝑣≥0.(4.26) Thus, for any 𝑁>𝑀, _ℝ𝑑𝜐1𝑡𝜂(𝑡,𝑥)𝑑𝑥=_{𝑧∈ℤ𝑑}_{[]−𝑁,𝑁𝑑}𝜐1≤𝑡𝜂(𝑡,2𝑁𝑧+𝑥)𝑑𝑥_{[−𝑁,𝑁]𝑑}𝜐1𝑡_{𝑧∈ℤ𝑑}𝜂(𝑡,2𝑁𝑧+𝑥)𝑑𝑥=_{[]−𝑁,𝑁𝑑}𝜐1𝑡𝜂̃(𝑡,𝑥)𝑑𝑥,(4.27) where 𝜂̃(𝑡,𝑥)=^𝑡0𝐿𝐵_𝜅𝑠−𝑥𝑑𝑠,𝐿(𝑥)=_{𝑧∈ℤ𝑑}𝐿(2𝑁𝑧+𝑥).(4.28) Denote by 𝑇^𝑁𝑑 the torus of the size 2𝑁, that is, the cube [−𝑁,𝑁]^𝑑 with the sides identified. Let us denote by 𝐽_𝑁 the projection on this torus: by definition, for 𝑥∈ℝ^𝑑 its projection 𝐽_𝑁(𝑥) is the unique point 𝑥̃∈𝑇^𝑁𝑑 such that 𝑥−𝑥̃∈2𝑁ℤ^𝑑. Denote 𝐵^𝑁𝑠=𝐽_𝑁(𝐵_⋅), the Brownian motion on the torus 𝑇^𝑁𝑑. With this notation in mind, we rewrite the right hand side term in (4.27) _{[−𝑁,𝑁]𝑑}𝜐1𝑡𝜂̃(𝑡,𝑥)𝑑𝑥=_𝑇^𝑁𝑑𝜐1𝑡𝜂^𝑁(𝑡,𝑥)𝑑𝑥,𝜂^𝑁(𝑡,𝑥)=^𝑡0𝐿∘𝐽_𝑁𝐵_𝜅𝑠−𝑥𝑑𝑠.(4.29) Consider the empirical measures for the Brownian motion on the torus 𝑇^𝑁𝑑𝑄^𝑁𝑡(𝐴)=1𝑡^𝑡01_𝐵^𝑁𝜅𝑠_∈𝐴𝑑𝑠,𝐴∈ℬ𝑇^𝑁𝑑.(4.30) Note that 1𝑡𝜂^𝑁(𝑡,𝑥)=_𝑇𝑁𝑑𝐿∘𝐽_𝑁(𝑦−𝑥)𝑄^𝑁𝑡(𝑑𝑦),(4.31) and the mapping 𝜇⟶_𝑇𝑁𝑑𝜐_𝑇𝑁𝑑𝐿∘𝐽_𝑁(𝑦−𝑥)𝜇(𝑑𝑦)𝑑𝑥(4.32) is continuous and bounded on the space of all probability distributions on 𝑇^𝑁𝑑 with the metrics of weak convergence. Hence, combination of (4.27), the large deviation principle for 𝑄^𝑁𝑡 (Theorem 3 in [52]), and Varadhan’s lemma (Proposition 3.8 in [53]) yields limsup_𝑡→∞1𝑡𝔼⁰exp𝑡_ℝ𝑑𝜐1≤sup_{𝑔∈ℱ𝑁𝑑}𝑡𝜂(𝑡,𝑥)𝑑𝑥_𝑇^𝑁𝑑𝜐_𝑇𝑁𝑑𝐿∘𝐽_𝑁(𝑥−𝑦)𝑔²(𝑦)𝑑𝑦𝑑𝑥−𝜅2_𝑇^𝑁𝑑||∇𝑔(𝑦)||²𝑑𝑦,(4.33) where ℱ^𝑁𝑑=𝑔∈𝑊¹²𝑇^𝑁𝑑∶_𝑇𝑁𝑑𝑔²(𝑥)𝑑𝑥=1.(4.34) By smooth truncation, it is easy to verify that sup_{𝑔∈ℱ𝑑}_ℝ𝑑𝜐_ℝ𝑑𝐿(𝑥−𝑦)𝑔²(𝑦)𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑≤liminf_𝑁→∞||∇𝑔(𝑦)||²𝑑𝑦sup_{𝑔∈ℱ𝑁𝑑}_𝑇^𝑁𝑑𝜐_𝑇𝑁𝑑𝐿∘𝐽_𝑁(𝑥−𝑦)𝑔²(𝑦)𝑑𝑦𝑑𝑥−𝜅2_𝑇^𝑁𝑑||∇𝑔(𝑦)||²𝑑𝑦,(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 𝐿_𝑛,𝑛≥1 to be pointwise increasing, because otherwise, one can take 𝐿_𝑛=max_𝑘≤𝑛𝐿_𝑛 instead.
Write Δ_𝑛=𝐿−𝐿_𝑛 and 𝜂(𝑡,𝑥)=𝜂_𝑛(𝑡,𝑥)+𝜁_𝑛(𝑡,𝑥),𝜁_𝑛(𝑡,𝑥)=^𝑡0Δ_𝑛𝐵_𝜅𝑠−𝑥𝑑𝑠.(4.36) For every 𝛾∈(0,1), we have by convexity 𝜐1𝑡𝜂(𝑡,𝑥)≤𝛾𝜐𝛾1𝑡𝜂^𝑛(𝑡,𝑥)+(1−𝛾)𝜐1(1−𝛾)𝑡𝜁^𝑛(𝑡,𝑥)(4.37) The Jensen inequality, analogously to (4.6), provides that _ℝ𝑑𝜐1(1−𝛾)𝑡𝜁^𝑛(𝑡,𝑥)𝑑𝑥≤_ℝ𝑑𝜐1(1−𝛾)Δ^𝑛(𝑥)𝑑𝑥.(4.38) Then, from the upper bound with a regular kernel 𝐿_𝑛, we obtain limsup_𝑡→∞1𝑡𝔼⁰exp𝑡_ℝ𝑑𝜐1≤sup_{𝑔∈ℱ𝑑}𝑡𝜂(𝑡,𝑥)𝑑𝑥𝛾_ℝ𝑑𝜐1𝛾_ℝ^𝑑𝐿_𝑛(𝑥−𝑦)𝑔²(𝑦)𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑+||∇𝑔(𝑦)||²𝑑𝑦_ℝ𝑑𝜐1(1−𝛾)Δ^𝑛(𝑥)𝑑𝑥,(4.39) for any 𝑛≥1 and 𝛾∈(0,1). Passing to the limit first as 𝑛→∞ and then as 𝛾→1 completes the proof.

4.2. Proof of Theorem 2.1: Critical Case

The kernel (1.5) has the following scaling property: 𝐾(𝑥)=𝜏^−𝑝/2𝐾(𝜏^−1/2𝑥) for any 𝜏>0. Then, by the scaling property of the Brownian motion, 𝜉(𝑡,𝑥)=^𝑑^𝑡0𝐾𝜏^−1/2𝐵_𝑠𝜏−𝑥=𝜏^𝑝/(2−1)^0𝑡𝜏𝐾𝐵_𝜅𝑠−𝜏^1/2𝑥𝑑𝑠=𝜏^𝑝/(2−1)𝜉𝑡𝜏,𝑥𝜏^1/2.(4.40) Henceforth, the integral under the exponent in the right-hand side of (1.21), after the variable change 𝜏^1/2𝑥↦𝑥, can be written as 𝜏^−𝑑/2_ℝ𝑑𝜓𝜏^𝑝/2−1𝛼_𝑡𝜉(𝑡𝜏,𝑥)𝑑𝑥.(4.41) We take 𝜏_𝑡=𝑡^−2/𝑑+2. Under such a choice, 𝜏^{𝑡−𝑑/2}=𝑡𝜏_𝑡=𝑡^𝑑/𝑑+2. Observe that 𝜏^𝑝/2−1𝛼_𝑡∼1𝛼,𝑡⟶∞,(4.42) because of (1.20). By monotonicity of 𝜓, we can change the variables 𝑡^𝑑/𝑑+2↦𝑡 and, applying Theorem 4.1 with 𝐿=𝐾,𝜐(𝑢)=𝜓(𝑢/(𝛼±𝜀)), obtain limsup_𝑡→∞𝑡^{−𝑑/(𝑑+2)}log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡𝑜≤sup_{𝑔∈ℱ𝑑}𝑉𝐵_𝜅𝑠𝑑𝑠𝜈_ℝ^𝑑𝜓𝜃(𝛼−𝜀)_ℝ^𝑑𝑔²(𝑦)||𝑥−𝑦||^𝑝𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑liminf_𝑡→∞||∇𝑔(𝑦)||²𝑑𝑦,𝑡^{−𝑑/(𝑑+2)}log𝔼⊗𝔼₀exp−1𝛼_𝑡^𝑡𝑜≥sup_{𝑔∈ℱ𝑑}𝑉𝐵_𝜅𝑠𝑑𝑠𝜈_ℝ^𝑑𝜓𝜃(𝛼+𝜀)_ℝ^𝑑𝑔²(𝑦)||𝑥−𝑦||^𝑝𝑑𝑦𝑑𝑥−𝜅2_ℝ^𝑑||∇𝑔(𝑦)||²𝑑𝑦.(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 𝜓_𝑐(𝑢)=𝑐⁻²𝜓(𝑐𝑢), 𝑐>0 and 𝜓₀(𝑢)=𝑢²/2.

It can be verified that 𝜓_𝑐(𝑢)↑𝜓₀(𝑢) when 𝑐↓0. In particular, 𝜓=𝜓₁≤𝜓₀, and hence 𝜏^−𝑑/2_ℝ𝑑𝜓𝜏^𝑝/