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

Authors: Xia Chen; Alexey Kulik
Journal: International Journal of Stochastic Analysis
 (2011)

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## 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.


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## Body

## 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.Letω(dx) be a Poisson field in ℝd with intensity measure νdx, and let B be an independent Brownian motion in ℝd. Throughout, ℙ and 𝔼 denote the probability law and the expectation, respectively, generated by the Poisson field ω(dx), while ℙx 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 ℝd (known as a shape function), define the respective random function (known as a Poisson potential) (1.1)V(x)=∫RdK(y-x)ω(dy),
which heuristically represents the net force at x∈ℝd 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 (1.2)dμt,ωdP0=1Zt,ωexp⁡{-∫0tV(Bκs)ds},Zt,ω=E0exp⁡{-∫0tV(Bκs)ds}.
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 μt is defined by (1.3)dμtd(P⊗P0)=1Ztexp⁡{-∫0tV(Bκs)ds},Zt=E⊗E0exp⁡{-∫0tV(Bκs)ds}.
Heuristically, the integral (1.4)∫0tV(Bκs)ds
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 functionK 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 (1.5)K(x)=θ|x|-p,x∈Rd,
which clearly is both locally unbounded and supported by whole ℝd. 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∈ℝd 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 (1.6)V¯(x)=∫RdK(y-x)[ω(dy)-νdy].
Assume for a while that K is locally bounded and compactly supported. Then, (1.7)V¯(x)=∫RdK(y-x)[ω(dy)-νdy]=∫RdK(y-x)ω(dy)-ν∫RdK(y-x)dy=V(x)-ν∫RdK(y)dy,that is,V¯-V=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 (1.8)dμ¯t,ωdP0=1Z¯t,ωexp⁡{-∫0tV¯(Bκs)ds},Z¯t,ω=E0exp⁡{-∫0tV¯(Bκs)ds},(1.9)dμ¯td(P⊗P0)=1Z¯texp⁡{-∫0tV¯(Bκs)ds},Z¯t=E⊗E0exp⁡{-∫0tV¯(Bκs)ds},
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 Vh, partially renormalized at the level h; see [7, Chapter 6]. By definition, (1.10)Vh(x)=∫Rd(K(y-x)-h)+ω(dy)+∫Rd(K(y-x)∧h)[ω(dy)-νdy],
where h∈[0,∞] is a renormalization level. Clearly, V0=V,V∞=V¯. It is known (see [7, Chapter 6]) that Vh is well defined for every h∈(0,+∞) as soon as Vh′ is well defined for some h′∈[0,+∞], and in that case, there exists a constant CK,h,h′ such that Vh-Vh′=νCK,h,h′. This makes it possible to define the respective Gibbs measures in a uniform way, replacing V in (1.2), (1.3) by Vh with (any) h∈(0,+∞). In addition, such a definition extends the class of shape functions: for K given by (1.5), Vh with h∈(0,+∞) is well defined for p>d/2.The main objective of this paper is to study the asymptotic behavior, ast→+∞, of the annealed exponential moments (1.11)E⊗E0exp⁡{-1αt∫0tV¯(Bκs)ds}.
This problem is clearly relevant with the model discussed above: in the particular case κ=1, αt≡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 Vh with h∈(0,+∞).The functionαt in (1.11) appears, on one hand, because of our further intent to study in further publications the a.s. behavior (1.12)∫0tV¯(Bκs)ds,t⟶∞.
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(1.13)∂tu(t,x)=κΔu(t,x)±Q(x)u(t,x),u(0,x)=1,x∈Rd.
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∈ℤd} 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 (1.14)u(t,x)=Exexp⁡{±∫0tQ(B2κs)ds},
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 calledLifshitz tails asymptotic behavior of the integrated density of states function N of a random Schrödinger operator of the type (1.15)H=-κ2Δ+Q.
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 (1.16)N(λ)=lim⁡U↑Rd1|U|∑k1λk,U≤λ,
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 ℝd, and the limit pass is made w.r.t. a sequence of cubes which has same center and extends to the whole ℝd. 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 functionN, and the problem discussed above is provided by the representation for the Laplace transform of N(1.17)∫Re-λtdN(λ)=(2πκt)-d/2E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds],t≥0.
Here, 𝔼0,0κt 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 𝔼0,0κt. 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 Theorem2.1. They are formulated for the shape function defined by (1.5). Depending on p in this definition, we separate three cases (1.18)αt=o(t(d+2-p)/(d+2)),t⟶∞,(1.19)t(d+2-p)/(d+2)=o(αt),t⟶∞,(1.20)αt~αt(d+2-p)/(d+2)withsomeα>0,t⟶∞,
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(1.21)E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=E0exp⁡[ν∫Rdψ(1αtξ(t,x))dx],
with (1.22)ψ(u)=e-u-1+u,(1.23)ξ(t,x)=∫0tK(Bκs-x)ds,
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 (1.24)ν∫Rdψ(1αt∫0tK(-x)ds)dx=ν∫Rdψ(tαtK(-x))dx,
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 potentialV and a partially renormalized Poisson potential Vh (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 (2.1)Fd={g∈W21(Rd):∫Rdg2(x)dx=1},
where W21(ℝd) is used for the Sobolev space of functions that belong to ℒ2 together with their first order derivatives. We also denote (2.2)φ(u)=1-e-u,Ξ(u,v)=ψ(u)-e-uφ(v)=e-u-v-1+u,u,v∈R,
(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.
Letp∈(d/2,d).(i)
In the “light-scale” case,(2.3)lim⁡t→∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=ν∫Rdψ(θ|x|-p)dx=νωdθd/p(pd-p)Γ(2p-dp)=-νωdθd/pΓ(p-dp).(ii)
In the “critical” case,(2.4)lim⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=sup⁡g∈Fd{ν∫Rdψ(θα∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy}.(iii)
In the “heavy-scale” case, under additional assumptionp<(d+2)/2,
(2.5)lim⁡t→∞αt4/(d+2-2p)t-(d+4-2p)/(d+2-2p)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=sup⁡g∈Fd{νθ22∫Rd(∫Rdg2(y)|x-y|pdy)2dx-κ2∫Rd|∇g(y)|2dy}.Remark 2.2.
The additional assumptionp<(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.
LetK be bounded and satisfy
(2.6)K(x)~θ|x|-p,|x|⟶∞,
with p>d.(i)
(see [41]) If p∈(d,d+2),
(2.7)lim⁡t→∞t-d/plog⁡E⊗E0exp⁡[-∫0tV(Bκs)ds]=-νωdθd/pΓ(p-dp).(ii)
(see [40]) If p=d+2,
(2.8)lim⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-∫0tV(Bκs)ds]=-inf⁡g∈Fd{ν∫Rdφ(θ∫Rdg2(y)|x-y|pdy)dx+κ2∫Rd|∇g(y)|2dy}.(iii)
(see [46]) If p>d+2,
(2.9)lim⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-∫0tV(Bκs)ds]=-inf⁡g∈Fd{ν∫Rd1g(x)>0dx+κ2∫Rd|∇g(y)|2dy}.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.Theorem2.1 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 αt≡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 αt≡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 αt) and shape dependent (i.e., involves p).It is a natural question whether Theorem2.1 can be extended to other types of potentials, like a Poisson potential V or a partially renormalized Poisson potential Vh. 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 functionK can be made very mild: instead of (1.5), we assume (2.6) with p>d/2 and, when p<d, (2.10)∫Rdψ(K(x))dx<+∞,
which is just the assumption for V¯ to be well defined.Theorem 2.4.
Let the shape functionK satisfy (2.6) and scale function αt satisfy (1.18).(i)
Statement (i) of Theorem2.1 holds true assuming K satisfies (2.10).(ii)
Forp>d,
(2.11)lim⁡t→∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV(Bκs)ds]=-ν∫Rdφ(θ|x|-p)dx=-νωdθd/pΓ(p-dp).(iii)
Forp=d and h>0,
(2.12)log⁡E⊗E0exp⁡[-1αt∫0tVh(Bκs)ds]=ν[∫Rd(min⁡(K(y),h)-αtt)+dy+ωdθEu](tαt)+o(tαt),t⟶∞,
where Eu=-Γ′(1)=0,57721⋯ is the Euler constant. In particular, when K has the form (1.5),
(2.13)log⁡E⊗E0exp⁡[-1αt∫0tVh(Bκs)ds]=νωdθ[log⁡(tαt)+log⁡h+Eu](tαt)+o(tαt),t⟶∞.The following theorem shows that statements of Theorems2.1 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 𝔼0,0κt, 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 fieldQ(x), x∈ℝd and a cube U⊂ℝd, denote by HUQ the random Schrödinger operator in U with the potential Q and the Dirichlet boundary conditions
(2.14)HUQf=-κ2Δf+Qf,f|∂U=0.
When the field Q is assumed to have locally bounded realizations, the operator HUQ is a.s. well defined as an operator on L2(U,dx) and is self-adjoint. In addition, respective semigroup Rt,UQ=e-tHUQ,t≥0 has a Feynman-Kac representation ([1, page 13])
(2.15)Rt,UQf(x)=Ext(exp⁡[-∫0tQ(Bκs)ds]χU,t(B⋅)f(Bt)),x∈U,t≥0,
where
(2.16)χU,t(B⋅)=lBκs∈U,s∈[0,t].
For general Q, we define HUQ by the following limit procedure. Consider truncations QN=(|Q|∧N)sgnQ. Under appropriate assumptions on Q, for almost every realization of this field, operators Rt,UQN converge strongly for every t≥0 as N→∞. In that case, HUQ is defined as the generator of the limit semigroup Rt,UQ, t≥0. Assuming the spectrum of HUQ to be discrete (we verify this assumption below), we denote this spectrum {λk,UQ} and define the function
(2.17)NUQ(λ)=1|U|∑klλk,UQ≤λ,λ∈R.Proposition 2.6.
Let the shape functionK 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 W21(ℝd).Consider either a partially renormalized potential Q=Vh 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 potentialQ and every cube U, the described above procedure well defines both the random Schrödinger operator HUQ and respective function NUQ,(b)
there exists an integrated density of statesNQ, that is, a deterministic monotonous function such that(2.18)NQ(λ)=lim⁡U↑RdNUQ(λ),
a.s. for every point of continuity ofNQ. Respective Laplace transform has the representation
(2.19)∫Re-λtdNQ(λ)=(2πκt)-(d/2)E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds],t≥0.Note that in the proof of Proposition2.6 (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 Theorem2.5 and representation (2.19), we deduce the following Lifshitz tails asymptotics for random Schrödinger operators with random potentials V¯ and Vh.Theorem 2.7.
LetK satisfy (2.6).(i)
Forp∈(d/2,d), assuming additionally (2.10), one has in limit λ→-∞(2.20)log⁡NV¯(λ)=-[νωdΓ(2p-dp)]-p/(d-p)(θ(d-p)d)d/(d-p)(-λ)d/(d-p)(1+o(1)).(ii)
Forp=d and h∈(0,∞), one has in limitλ→-∞(2.21)log⁡NVh(λ)=-νωdθexp⁡[-λνωdθ-log⁡⁡h-Eu-1](1+o(1))=-νωdθhexp⁡[-λνωdθ-Eu-1](1+o(1)).Theorem2.7 involves the asymptotic results for exponential moments (Theorem 2.5) only in a partial form, for the trivial scale function αt≡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(2.22)Hγ=-κ2Δ+γQ,γ>0.
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 (2.23)NQ(λ,γ),λ∈R,γ>0,theintegrated 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 Theorem2.8 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.
LetK be of the form (1.5) with p∈(d/2,d).(i)
When(-λ)p/d/γ→∞ and (-λ)(d+2-p)/2/γ→∞,
(2.24)log⁡NV¯(λ,γ)=-[νωdΓ(2p-dp)]-p/(d-p)(θ(d-p)d)d/(d-p)×(-λ)d/(d-p)γ-d/(d-p)(1+o(1)).(ii)
When(-λ)d+4-2p/4/γ→∞ and (-λ)(d+2-p)/2/γ→0, under additional assumption p<(d+2)/2,
(2.25)log⁡NV¯(λ,γ)=-(2C2d+2-2p)-(d+2-2p)/2(2d+4-2p)(d+4-2p)/2(-λ)(d+4-2p)/2γ-2(1+o(1)),
where C2 denotes the constant in the r.h.s of (2.5).(iii)
Whenλ→0- and (-λ)(d+2-p)/2/γ is bounded away both from 0 and from ∞,
(2.26)log⁡NV¯(λ,γ)=-((d-p)Cψp)-p/(d-p)((d-p)d)d/(d-p)(-λ)d/(d-p)γ-p/(d-p)(1+o(1)),
where Cψ denotes the constant in the right hand side of (2.4) with α=1.Note that under the assumptions of Theorem2.8, 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 Theorem2.4, which includes Statement (i) of Theorem 2.1 as a partial case. For a given h>0, denote (3.1)ξh(t,x)=∫0t(K(y-x)-h)+ds,ξh(t,x)=∫0t(K(y-x)∧h)ds.Similarly to (1.21), we have (3.2)E⊗E0exp⁡[-1αt∫0tV(Bκs)ds]=E0exp⁡[-∫Rdφ(1αtξ(t,x))dx],(3.3)E⊗E0exp⁡[-1αt∫0tVh(Bκs)ds]=E0exp⁡[∫RdΞ(1αtξh(t,x),1αtξh(t,x))dx].
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 Vh, we consider a partially renormalized Poisson potential with the properly chosen renormalization level, dependent on t. Let g>0 and ht=gαt/t. Then, assuming p=d, (2.6) and (1.18), we will prove that (3.4)lim⁡t→∞(αtt)log⁡E⊗E0exp⁡[-1αt∫0tVht(Bκs)ds]=ν∫RdΞ((θ|x|-d)∧g,(θ|x|-d-g)+)dx=νωdθ[log⁡g+Eu].
Note that by Proposition  6.1 in [7], (3.5)Vh′(x)-Vh(x)=ν∫Rd(min⁡(K(y),h)-h′)+dy,
for any h≥h′ such that Vh, Vh′ 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 Sections3.1 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 functionϱ, by the Jensen inequality, we have (3.6)ϱ(1αtξ(t,x))=ϱ(∫0t(1αtK(Bκs-x))ds)≤1t∫0tϱ(tαtK(Bκs-x))ds.
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: (3.7)∫Rdϱ(1αtξ(t,x))dx≤1t∫Rd∫0tϱ(tαtK(Bκs-x))dsdx=∫Rdϱ(tαtK(-x))dx=(tαt)d/p∫Rdϱ(K(x,λt))dx.
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, (3.8)∫Rdψ(K(x,λ))dx⟶∫Rdψ(K(x))dxor∫Rdφ(K(x,λ))dx⟶∫Rdφ(K(x))dx,λ→∞. By assumption (2.6), for every ɛ>0, there exists λɛ such that (3.9)K(x,λ)|x|p∈[θ-ɛ,θ+ɛ],|x|>ɛ,λ>λɛ.Whenp>d, this easily provides (3.10)lim⁡λ→∞∫|x|>ɛφ(K(x,λ))dx=∫|x|>ɛφ(K(x))dx,ɛ>0.
Since φ is bounded on ℝ+, (3.10) provides the second relation in (3.8).Whenp∈(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 (3.11)lim⁡ɛ→0limsup⁡λ→∞∫|x|≤ɛψ(K(x,λ))dx=0.
To that end, we choose r1, θ1 such that K(x)≤θ1|x|-p, |x|>r and write for λ large enough (3.12)∫|x|≤ɛψ(K(x,λ))dx=[∫|x|≤r1/λ+∫r1/λ<|x|≤ɛ]ψ(K(x,λ))dx≤λ-d∫|x|≤r1ψ(λpK(x))dx+∫|x|≤ɛψ(θ1|x|-p)dx.
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, forp=d from the Jensen’s inequality for the convex function Ξ:ℝ2→ℝ, we have (3.13)Ξ(1αtξh(t,x),1αtξh(t,x))≤1t∫0tΞ(tαt(K(Bκs-x)∧h),tαt(K(Bκs-x)-h)+)ds,and consequently, forht=gαt/t(3.14)∫RdΞ(1αtξht(t,x),1αtξht(t,x))dx≤∫RdΞ(tαt(K(-x)∧ht),tαt(K(-x)-ht)+)dx=(tαt)d/p∫RdΞ(K(x,λt)∧g,(K(x,λt)-g)+)dx.
Similarly to (3.8), one can prove (3.15)∫RdΞ(K(x,λ)∧g,(K(x,λ)-g)+)dx⟶∫RdΞ(K(x,λ)∧g,(K(x,λ)-g)+)dx,λ⟶∞,
which provides the upper bound in (3.4).


### 3.2. Proof of the Lower Bound

For a fixedɛ>0, take R fixed but large enough so that (3.16)(θ-ɛ)|x|-p≤K(x)≤(θ+ɛ)|x|-p,|x|≥R.
Take β>0 and consider the set (3.17)At,β={sup⁡s≤t|Bκs|≤βλt},
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, (3.18)log⁡P0(At,β)≥-ct(βλt)-2
with some constant c>0. Therefore, condition αt=o(t(d+2-p)/(d+2)) yields (3.19)(αtt)d/plog⁡P0(At,β)⟶0,t⟶+∞.
Take γ>2β. On the set At,β, one has (3.20)|Bκs-x|≥β(tαt)1/p,s∈[0,t],|x|≥γλt.
Then, for t large enough to provide βλt>R, we have (3.21)(θ-ɛ)|Bκs-x|-p≤K(Bκs-x)≤(θ+ɛ)|Bκs-x|-p,s∈[0,t],|x|≥γλt.
Therefore, a two-sided estimate (3.22)(θ-ɛ)(1+βγ)-p|x|-p≤K(Bκs-x)≤(θ+ɛ)(1-βγ)-p|x|-p,s∈[0,t]
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).Sinceψ is nonnegative and nondecreasing, (3.22) yields (3.23)∫Rdψ(1αtξ(t,x))dx≥∫|x|>γλtψ(tαt(θ-ɛ)(1+βγ)-p|x|-p)dx=(tαt)d/pIɛ,β,γψ,
on At,β with (3.24)Iɛ,β,γψ=∫|x|>γψ((θ-ɛ)(1+βγ)-p|x|-p)dx.
Together with (1.21) and (3.19), this inequality provides (3.25)liminf⁡t→+∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥Iɛ,β,γψ,
for every ɛ>0, β>0, γ>0. Since (3.26)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γψ=∫Rdψ(θ|x|-p)dx,
this completes the proof of the lower bound in (2.3).Since(-φ) is nonincreasing and satisfies -φ≥-1, (3.22) yields (3.27)-∫Rdφ(1αtξ(t,x))dx≥-∫|x|≤γλtdx-∫|x|>γλtφ(tαt(θ+ɛ)(1-βγ)-p|x|-p)dx=(tαt)d/pIɛ,β,γφ,
on At,β with (3.28)Iɛ,β,γφ=-∫|x|≤γdx-∫|x|>γφ((θ+ɛ)(1-βγ)-p|x|-p)dx.Since(3.29)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γφ=-∫Rdφ(θ|x|-p)dx,
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(3.30)∫RdΞ(1αtξht(t,x),1αtξht(t,x))dx≥∫|x|>γλtΞ(tαt[{(θ-ɛ)(1+βγ)-d|x|-d}∧{gαtt}],tαt[{(θ+ɛ)(1-βγ)-d|x|-d}-tgαtt]+)dx=(tαt)Iɛ,β,γΞ,
on At,β with (3.31)Iɛ,β,γΞ=-∫|x|≤γdx+∫|x|>γΞ([(θ-ɛ)(1+βγ)-d|x|-d]∧g,[(θ+ɛ)(1-βγ)-d|x|-d-g]+)dx.
Together with (3.2) and (3.19), this inequality provides (3.32)lim⁡inf⁡t→+∞(αtt)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥Iɛ,β,γΞ,
for every ɛ>0, β>0, γ>0. Since (3.33)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γΞ=∫RdΞ((θ|x|-d)∧g,(θ|x|-d-g)+)dx,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,(3.34)∫RdΞ((θ|x|-d)∧g,(θ|x|-d-g)+)dx=ωd∫0∞Ξ((θr)∧g,(θr-g)+)dx=ωd∫0∞[e-θ/r-1-(θr)∧g]dr=θωd∫0∞e-s-1-s∧gs2ds=ωdθ[∫0∞e-s-1-s∧1s2ds+log⁡g],
in the last identity we have used an elementary relation (3.35)∫0∞s∧g-s∧1s2ds=log⁡g.
Integration by parts and n. 538 in [47] gives (3.36)∫0∞e-s-1-s∧1s2ds=∫0t1-e-ssds-∫0∞e-ssds=Eu,
which completes calculation of the integral in (3.4).Finally, letK has the form (1.5). Take ht=αt/t, then ht<h for t large enough, and (3.37)∫Rd(min⁡(K(y),h)-αtt)+dy=∫Rd(θ|x|-d∧h-ht)+dx=ωdθ∫0∞s∧h-s∧hts2ds=ωdθ[log⁡h-log⁡ht]=ωdθ[log⁡h+log⁡(tαt)].
Combined with (2.12), this calculation provides (2.13).


## 3.1. Proof of the Upper Bound

For any convex functionϱ, by the Jensen inequality, we have (3.6)ϱ(1αtξ(t,x))=ϱ(∫0t(1αtK(Bκs-x))ds)≤1t∫0tϱ(tαtK(Bκs-x))ds.
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: (3.7)∫Rdϱ(1αtξ(t,x))dx≤1t∫Rd∫0tϱ(tαtK(Bκs-x))dsdx=∫Rdϱ(tαtK(-x))dx=(tαt)d/p∫Rdϱ(K(x,λt))dx.
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, (3.8)∫Rdψ(K(x,λ))dx⟶∫Rdψ(K(x))dxor∫Rdφ(K(x,λ))dx⟶∫Rdφ(K(x))dx,λ→∞. By assumption (2.6), for every ɛ>0, there exists λɛ such that (3.9)K(x,λ)|x|p∈[θ-ɛ,θ+ɛ],|x|>ɛ,λ>λɛ.Whenp>d, this easily provides (3.10)lim⁡λ→∞∫|x|>ɛφ(K(x,λ))dx=∫|x|>ɛφ(K(x))dx,ɛ>0.
Since φ is bounded on ℝ+, (3.10) provides the second relation in (3.8).Whenp∈(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 (3.11)lim⁡ɛ→0limsup⁡λ→∞∫|x|≤ɛψ(K(x,λ))dx=0.
To that end, we choose r1, θ1 such that K(x)≤θ1|x|-p, |x|>r and write for λ large enough (3.12)∫|x|≤ɛψ(K(x,λ))dx=[∫|x|≤r1/λ+∫r1/λ<|x|≤ɛ]ψ(K(x,λ))dx≤λ-d∫|x|≤r1ψ(λpK(x))dx+∫|x|≤ɛψ(θ1|x|-p)dx.
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, forp=d from the Jensen’s inequality for the convex function Ξ:ℝ2→ℝ, we have (3.13)Ξ(1αtξh(t,x),1αtξh(t,x))≤1t∫0tΞ(tαt(K(Bκs-x)∧h),tαt(K(Bκs-x)-h)+)ds,and consequently, forht=gαt/t(3.14)∫RdΞ(1αtξht(t,x),1αtξht(t,x))dx≤∫RdΞ(tαt(K(-x)∧ht),tαt(K(-x)-ht)+)dx=(tαt)d/p∫RdΞ(K(x,λt)∧g,(K(x,λt)-g)+)dx.
Similarly to (3.8), one can prove (3.15)∫RdΞ(K(x,λ)∧g,(K(x,λ)-g)+)dx⟶∫RdΞ(K(x,λ)∧g,(K(x,λ)-g)+)dx,λ⟶∞,
which provides the upper bound in (3.4).


## 3.2. Proof of the Lower Bound

For a fixedɛ>0, take R fixed but large enough so that (3.16)(θ-ɛ)|x|-p≤K(x)≤(θ+ɛ)|x|-p,|x|≥R.
Take β>0 and consider the set (3.17)At,β={sup⁡s≤t|Bκs|≤βλt},
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, (3.18)log⁡P0(At,β)≥-ct(βλt)-2
with some constant c>0. Therefore, condition αt=o(t(d+2-p)/(d+2)) yields (3.19)(αtt)d/plog⁡P0(At,β)⟶0,t⟶+∞.
Take γ>2β. On the set At,β, one has (3.20)|Bκs-x|≥β(tαt)1/p,s∈[0,t],|x|≥γλt.
Then, for t large enough to provide βλt>R, we have (3.21)(θ-ɛ)|Bκs-x|-p≤K(Bκs-x)≤(θ+ɛ)|Bκs-x|-p,s∈[0,t],|x|≥γλt.
Therefore, a two-sided estimate (3.22)(θ-ɛ)(1+βγ)-p|x|-p≤K(Bκs-x)≤(θ+ɛ)(1-βγ)-p|x|-p,s∈[0,t]
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).Sinceψ is nonnegative and nondecreasing, (3.22) yields (3.23)∫Rdψ(1αtξ(t,x))dx≥∫|x|>γλtψ(tαt(θ-ɛ)(1+βγ)-p|x|-p)dx=(tαt)d/pIɛ,β,γψ,
on At,β with (3.24)Iɛ,β,γψ=∫|x|>γψ((θ-ɛ)(1+βγ)-p|x|-p)dx.
Together with (1.21) and (3.19), this inequality provides (3.25)liminf⁡t→+∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥Iɛ,β,γψ,
for every ɛ>0, β>0, γ>0. Since (3.26)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γψ=∫Rdψ(θ|x|-p)dx,
this completes the proof of the lower bound in (2.3).Since(-φ) is nonincreasing and satisfies -φ≥-1, (3.22) yields (3.27)-∫Rdφ(1αtξ(t,x))dx≥-∫|x|≤γλtdx-∫|x|>γλtφ(tαt(θ+ɛ)(1-βγ)-p|x|-p)dx=(tαt)d/pIɛ,β,γφ,
on At,β with (3.28)Iɛ,β,γφ=-∫|x|≤γdx-∫|x|>γφ((θ+ɛ)(1-βγ)-p|x|-p)dx.Since(3.29)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γφ=-∫Rdφ(θ|x|-p)dx,
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(3.30)∫RdΞ(1αtξht(t,x),1αtξht(t,x))dx≥∫|x|>γλtΞ(tαt[{(θ-ɛ)(1+βγ)-d|x|-d}∧{gαtt}],tαt[{(θ+ɛ)(1-βγ)-d|x|-d}-tgαtt]+)dx=(tαt)Iɛ,β,γΞ,
on At,β with (3.31)Iɛ,β,γΞ=-∫|x|≤γdx+∫|x|>γΞ([(θ-ɛ)(1+βγ)-d|x|-d]∧g,[(θ+ɛ)(1-βγ)-d|x|-d-g]+)dx.
Together with (3.2) and (3.19), this inequality provides (3.32)lim⁡inf⁡t→+∞(αtt)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥Iɛ,β,γΞ,
for every ɛ>0, β>0, γ>0. Since (3.33)lim⁡ɛ→0lim⁡γ→0lim⁡β→0Iɛ,β,γΞ=∫RdΞ((θ|x|-d)∧g,(θ|x|-d-g)+)dx,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,(3.34)∫RdΞ((θ|x|-d)∧g,(θ|x|-d-g)+)dx=ωd∫0∞Ξ((θr)∧g,(θr-g)+)dx=ωd∫0∞[e-θ/r-1-(θr)∧g]dr=θωd∫0∞e-s-1-s∧gs2ds=ωdθ[∫0∞e-s-1-s∧1s2ds+log⁡g],
in the last identity we have used an elementary relation (3.35)∫0∞s∧g-s∧1s2ds=log⁡g.
Integration by parts and n. 538 in [47] gives (3.36)∫0∞e-s-1-s∧1s2ds=∫0t1-e-ssds-∫0∞e-ssds=Eu,
which completes calculation of the integral in (3.4).Finally, letK has the form (1.5). Take ht=αt/t, then ht<h for t large enough, and (3.37)∫Rd(min⁡(K(y),h)-αtt)+dy=∫Rd(θ|x|-d∧h-ht)+dx=ωdθ∫0∞s∧h-s∧hts2ds=ωdθ[log⁡h-log⁡ht]=ωdθ[log⁡h+log⁡(tαt)].
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 functionL:ℝd→ℝ+ and denote (4.1)η(t,x)=∫0tL(Bκs-x)dx.Theorem 4.1.
Let, for some sequenceLn,n≥1 of nonnegative continuous compactly supported functions,
(4.2)L(x)=sup⁡n≥1Ln(x),
for a.a. x∈ℝd. Let υ:ℝ+→ℝ be an increasing convex function with υ(0)=0, and
(4.3)∫Rdυ(L(x))dx<+∞.
Then,
(4.4)lim⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}=sup⁡g∈Fd{∫Rdυ(∫Rdg2(y)L(x-y)dy)dx-κ2∫Rd|∇g(y)|2dy}.
Proof of Theorem 4.1: the lower bound. By Jensen’s inequality,(4.5)υ(1tη(t,x))=υ(1t∫0tL(Bκs-x)ds)≤1t∫0tυ(L(Bκs-x))ds,
and therefore, (4.6)∫Rdυ(1tη(t,x))dx≤1t∫0t∫Rdυ(L(Bκs-x))dxds=∫Rdυ(L(x))dx<+∞.
For every R>0, we write (4.7)∫Rdυ(1tη(t,x))dx≥∫[-R,R]dυ(1tη(t,x))dx,and note that (4.6) provides (1/t)η(t,·)∈ℒ1([-R,R]d), because υ has at least linear growth at +∞.For a fixedR, denote (4.8)L1,R+={h∈L1([-R,R]d),h≥0},
and consider a convex function ΥR:ℒ1,R+→[0,+∞](4.9)ΥR(h)=∫[-R,R]dυ(h(x))dx.
Denote by BR the class of bounded measurable functions f:[-R,R]d→ℝ, and put (4.10)CΥ,f,R=sup⁡{C:C+∫[-R,R]df(x)h(x)dx≤ΥR(h),h∈L1,R+},f∈BR.Lemma 4.2.
For everyh∈ℒ1,R+ with ΥR(h)<+∞,
(4.11)ΥR(h)=sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h(x)dx).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(4.12)epi⁡ΥR={(h,t):h∈L1,R+,t≥ΥR(h)},
clearly, epiΥR 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 epiΥR is the intersection of all the closed half-spaces containing epiΥR. Note that every continuous linear functional on the space ℒ1([-R,R]d)×ℝ has the form (f,a) with f∈BR, a∈ℝ and
(4.13)〈(h,t),(f,a)〉=∫[-R,R]dh(x)f(x)dx+at.
Take h*∈ℒυ+ with ΥR(h*)<+∞, and t*<ΥR(h*). Then (h*,t*)∉epiΥR, and therefore, there exists (f,a) and c∈ℝ such that
(4.14)〈(h*,t*),(f,a)〉<c,〈(h,t),(f,a)〉≥c,(h,t)∈epi⁡ΥR.
By the definition of epiΥR, if (h,t)∈epiΥR, then (h,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,
(4.15)ca+∫[-R,R]dh*(x)fa(x)dx>t*,ca+∫[-R,R]dh(x)fa(x)dx≤t,(h,t)∈epi⁡ΥR.
Take t=ΥR(h) in the second inequality in (4.15); this yields ca≤CΥ,fa,R. Consequently,
(4.16)t*≤sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h*(x)dx),
which means that
(4.17)ΥR(h*)≤sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h*(x)dx),
because t*<ΥR(h*) is arbitrary. The inverse inequality is obvious.Takef∈BR, then
(4.18)E0exp⁡{t∫[-R,R]dυ(1tη(t,x))dx}≥eCΥ,f,RtE0exp⁡{∫[-R,R]df(x)η(t,x)dx}.
Note that
(4.19)∫[-R,R]df(x)η(t,x)dx=∫0tf̂(Bκs)ds,
where
(4.20)f̂(y)=∫[-R,R]df(x)L(y-x)dx
is a bounded function. Henceforth, by the large deviations result by Kac [50] (see also Theorem  4.1.6 in [51]), we have
(4.21)liminf⁡t→∞1tlog⁡E0exp⁡{t∫[-R,R]dυ(1tη(t,x))dx}≥CΥ,f,R+lim⁡t→∞1tlog⁡E0exp⁡{∫0tf̂(Bκs)ds}≥CΥ,f,R+sup⁡g∈Fd{∫Rdf̂(x)g2(x)dx-12∫Rd|∇g(x)|2dx}.
Note that
(4.22)∫Rdf̂(x)g2(x)dx=∫[-R,R]df(x)[∫Rdg2(y)L(x-y)dy]dx.
Summarizing our proof,
(4.23)liminf⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}≥sup⁡g∈Fd{CΥ,f,R+∫[-R,R]df(x)[∫Rdg2(y)L(x-y)dy]dx-12∫Rd|∇g(x)|2dx},
for every R>0, f∈BR. We take supremum over f∈BR and get, by Lemma 4.2,
(4.24)liminf⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}≥sup⁡g∈Fd{ΥR(∫Rdg2(y)L(⋅-y)dy)-12∫Rd|∇g(x)|2dx}.
Note that by Jensen’s inequality, for every g∈ℱd,
(4.25)ΥR(∫Rdg2(y)L(⋅-y)dy)=∫[-R,R]dυ(∫Rdg2(y)L(x-y)dy)dx≤∫[-R,R]d∫Rdg2(y)υ(L(x-y))dydx≤∫Rdυ(L(x))dx<+∞,
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 Theorem4.1 (the upper bound).
Assume first thatL 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 thatυ(u+v)-υ(u)≥υ(v)-υ(0) because of the convexity, and υ(0)=0. Hence, the function υ satisfies
(4.26)υ(u+v)≥υ(u)+υ(v),u,v≥0.
Thus, for any N>M,
(4.27)∫Rdυ(1tη(t,x))dx=∑z∈Zd∫[-N,N]dυ(1tη(t,2Nz+x))dx≤∫[-N,N]dυ(1t∑z∈Zdη(t,2Nz+x))dx=∫[-N,N]dυ(1tη̃(t,x))dx,
where
(4.28)η̃(t,x)=∫0tL̃(Bκs-x)ds,L̃(x)=∑z∈ZdL(2Nz+x).
Denote by TdN 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 x∈ℝd its projection JN(x) is the unique point x̃∈TdN such that x-x̃∈2Nℤd. Denote BsN=JN(B·), the Brownian motion on the torus TdN. With this notation in mind, we rewrite the right hand side term in (4.27)
(4.29)∫[-N,N]dυ(1tη̃(t,x))dx=∫TdNυ(1tηN(t,x))dx,ηN(t,x)=∫0t(L∘JN)(Bκs-x)ds.
Consider the empirical measures for the Brownian motion on the torus TdN(4.30)QtN(A)=1t∫0t1BκsN∈Ads,A∈B(TdN).
Note that
(4.31)1tηN(t,x)=∫TdN(L∘JN)(y-x)QtN(dy),
and the mapping
(4.32)μ⟶∫TdNυ(∫TdN(L∘JN)(y-x)μ(dy))dx
is continuous and bounded on the space of all probability distributions on TdN with the metrics of weak convergence. Hence, combination of (4.27), the large deviation principle for QtN (Theorem  3 in [52]), and Varadhan’s lemma (Proposition  3.8 in [53]) yields
(4.33)limsup⁡t→∞1tE0exp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈FdN{∫TdNυ(∫TdN(L∘JN)(x-y)g2(y)dy)dx-κ2∫TdN|∇g(y)|2dy},
where
(4.34)FdN={g∈W21(TdN):∫TdNg2(x)dx=1}.
By smooth truncation, it is easy to verify that
(4.35)sup⁡g∈Fd{∫Rdυ(∫RdL(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}≤liminf⁡N→∞sup⁡g∈FdN{∫TdNυ(∫TdN(L∘JN)(x-y)g2(y)dy)dx-κ2∫TdN|∇g(y)|2dy},
which completes the proof.
Finally, we remove the additional regularity assumption onL. Recall the assumption (4.2) and note that one can assume the sequence Ln,n≥1 to be pointwise increasing, because otherwise, one can take L̃n=maxk≤nLn instead.
WriteΔn=L-Ln and
(4.36)η(t,x)=ηn(t,x)+ζn(t,x),ζn(t,x)=∫0tΔn(Bκs-x)ds.
For every γ∈(0,1), we have by convexity
(4.37)υ(1tη(t,x))≤γυ(1γtηn(t,x))+(1-γ)υ(1(1-γ)tζn(t,x))
The Jensen inequality, analogously to (4.6), provides that
(4.38)∫Rdυ(1(1-γ)tζn(t,x))dx≤∫Rdυ(1(1-γ)Δn(x))dx.
Then, from the upper bound with a regular kernel Ln, we obtain
(4.39)limsup⁡t→∞1tE0exp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈Fd{γ∫Rdυ(1γ∫RdLn(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}+∫Rdυ(1(1-γ)Δn(x))dx,
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 Theorem2.1: 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, (4.40)ξ(t,x)=d∫0tK(τ-1/2Bsτ-x)=τp/(2-1)∫0tτK(Bκs-τ1/2x)ds=τp/(2-1)ξ(tτ,xτ1/2).
Henceforth, the integral under the exponent in the right-hand side of (1.21), after the variable change τ1/2x↦x, can be written as (4.41)τ-d/2∫Rdψ(τp/2-1αtξ(tτ,x))dx.
We take τt=t-2/d+2. Under such a choice, τt-d/2=tτt=td/d+2. Observe that (4.42)τp/2-1αt~1α,t⟶∞,
because of (1.20). By monotonicity of ψ, we can change the variables td/d+2↦t and, applying Theorem 4.1 with L=K,υ(u)=ψ(u/(α±ɛ)), obtain (4.43)limsup⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-1αt∫otV¯(Bκs)ds]≤sup⁡g∈Fd{ν∫Rdψ(θ(α-ɛ)∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy},liminf⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-1αt∫otV¯(Bκs)ds]≥sup⁡g∈Fd{ν∫Rdψ(θ(α+ɛ)∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy}.Passing to the limit asɛ→∞ completes the proof of Statement (ii) of Theorem 2.1.


### 4.3. Proof of Theorem2.1: 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)=c-2ψ(cu), c>0 and ψ0(u)=u2/2.It can be verified thatψc(u)↑ψ0(u) when c↓0. In particular, ψ=ψ1≤ψ0, and hence (4.44)τ-d/2∫Rdψ(τp/2-1αtξ(tτ,x))dx≤12τp-2-d/2αt2∫Rdξ2(tτ,x)dx=12t2τp-d/2αt2∫Rd(1tτξ(tτ,x))2dx.
Choose τt in such a way that (4.45)t2τtp-d/2αt2=tτt,
that is, (4.46)τt=αt-4/(d+2-2p)t2/(d+2-2p).
Under such a choice, (4.47)tτt=αt-4/(d+2-2p)t(d+4-2p)/(d+2-2p),
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, takeτt from (4.46) and write (4.48)τt-d/2∫Rdψ(τtp/2-1αtξ(tτt,x))dx=τtp-2-d/2αt2τ-d/2∫Rdψct(1tτtξ(tτt,x))dx,
with (4.49)ct=αt-1tτtp/2=αt2p/(d+2-2p)-1t1-p/(d+2-2p)=t(d+2-p)/(d+2-2p)αt-(d+2)/(d+2-2p).
Condition (1.19) and assumption p<(d+2)/2 provide ct→0. Then, for every fixed c>0, we have ct<c for t large enough, and therefore, (4.50)liminf⁡t→∞αt4/(d+2-2p)t-(d+4-2p)/(d+2-2p)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥sup⁡g∈Fd{ν∫Rdψc(θα∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy},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 ℝd, independent on B. Write, for a given shape function K, (4.51)U(x)=∫RdK(y-x)W(dy),Ut=∫0tU(Bκs)ds=∫Rdξ(t,x)W(dx),
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 K∉ℒ2(ℝd) for any p. On the other hand, for p∈(d/2,(d+2)/2), one has (4.52)E0∫Rdξ2(t,x)dx<+∞,and consequentlyUt 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, (4.53)lim⁡t→∞αt4/(d+2-2p)t-(d+4-2p)/(d+2-2p)log⁡E⊗E0exp⁡[-1αtUt]=sup⁡g∈Fd{θ22∫Rd(∫Rdg2(y)|x-y|pdy)2dx-κ2∫Rd|∇g(y)|2dy}.
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.
LetK have the form (1.5) with p∈(d/2,(d+2)/2). Then, for every t>0, the distribution of
(4.54)ν-1/2∫0tV¯(Bκs)ds,
w.r.t. ℙ⊗ℙ0 weakly converges as ν→∞ to the distribution of Ut w.r.t. ℙW⊗ℙ0, where ℙW denotes the distribution of the Wiener sheet W.Proof.
Characteristic function of (4.54) is equal
(4.55)E0exp⁡[ν∫Rd(eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x))dx],
see Proposition 2.2 in [7]. For every z∈ℂ,t>0,x∈ℝd,
(4.56)ν(eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x))⟶-z22ξ2(t,x),ν⟶∞.
In addition,
(4.57)ν|eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x)|≤C0z2ξ2(t,x),
with some constant C0. Since for p∈(d/2,(d+2)/2)(4.58)E0eC∫Rdξ2(t,x)dx<∞,t>0,C>0,
(see [45]), we have by the dominated convergence theorem that the characteristic functions (4.55) converge pointwise to
(4.59)E0exp⁡[-z22∫Rdξ2(t,x)dx],
which is just the characteristic function of Ut.


## 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 functionL:ℝd→ℝ+ and denote (4.1)η(t,x)=∫0tL(Bκs-x)dx.Theorem 4.1.
Let, for some sequenceLn,n≥1 of nonnegative continuous compactly supported functions,
(4.2)L(x)=sup⁡n≥1Ln(x),
for a.a. x∈ℝd. Let υ:ℝ+→ℝ be an increasing convex function with υ(0)=0, and
(4.3)∫Rdυ(L(x))dx<+∞.
Then,
(4.4)lim⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}=sup⁡g∈Fd{∫Rdυ(∫Rdg2(y)L(x-y)dy)dx-κ2∫Rd|∇g(y)|2dy}.
Proof of Theorem 4.1: the lower bound. By Jensen’s inequality,(4.5)υ(1tη(t,x))=υ(1t∫0tL(Bκs-x)ds)≤1t∫0tυ(L(Bκs-x))ds,
and therefore, (4.6)∫Rdυ(1tη(t,x))dx≤1t∫0t∫Rdυ(L(Bκs-x))dxds=∫Rdυ(L(x))dx<+∞.
For every R>0, we write (4.7)∫Rdυ(1tη(t,x))dx≥∫[-R,R]dυ(1tη(t,x))dx,and note that (4.6) provides (1/t)η(t,·)∈ℒ1([-R,R]d), because υ has at least linear growth at +∞.For a fixedR, denote (4.8)L1,R+={h∈L1([-R,R]d),h≥0},
and consider a convex function ΥR:ℒ1,R+→[0,+∞](4.9)ΥR(h)=∫[-R,R]dυ(h(x))dx.
Denote by BR the class of bounded measurable functions f:[-R,R]d→ℝ, and put (4.10)CΥ,f,R=sup⁡{C:C+∫[-R,R]df(x)h(x)dx≤ΥR(h),h∈L1,R+},f∈BR.Lemma 4.2.
For everyh∈ℒ1,R+ with ΥR(h)<+∞,
(4.11)ΥR(h)=sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h(x)dx).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(4.12)epi⁡ΥR={(h,t):h∈L1,R+,t≥ΥR(h)},
clearly, epiΥR 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 epiΥR is the intersection of all the closed half-spaces containing epiΥR. Note that every continuous linear functional on the space ℒ1([-R,R]d)×ℝ has the form (f,a) with f∈BR, a∈ℝ and
(4.13)〈(h,t),(f,a)〉=∫[-R,R]dh(x)f(x)dx+at.
Take h*∈ℒυ+ with ΥR(h*)<+∞, and t*<ΥR(h*). Then (h*,t*)∉epiΥR, and therefore, there exists (f,a) and c∈ℝ such that
(4.14)〈(h*,t*),(f,a)〉<c,〈(h,t),(f,a)〉≥c,(h,t)∈epi⁡ΥR.
By the definition of epiΥR, if (h,t)∈epiΥR, then (h,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,
(4.15)ca+∫[-R,R]dh*(x)fa(x)dx>t*,ca+∫[-R,R]dh(x)fa(x)dx≤t,(h,t)∈epi⁡ΥR.
Take t=ΥR(h) in the second inequality in (4.15); this yields ca≤CΥ,fa,R. Consequently,
(4.16)t*≤sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h*(x)dx),
which means that
(4.17)ΥR(h*)≤sup⁡f∈BR(CΥ,f,R+∫[-R,R]df(x)h*(x)dx),
because t*<ΥR(h*) is arbitrary. The inverse inequality is obvious.Takef∈BR, then
(4.18)E0exp⁡{t∫[-R,R]dυ(1tη(t,x))dx}≥eCΥ,f,RtE0exp⁡{∫[-R,R]df(x)η(t,x)dx}.
Note that
(4.19)∫[-R,R]df(x)η(t,x)dx=∫0tf̂(Bκs)ds,
where
(4.20)f̂(y)=∫[-R,R]df(x)L(y-x)dx
is a bounded function. Henceforth, by the large deviations result by Kac [50] (see also Theorem  4.1.6 in [51]), we have
(4.21)liminf⁡t→∞1tlog⁡E0exp⁡{t∫[-R,R]dυ(1tη(t,x))dx}≥CΥ,f,R+lim⁡t→∞1tlog⁡E0exp⁡{∫0tf̂(Bκs)ds}≥CΥ,f,R+sup⁡g∈Fd{∫Rdf̂(x)g2(x)dx-12∫Rd|∇g(x)|2dx}.
Note that
(4.22)∫Rdf̂(x)g2(x)dx=∫[-R,R]df(x)[∫Rdg2(y)L(x-y)dy]dx.
Summarizing our proof,
(4.23)liminf⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}≥sup⁡g∈Fd{CΥ,f,R+∫[-R,R]df(x)[∫Rdg2(y)L(x-y)dy]dx-12∫Rd|∇g(x)|2dx},
for every R>0, f∈BR. We take supremum over f∈BR and get, by Lemma 4.2,
(4.24)liminf⁡t→∞1tlog⁡E0exp⁡{t∫Rdυ(1tη(t,x))dx}≥sup⁡g∈Fd{ΥR(∫Rdg2(y)L(⋅-y)dy)-12∫Rd|∇g(x)|2dx}.
Note that by Jensen’s inequality, for every g∈ℱd,
(4.25)ΥR(∫Rdg2(y)L(⋅-y)dy)=∫[-R,R]dυ(∫Rdg2(y)L(x-y)dy)dx≤∫[-R,R]d∫Rdg2(y)υ(L(x-y))dydx≤∫Rdυ(L(x))dx<+∞,
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 Theorem4.1 (the upper bound).
Assume first thatL 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 thatυ(u+v)-υ(u)≥υ(v)-υ(0) because of the convexity, and υ(0)=0. Hence, the function υ satisfies
(4.26)υ(u+v)≥υ(u)+υ(v),u,v≥0.
Thus, for any N>M,
(4.27)∫Rdυ(1tη(t,x))dx=∑z∈Zd∫[-N,N]dυ(1tη(t,2Nz+x))dx≤∫[-N,N]dυ(1t∑z∈Zdη(t,2Nz+x))dx=∫[-N,N]dυ(1tη̃(t,x))dx,
where
(4.28)η̃(t,x)=∫0tL̃(Bκs-x)ds,L̃(x)=∑z∈ZdL(2Nz+x).
Denote by TdN 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 x∈ℝd its projection JN(x) is the unique point x̃∈TdN such that x-x̃∈2Nℤd. Denote BsN=JN(B·), the Brownian motion on the torus TdN. With this notation in mind, we rewrite the right hand side term in (4.27)
(4.29)∫[-N,N]dυ(1tη̃(t,x))dx=∫TdNυ(1tηN(t,x))dx,ηN(t,x)=∫0t(L∘JN)(Bκs-x)ds.
Consider the empirical measures for the Brownian motion on the torus TdN(4.30)QtN(A)=1t∫0t1BκsN∈Ads,A∈B(TdN).
Note that
(4.31)1tηN(t,x)=∫TdN(L∘JN)(y-x)QtN(dy),
and the mapping
(4.32)μ⟶∫TdNυ(∫TdN(L∘JN)(y-x)μ(dy))dx
is continuous and bounded on the space of all probability distributions on TdN with the metrics of weak convergence. Hence, combination of (4.27), the large deviation principle for QtN (Theorem  3 in [52]), and Varadhan’s lemma (Proposition  3.8 in [53]) yields
(4.33)limsup⁡t→∞1tE0exp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈FdN{∫TdNυ(∫TdN(L∘JN)(x-y)g2(y)dy)dx-κ2∫TdN|∇g(y)|2dy},
where
(4.34)FdN={g∈W21(TdN):∫TdNg2(x)dx=1}.
By smooth truncation, it is easy to verify that
(4.35)sup⁡g∈Fd{∫Rdυ(∫RdL(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}≤liminf⁡N→∞sup⁡g∈FdN{∫TdNυ(∫TdN(L∘JN)(x-y)g2(y)dy)dx-κ2∫TdN|∇g(y)|2dy},
which completes the proof.
Finally, we remove the additional regularity assumption onL. Recall the assumption (4.2) and note that one can assume the sequence Ln,n≥1 to be pointwise increasing, because otherwise, one can take L̃n=maxk≤nLn instead.
WriteΔn=L-Ln and
(4.36)η(t,x)=ηn(t,x)+ζn(t,x),ζn(t,x)=∫0tΔn(Bκs-x)ds.
For every γ∈(0,1), we have by convexity
(4.37)υ(1tη(t,x))≤γυ(1γtηn(t,x))+(1-γ)υ(1(1-γ)tζn(t,x))
The Jensen inequality, analogously to (4.6), provides that
(4.38)∫Rdυ(1(1-γ)tζn(t,x))dx≤∫Rdυ(1(1-γ)Δn(x))dx.
Then, from the upper bound with a regular kernel Ln, we obtain
(4.39)limsup⁡t→∞1tE0exp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈Fd{γ∫Rdυ(1γ∫RdLn(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}+∫Rdυ(1(1-γ)Δn(x))dx,
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 Theorem2.1: 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, (4.40)ξ(t,x)=d∫0tK(τ-1/2Bsτ-x)=τp/(2-1)∫0tτK(Bκs-τ1/2x)ds=τp/(2-1)ξ(tτ,xτ1/2).
Henceforth, the integral under the exponent in the right-hand side of (1.21), after the variable change τ1/2x↦x, can be written as (4.41)τ-d/2∫Rdψ(τp/2-1αtξ(tτ,x))dx.
We take τt=t-2/d+2. Under such a choice, τt-d/2=tτt=td/d+2. Observe that (4.42)τp/2-1αt~1α,t⟶∞,
because of (1.20). By monotonicity of ψ, we can change the variables td/d+2↦t and, applying Theorem 4.1 with L=K,υ(u)=ψ(u/(α±ɛ)), obtain (4.43)limsup⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-1αt∫otV¯(Bκs)ds]≤sup⁡g∈Fd{ν∫Rdψ(θ(α-ɛ)∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy},liminf⁡t→∞t-d/(d+2)log⁡E⊗E0exp⁡[-1αt∫otV¯(Bκs)ds]≥sup⁡g∈Fd{ν∫Rdψ(θ(α+ɛ)∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy}.Passing to the limit asɛ→∞ completes the proof of Statement (ii) of Theorem 2.1.


## 4.3. Proof of Theorem2.1: 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)=c-2ψ(cu), c>0 and ψ0(u)=u2/2.It can be verified thatψc(u)↑ψ0(u) when c↓0. In particular, ψ=ψ1≤ψ0, and hence (4.44)τ-d/2∫Rdψ(τp/2-1αtξ(tτ,x))dx≤12τp-2-d/2αt2∫Rdξ2(tτ,x)dx=12t2τp-d/2αt2∫Rd(1tτξ(tτ,x))2dx.
Choose τt in such a way that (4.45)t2τtp-d/2αt2=tτt,
that is, (4.46)τt=αt-4/(d+2-2p)t2/(d+2-2p).
Under such a choice, (4.47)tτt=αt-4/(d+2-2p)t(d+4-2p)/(d+2-2p),
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, takeτt from (4.46) and write (4.48)τt-d/2∫Rdψ(τtp/2-1αtξ(tτt,x))dx=τtp-2-d/2αt2τ-d/2∫Rdψct(1tτtξ(tτt,x))dx,
with (4.49)ct=αt-1tτtp/2=αt2p/(d+2-2p)-1t1-p/(d+2-2p)=t(d+2-p)/(d+2-2p)αt-(d+2)/(d+2-2p).
Condition (1.19) and assumption p<(d+2)/2 provide ct→0. Then, for every fixed c>0, we have ct<c for t large enough, and therefore, (4.50)liminf⁡t→∞αt4/(d+2-2p)t-(d+4-2p)/(d+2-2p)log⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]≥sup⁡g∈Fd{ν∫Rdψc(θα∫Rdg2(y)|x-y|pdy)dx-κ2∫Rd|∇g(y)|2dy},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 ℝd, independent on B. Write, for a given shape function K, (4.51)U(x)=∫RdK(y-x)W(dy),Ut=∫0tU(Bκs)ds=∫Rdξ(t,x)W(dx),
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 K∉ℒ2(ℝd) for any p. On the other hand, for p∈(d/2,(d+2)/2), one has (4.52)E0∫Rdξ2(t,x)dx<+∞,and consequentlyUt 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, (4.53)lim⁡t→∞αt4/(d+2-2p)t-(d+4-2p)/(d+2-2p)log⁡E⊗E0exp⁡[-1αtUt]=sup⁡g∈Fd{θ22∫Rd(∫Rdg2(y)|x-y|pdy)2dx-κ2∫Rd|∇g(y)|2dy}.
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.
LetK have the form (1.5) with p∈(d/2,(d+2)/2). Then, for every t>0, the distribution of
(4.54)ν-1/2∫0tV¯(Bκs)ds,
w.r.t. ℙ⊗ℙ0 weakly converges as ν→∞ to the distribution of Ut w.r.t. ℙW⊗ℙ0, where ℙW denotes the distribution of the Wiener sheet W.Proof.
Characteristic function of (4.54) is equal
(4.55)E0exp⁡[ν∫Rd(eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x))dx],
see Proposition 2.2 in [7]. For every z∈ℂ,t>0,x∈ℝd,
(4.56)ν(eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x))⟶-z22ξ2(t,x),ν⟶∞.
In addition,
(4.57)ν|eizν-1/2ξ(t,x)-1-izν-1/2ξ(t,x)|≤C0z2ξ2(t,x),
with some constant C0. Since for p∈(d/2,(d+2)/2)(4.58)E0eC∫Rdξ2(t,x)dx<∞,t>0,C>0,
(see [45]), we have by the dominated convergence theorem that the characteristic functions (4.55) converge pointwise to
(4.59)E0exp⁡[-z22∫Rdξ2(t,x)dx],
which is just the characteristic function of Ut.


## 5. The integrated Density of States

### 5.1. Proof of Proposition2.6

Statement A.
We proceed in two steps. First, we show that under conditions of the proposition, almost all realizations ofQ 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(5.1)V¯(x)=∫RdKg(x-y)[ω(dy)-dy]+∫RdKg(x-y)[ω(dy)-dy].
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 W21(ℝd).It is a simple observation that for a smooth compactly supported functionL:ℝd→ℝ, every realization of respective renormalized Poisson potential belongs to W21(ℝd), and (5.2)∇∫RdL(x-y)[ω(dx)-dy]=∫Rd∇L(x-y)[ω(dy)-dy].
This, by usual approximation argument, provides that almost all realizations of the renormalized Poisson potential with kernel K belong to W21(ℝd), and (5.2) holds true in Sobolev sense for K=L.SinceK is Lipschitz, ∇K is bounded; the function K is bounded, as well. Then, Proposition 2.7 in [7] provides (5.3)Eexp⁡(|V¯(0)|+|∇V¯(0)|)<+∞.
By shift invariance, this gives (5.4)E∫Uexp⁡(|V¯(x)|+|∇V¯(x)|)dx<+∞.
Therefore, almost all realizations of V¯ belong to W∞1(U)=⋂p>1Wp1(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.
LetQ:ℝd→ℝ be a function bounded from below and U a given cube.
Denote(5.5)QN=Q∧N,
and consider the Schrödinger operators HUQN, N≥1 with the Dirichlet boundary conditions, defined by (2.14) with Q replaced by QN.
Then,(i)
Rt,UQN=e-tHUQN,t≥0 converge strongly as N→∞ to a continuous semigroup Rt,UQ,t≥0 of self-adjoint operators in L2(U,dx). Rt,UQ, t≥0 admits the Feynman-Kac representation (2.15),(ii)
every operatorRt,UQ, t≥0 is of trace class,(iii)
for the generatorHUQ of the semigroup Rt,UQ, t≥0, the function (2.17) is well defined, and its Laplace transform admits the representation
(5.6)∫Re-λtdNUQ(λ)=(2πκt)-d/21|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds]χU,t(B⋅+x))dx.Proof.
For everyN, the semigroup Rt,UQN=e-tHUQN,t≥0 admits the Feynman-Kac representation (2.15). An alternative form of this representation ([1, page 13]) is that Rt,UQN is an integral operator with the kernel
(5.7)rt,UQN(x,y)=pt(x,y)Ex,yt(exp⁡[-∫0tQN(Bκs)ds]χU,t(B⋅)),
where
(5.8)pt(x,y)=(2πκt)-d/2e-|x-y|2/2κt
is the transition probability density for the process Bκs, s≥0, and 𝔼x,yt 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 rt,UQN is continuous and symmetric. Then, by the monotone convergence theorem, there exists a monotonous limit
(5.9)rt,UQ(x,y)=lim⁡N→∞rt,UQN(x,y)=(2πκt)-d/2e-|x-y|2/(2κt)Ex,yt(exp⁡[-∫0tQ(Bκs)ds]χU,t(B⋅)),
which is bounded and symmetric. Integral operators Rt,UQN converge to the operator Rt,UQ with kernel rt,UQ in the Hilbert-Schmidt norm, because respective kernels converge in L2(U×U,dx). This proves Statement (i) immediately. Since
(5.10)Rt,UQ=Rt/2,UQRt/2,UQ,
and every Rt/2,UQ is a Hilbert-Schmidt operator, the operator Rt,UQ is of trace class. This proves Statement (ii).
We have just proved thatRt,UQ is of trace class, which means that it has a purely point spectrum and
(5.11)TraceRt,UQ=∑kλk,Q,t<+∞,
where {λk,Q,t} denote eigenvalues of Rt,UQ, counted with their multiplicities. By the spectral decomposition theorem, this yields that the generator HUQ of the semigroup Rt,UQ, t≥0 has a purely point spectrum locally finite on every interval (-∞,λ]. In addition, λk,Q,t=e-λk,Qt, where {λk,Q} are respective eigenvalues of HUQ, counted with their multiplicities. Therefore, the function (2.17) is well defined, and its Laplace transform has the form
(5.12)∫Re-λtdNUQ(λ)=1|U|∑ke-tλk,U=1|U|TraceRt,UQ=1|U|∫U×Urt/2,UQ(x,y)rt/2,UQ(y,x)dxdy,
in the last equality, we have used the standard relation (e.g., [56, Chapter III, Section  9])
(5.13)TraceA*A=‖A‖HS2,
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
(5.14)(2πκt)-(d/2)1|U|∫UEx,xκt(exp⁡[-∫0tQ(Bκs)ds]χU,t(B⋅))dx.
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 fieldsQ=V¯ and Q=Vh are ergodic (or metrically transitive) in the sense that the σ-algebra generated by functionals, invariant w.r.t. the transformations
(5.15)Sh:Q(⋅)⟼Q(⋅+h),h∈Rd
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,
(5.16)lim⁡U↑Rd1|U|∫Uf(Q(⋅+x))dx=Ef(Q),
both almost surely and in mean sense.
By Proposition 2.7 in [7], 𝔼e-cQ(0)<+∞ for every c>0. This, by the Jensen inequality, provides
(5.17)E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds]<+∞,t≥0,
the argument here is the same as at the beginning of Section 3.1. Then, (5.16) applied to the function
(5.18)f:q⟼E0,0κtexp⁡[-∫0tq(Bκs)ds]
yields
(5.19)lim⁡U↑Rd1|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds])dx=E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds]
both in mean and in a.s. sense. Straightforward calculation shows that with probability 1,
(5.20)lim⁡U↑Rd1|U|∫UχU,t(B⋅+x)dx=1.
Together with mean convergence (5.19), this provides
(5.21)lim⁡U↑Rd1|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds](1-χU,t(B⋅+x)))dx=0,
which completes the proof.


### 5.2. Proof of Theorem2.5

#### 5.2.1. Classic Regime

Arguments in Section3.1 are process insensitive. Henceforth, the upper bounds in (2.3), (2.11), and (3.4) hold true with 𝔼0 replaced by 𝔼0,0κt.On the other hand, Brownian bridge measures enjoy the scaling property similar to the Brownian one ([1, page 140]): its law ℙ0,0t is the image of measure of ℙ0,01 under the map (5.22)w(⋅)⟼tw(⋅t).
In addition, the Brownian bridge measure ℙ0,01 has the small balls asymptotics similar to the Brownian one [59] (5.23)ɛ2log⁡P0,01(sup⁡s∈[0,1]|B(s)|≤ɛ)⟶-12j(d-2)/22,
where j(d-2)/2 is the smallest positive root of the Bessel function J(d-2)/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 𝔼0,0κt 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 𝔼0,0κt 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(5.24)η(t,x)=η(t-1,x)+ζ(t,x),ζ(t,x)=∫t-1tL(Bκs-x)ds.
For every γ∈(0,1), we have by convexity (5.25)υ(1tη(t,x))≤γυ(1γtη(t-1,x))+(1-γ)υ(1(1-γ)tζ(t,x)).
Analogously to (4.6), we have (5.26)∫Rdυ(1(1-γ)tζ(t,x))dx≤∫t-1t∫Rdυ(1(1-γ)tL(Bκs-x))dxds=∫Rdυ(1(1-γ)tL(x))dx.
The last term vanishes when t→∞. Therefore, for every γ∈(0,1), (5.27)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤limsup⁡t→∞1tE0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}.
On the other hand, applying the Markov property at time t-1, we arrive at (5.28)E0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}=E0exp⁡{γt∫Rdυ(1γtη(t-1,x))dx}p1(Bκ(t-1),0)≤E0exp⁡{γt∫Rdυ(1γtη(t,x))dx}p1(Bκ(t-1),0).
Because (5.29)p1(x,y)≤(2πκ)-d/2,x,y∈Rd,we get from the upper bound in (4.4) that for every γ∈(0,1), (5.30)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈Fd{γ∫Rdυ(1γ∫RdL(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}.
Passing to the limit γ→1 completes the proof of the upper bound.The lower bound in (4.4) with 𝔼0,0κt 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(5.31)lim⁡t→∞1tlog⁡E0exp⁡{∫0tf̂(Bκs)ds}≥CΥ,f,R+sup⁡g∈Fd{∫Rdf̂(x)g2(x)dx-12∫Rd|∇g(x)|2dx}.
Note that under ℙ0, (5.32)Bκs-sκtBκt
is a Gaussian process with the covariance κ(s∧s′)-κ(ss′/t) and, therefore, has the distribution ℙ0,0t ([1, page 140]). Then, for any Lipschiz continuous bounded function f̂, (5.31) holds true with 𝔼0,0κt instead of 𝔼0.Note that the statement of the Lemma4.2 still holds true when BR is replaced with any class of functions K⊂BR separating points in ℒ1([-R,R]d); in particular, one can take K=BLR, the class of Lipschitz continuous bounded functions. Clearly, for f∈BLR the function (5.33)f̂(y)=∫[-R,R]df(x)L(y-x)dx
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 𝔼0,0κt.


### 5.3. Proof of Theorem2.7

Theorem  9.7 in [42], Chapter IV gives (2.20) as a straightforward corollary of (2.3). Because t↦tlogt is a regularly varying function of the order 1, this theorem is not applicable when (2.21) is considered.From (2.13) (with 𝔼0,0κt instead of 𝔼0), we have (5.34)log⁡∫Re-(λ+νωdθ(log⁡h+Eu))tdNVh(λ)=νωdθtlog⁡t+o(t),t⟶+∞.
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 functionN(λ), λ∈ℝ,
(5.35)log⁡∫Re-λtdN(λ)=atlog⁡t+o(t),t⟶+∞,
with some a>0.
Then,(5.36)log⁡N(λ)=-aexp⁡[-λa-1](1+o(1)),λ⟶-∞.Proof.
The upper bound, in a standard way, is provided by the Chebyshev inequality: for everyt>0, λ<0,
(5.37)log⁡N(λ)≤λt+log⁡∫Re-χtdN(χ).
Take tλ=exp[-λ/a-1], the solution of the minimization problem
(5.38)tλ=argmin(λt+atlog⁡t).
Clearly, tλ→+∞, λ→-∞. By (5.35) and (5.37),
(5.39)log⁡N(λ)≤-atλ+o(tλ),λ⟶-∞,
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
(5.40)log⁡N(λn)≤-bexp⁡[-λna-1],n≥1.
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
(5.41)log⁡N(λ)≤-cexp⁡[-λa-1],λ<Λc.
Let n be large enough for λn<Λc. Denote tn=e-(λn-δ)/a-1 and write
(5.42)∫Re-λtndN(λ)=∫-∞Λce-λtnN(λ)dλ+[e-Λctn+∫Λc∞e-λtndN(λ)]=∫-∞λn-2δe-λtnN(λ)dλ+∫λn-2δλne-λtnN(λ)dλ+∫λnΛce-λtnN(λ)dλ+[e-Λctn+∫Λc∞e-λtndN(λ)]=In1+In2+In3+In4.
Let us estimate In1 – In4. Clearly,
(5.43)In4≤Ce-Λctn,
with appropriate constant C. Assumption (5.40) yields, via monotonicity,
(5.44)In2≤2δexp⁡[-(λn-2δ)tn-be-λn/(a-1)]=2δexp⁡[(δ+alog⁡tn+a)tn-be-δ/atn]=2δexp⁡[atnlog⁡tn-(be-δ/a-a-δ)tn],
here we have used the relation
(5.45)λn=-alog⁡tn-a+δ,
which comes from the definition of tn.
Consider the functionΘ:λ↦-λtn-ce-λ/a-1. Straightforward computation shows that assuming
(5.46)c>ae-δ/a,
its derivative is increasing on (-∞,λn-2δ], and
(5.47)Θn′(λn-2δ)=(caeδ/a-1)tn>1,
for n large enough. Then, with (5.41) in mind, we get
(5.48)In1≤∫-∞λn-2δeλ-λn+2δeΘn(λn-2δ)dλ=eΘn(λn-2δ)=exp⁡[atnlog⁡tn-(ceδ/a-a-δ)tn].
Similar argument leads to
(5.49)In3≤∫λn+∞e-λ+λneΘn(λn)dλ=eΘn(λn)=exp⁡[atnlog⁡tn-(δ+ceδ/a-a)tn].
Now, we can finalize the proof. Take δ>0 such that be-δ/a-a-δ>0. Note that
(5.50)aeδ/a-a-δ>0,δ+aeδ/a-a>0.
Therefore, c<a can be chosen in such a way that (5.46) holds true and
(5.51)ceδ/a-a-δ>0,δ+ceδ/a-a>0.
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,
(5.52)log⁡∫Re-λtndN(λ)≤atnlog⁡tn-ɛtn,
with some positive ɛ. This contradicts to (5.35) and proves that assumption (5.40) is impossible.Remark 5.3.
In the proof of the lower bound in Lemma5.2, 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 Theorem2.8

Our argument is based on the following version of the Gärtner-Ellis theorem.Lemma 5.4.
Consider a sequenceNn, n≥1 of nonnegative monotonous functions on ℝ, which vanish at -∞, and assume that there exist a>1, c>0, and a sequence Υn→+∞ such that
(5.53)1Υnlog⁡∫Re-μΥnxdNn(x)⟶I(μ)=cμa,n⟶∞,μ>0.
Then,
(5.54)1Υnlog⁡Nn(-x)=-I*(x)(1+o(1)),n⟶∞,
uniformly by x∈[A,B] for every [A,B]⊂(0,+∞). Here,
(5.55)I*(x)=sup⁡μ>0[μx-I(μ)]=(c(a-1))-1/(a-1)(a-1a)a/(a-1)xa/(a-1).The only difference between conditions of Lemma5.4 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∈ℝ,γ>0} be such that(a)
every functionN(·,γ), γ>0 is nonnegative and nondecreasing,(b)
for everyt>0,
(5.56)Ñ(t,γ):=∫Re-λtN(dλ,γ)<∞,(c)
for givena>1, b∈ℝ, c>0, and given sequences λn>0, γn>0, n≥1 with λnaγn-b→∞,
(5.57)log⁡Ñ(μλn1/(a-1)γn-b/(a-1),γn)=cμaλna/(a-1)γn-b/(a-1)(1+o(1)),n→∞,μ>0.
Then,(5.58)log⁡N(-λnx,γn)=(c(a-1))-1/(a-1)(a-1a)a/(a-1)λna/(a-1)γn-b/(a-1)xa/(a-1)(1+o(1)),n→∞,
uniformly by x∈[A,B] for every [A,B]⊂(0,+∞).Proof.
PutΥn=λna/(a-1)γn-b/(a-1),
(5.59)Nn(x)=N(λnx,γn),x∈R.
Because a>1 and λnaγn-b→∞, we have Υn→∞. Condition (5.57) provides (5.53). Henceforth, (5.58) follows by Lemma 5.4.Now, we can finalize the proof of Theorem2.8.Statement  (i)
Take(5.60)a=b=dp,c=νωdθd/p(pd-p)Γ(2p-dp).
For given sequences λn<0, γn>0, n≥1 and arbitrary μ>0 denote
(5.61)tn=μ(-λn)1/(a-1)γn-b/(a-1),αtn=1γn.
Condition (-λn)p/d/γn→∞ yields tn→∞, and condition (-λn)(d+2-p)/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+2-p)/2>1, and consequently (-λn)aγn-b→∞. Applying Corollary 5.5 with x=-1 and λn replaced by -λn, we obtain the required statement.Statement  (ii)
Take(5.62)a=d+4-2pd+2-2p,b=4d+2-2p,c=C2,
and keep the notation (5.58). Condition (-λn)(d+4-2p)/4/γn→∞ yields (-λn)aγn-b→∞. Condition (-λn)(d+2-p)/2/γn→0 yields tn(d+2-p)/(d+2)=o(αtn). Finally, these two conditions yield λn→0-, and consequently (-λn)(d+2-2p)/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
(5.63)lim⁡t→∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=Cψ.
Using this relation and following the proof of Statement (i) with appropriate modifications, we obtain the required statement.


## 5.1. Proof of Proposition2.6

Statement A.
We proceed in two steps. First, we show that under conditions of the proposition, almost all realizations ofQ 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(5.1)V¯(x)=∫RdKg(x-y)[ω(dy)-dy]+∫RdKg(x-y)[ω(dy)-dy].
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 W21(ℝd).It is a simple observation that for a smooth compactly supported functionL:ℝd→ℝ, every realization of respective renormalized Poisson potential belongs to W21(ℝd), and (5.2)∇∫RdL(x-y)[ω(dx)-dy]=∫Rd∇L(x-y)[ω(dy)-dy].
This, by usual approximation argument, provides that almost all realizations of the renormalized Poisson potential with kernel K belong to W21(ℝd), and (5.2) holds true in Sobolev sense for K=L.SinceK is Lipschitz, ∇K is bounded; the function K is bounded, as well. Then, Proposition 2.7 in [7] provides (5.3)Eexp⁡(|V¯(0)|+|∇V¯(0)|)<+∞.
By shift invariance, this gives (5.4)E∫Uexp⁡(|V¯(x)|+|∇V¯(x)|)dx<+∞.
Therefore, almost all realizations of V¯ belong to W∞1(U)=⋂p>1Wp1(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.
LetQ:ℝd→ℝ be a function bounded from below and U a given cube.
Denote(5.5)QN=Q∧N,
and consider the Schrödinger operators HUQN, N≥1 with the Dirichlet boundary conditions, defined by (2.14) with Q replaced by QN.
Then,(i)
Rt,UQN=e-tHUQN,t≥0 converge strongly as N→∞ to a continuous semigroup Rt,UQ,t≥0 of self-adjoint operators in L2(U,dx). Rt,UQ, t≥0 admits the Feynman-Kac representation (2.15),(ii)
every operatorRt,UQ, t≥0 is of trace class,(iii)
for the generatorHUQ of the semigroup Rt,UQ, t≥0, the function (2.17) is well defined, and its Laplace transform admits the representation
(5.6)∫Re-λtdNUQ(λ)=(2πκt)-d/21|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds]χU,t(B⋅+x))dx.Proof.
For everyN, the semigroup Rt,UQN=e-tHUQN,t≥0 admits the Feynman-Kac representation (2.15). An alternative form of this representation ([1, page 13]) is that Rt,UQN is an integral operator with the kernel
(5.7)rt,UQN(x,y)=pt(x,y)Ex,yt(exp⁡[-∫0tQN(Bκs)ds]χU,t(B⋅)),
where
(5.8)pt(x,y)=(2πκt)-d/2e-|x-y|2/2κt
is the transition probability density for the process Bκs, s≥0, and 𝔼x,yt 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 rt,UQN is continuous and symmetric. Then, by the monotone convergence theorem, there exists a monotonous limit
(5.9)rt,UQ(x,y)=lim⁡N→∞rt,UQN(x,y)=(2πκt)-d/2e-|x-y|2/(2κt)Ex,yt(exp⁡[-∫0tQ(Bκs)ds]χU,t(B⋅)),
which is bounded and symmetric. Integral operators Rt,UQN converge to the operator Rt,UQ with kernel rt,UQ in the Hilbert-Schmidt norm, because respective kernels converge in L2(U×U,dx). This proves Statement (i) immediately. Since
(5.10)Rt,UQ=Rt/2,UQRt/2,UQ,
and every Rt/2,UQ is a Hilbert-Schmidt operator, the operator Rt,UQ is of trace class. This proves Statement (ii).
We have just proved thatRt,UQ is of trace class, which means that it has a purely point spectrum and
(5.11)TraceRt,UQ=∑kλk,Q,t<+∞,
where {λk,Q,t} denote eigenvalues of Rt,UQ, counted with their multiplicities. By the spectral decomposition theorem, this yields that the generator HUQ of the semigroup Rt,UQ, t≥0 has a purely point spectrum locally finite on every interval (-∞,λ]. In addition, λk,Q,t=e-λk,Qt, where {λk,Q} are respective eigenvalues of HUQ, counted with their multiplicities. Therefore, the function (2.17) is well defined, and its Laplace transform has the form
(5.12)∫Re-λtdNUQ(λ)=1|U|∑ke-tλk,U=1|U|TraceRt,UQ=1|U|∫U×Urt/2,UQ(x,y)rt/2,UQ(y,x)dxdy,
in the last equality, we have used the standard relation (e.g., [56, Chapter III, Section  9])
(5.13)TraceA*A=‖A‖HS2,
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
(5.14)(2πκt)-(d/2)1|U|∫UEx,xκt(exp⁡[-∫0tQ(Bκs)ds]χU,t(B⋅))dx.
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 fieldsQ=V¯ and Q=Vh are ergodic (or metrically transitive) in the sense that the σ-algebra generated by functionals, invariant w.r.t. the transformations
(5.15)Sh:Q(⋅)⟼Q(⋅+h),h∈Rd
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,
(5.16)lim⁡U↑Rd1|U|∫Uf(Q(⋅+x))dx=Ef(Q),
both almost surely and in mean sense.
By Proposition 2.7 in [7], 𝔼e-cQ(0)<+∞ for every c>0. This, by the Jensen inequality, provides
(5.17)E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds]<+∞,t≥0,
the argument here is the same as at the beginning of Section 3.1. Then, (5.16) applied to the function
(5.18)f:q⟼E0,0κtexp⁡[-∫0tq(Bκs)ds]
yields
(5.19)lim⁡U↑Rd1|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds])dx=E⊗E0,0κtexp⁡[-∫0tQ(Bκs)ds]
both in mean and in a.s. sense. Straightforward calculation shows that with probability 1,
(5.20)lim⁡U↑Rd1|U|∫UχU,t(B⋅+x)dx=1.
Together with mean convergence (5.19), this provides
(5.21)lim⁡U↑Rd1|U|∫UE0,0κt(exp⁡[-∫0tQ(Bκs+x)ds](1-χU,t(B⋅+x)))dx=0,
which completes the proof.


## 5.2. Proof of Theorem2.5

### 5.2.1. Classic Regime

Arguments in Section3.1 are process insensitive. Henceforth, the upper bounds in (2.3), (2.11), and (3.4) hold true with 𝔼0 replaced by 𝔼0,0κt.On the other hand, Brownian bridge measures enjoy the scaling property similar to the Brownian one ([1, page 140]): its law ℙ0,0t is the image of measure of ℙ0,01 under the map (5.22)w(⋅)⟼tw(⋅t).
In addition, the Brownian bridge measure ℙ0,01 has the small balls asymptotics similar to the Brownian one [59] (5.23)ɛ2log⁡P0,01(sup⁡s∈[0,1]|B(s)|≤ɛ)⟶-12j(d-2)/22,
where j(d-2)/2 is the smallest positive root of the Bessel function J(d-2)/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 𝔼0,0κt 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 𝔼0,0κt 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(5.24)η(t,x)=η(t-1,x)+ζ(t,x),ζ(t,x)=∫t-1tL(Bκs-x)ds.
For every γ∈(0,1), we have by convexity (5.25)υ(1tη(t,x))≤γυ(1γtη(t-1,x))+(1-γ)υ(1(1-γ)tζ(t,x)).
Analogously to (4.6), we have (5.26)∫Rdυ(1(1-γ)tζ(t,x))dx≤∫t-1t∫Rdυ(1(1-γ)tL(Bκs-x))dxds=∫Rdυ(1(1-γ)tL(x))dx.
The last term vanishes when t→∞. Therefore, for every γ∈(0,1), (5.27)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤limsup⁡t→∞1tE0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}.
On the other hand, applying the Markov property at time t-1, we arrive at (5.28)E0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}=E0exp⁡{γt∫Rdυ(1γtη(t-1,x))dx}p1(Bκ(t-1),0)≤E0exp⁡{γt∫Rdυ(1γtη(t,x))dx}p1(Bκ(t-1),0).
Because (5.29)p1(x,y)≤(2πκ)-d/2,x,y∈Rd,we get from the upper bound in (4.4) that for every γ∈(0,1), (5.30)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈Fd{γ∫Rdυ(1γ∫RdL(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}.
Passing to the limit γ→1 completes the proof of the upper bound.The lower bound in (4.4) with 𝔼0,0κt 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(5.31)lim⁡t→∞1tlog⁡E0exp⁡{∫0tf̂(Bκs)ds}≥CΥ,f,R+sup⁡g∈Fd{∫Rdf̂(x)g2(x)dx-12∫Rd|∇g(x)|2dx}.
Note that under ℙ0, (5.32)Bκs-sκtBκt
is a Gaussian process with the covariance κ(s∧s′)-κ(ss′/t) and, therefore, has the distribution ℙ0,0t ([1, page 140]). Then, for any Lipschiz continuous bounded function f̂, (5.31) holds true with 𝔼0,0κt instead of 𝔼0.Note that the statement of the Lemma4.2 still holds true when BR is replaced with any class of functions K⊂BR separating points in ℒ1([-R,R]d); in particular, one can take K=BLR, the class of Lipschitz continuous bounded functions. Clearly, for f∈BLR the function (5.33)f̂(y)=∫[-R,R]df(x)L(y-x)dx
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 𝔼0,0κt.


## 5.2.1. Classic Regime

Arguments in Section3.1 are process insensitive. Henceforth, the upper bounds in (2.3), (2.11), and (3.4) hold true with 𝔼0 replaced by 𝔼0,0κt.On the other hand, Brownian bridge measures enjoy the scaling property similar to the Brownian one ([1, page 140]): its law ℙ0,0t is the image of measure of ℙ0,01 under the map (5.22)w(⋅)⟼tw(⋅t).
In addition, the Brownian bridge measure ℙ0,01 has the small balls asymptotics similar to the Brownian one [59] (5.23)ɛ2log⁡P0,01(sup⁡s∈[0,1]|B(s)|≤ɛ)⟶-12j(d-2)/22,
where j(d-2)/2 is the smallest positive root of the Bessel function J(d-2)/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 𝔼0,0κt 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 𝔼0,0κt 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(5.24)η(t,x)=η(t-1,x)+ζ(t,x),ζ(t,x)=∫t-1tL(Bκs-x)ds.
For every γ∈(0,1), we have by convexity (5.25)υ(1tη(t,x))≤γυ(1γtη(t-1,x))+(1-γ)υ(1(1-γ)tζ(t,x)).
Analogously to (4.6), we have (5.26)∫Rdυ(1(1-γ)tζ(t,x))dx≤∫t-1t∫Rdυ(1(1-γ)tL(Bκs-x))dxds=∫Rdυ(1(1-γ)tL(x))dx.
The last term vanishes when t→∞. Therefore, for every γ∈(0,1), (5.27)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤limsup⁡t→∞1tE0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}.
On the other hand, applying the Markov property at time t-1, we arrive at (5.28)E0,0κtexp⁡{γt∫Rdυ(1γtη(t-1,x))dx}=E0exp⁡{γt∫Rdυ(1γtη(t-1,x))dx}p1(Bκ(t-1),0)≤E0exp⁡{γt∫Rdυ(1γtη(t,x))dx}p1(Bκ(t-1),0).
Because (5.29)p1(x,y)≤(2πκ)-d/2,x,y∈Rd,we get from the upper bound in (4.4) that for every γ∈(0,1), (5.30)limsup⁡t→∞1tE0,0κtexp⁡{t∫Rdυ(1tη(t,x))dx}≤sup⁡g∈Fd{γ∫Rdυ(1γ∫RdL(x-y)g2(y)dy)dx-κ2∫Rd|∇g(y)|2dy}.
Passing to the limit γ→1 completes the proof of the upper bound.The lower bound in (4.4) with 𝔼0,0κt 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(5.31)lim⁡t→∞1tlog⁡E0exp⁡{∫0tf̂(Bκs)ds}≥CΥ,f,R+sup⁡g∈Fd{∫Rdf̂(x)g2(x)dx-12∫Rd|∇g(x)|2dx}.
Note that under ℙ0, (5.32)Bκs-sκtBκt
is a Gaussian process with the covariance κ(s∧s′)-κ(ss′/t) and, therefore, has the distribution ℙ0,0t ([1, page 140]). Then, for any Lipschiz continuous bounded function f̂, (5.31) holds true with 𝔼0,0κt instead of 𝔼0.Note that the statement of the Lemma4.2 still holds true when BR is replaced with any class of functions K⊂BR separating points in ℒ1([-R,R]d); in particular, one can take K=BLR, the class of Lipschitz continuous bounded functions. Clearly, for f∈BLR the function (5.33)f̂(y)=∫[-R,R]df(x)L(y-x)dx
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 𝔼0,0κt.


## 5.3. Proof of Theorem2.7

Theorem  9.7 in [42], Chapter IV gives (2.20) as a straightforward corollary of (2.3). Because t↦tlogt is a regularly varying function of the order 1, this theorem is not applicable when (2.21) is considered.From (2.13) (with 𝔼0,0κt instead of 𝔼0), we have (5.34)log⁡∫Re-(λ+νωdθ(log⁡h+Eu))tdNVh(λ)=νωdθtlog⁡t+o(t),t⟶+∞.
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 functionN(λ), λ∈ℝ,
(5.35)log⁡∫Re-λtdN(λ)=atlog⁡t+o(t),t⟶+∞,
with some a>0.
Then,(5.36)log⁡N(λ)=-aexp⁡[-λa-1](1+o(1)),λ⟶-∞.Proof.
The upper bound, in a standard way, is provided by the Chebyshev inequality: for everyt>0, λ<0,
(5.37)log⁡N(λ)≤λt+log⁡∫Re-χtdN(χ).
Take tλ=exp[-λ/a-1], the solution of the minimization problem
(5.38)tλ=argmin(λt+atlog⁡t).
Clearly, tλ→+∞, λ→-∞. By (5.35) and (5.37),
(5.39)log⁡N(λ)≤-atλ+o(tλ),λ⟶-∞,
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
(5.40)log⁡N(λn)≤-bexp⁡[-λna-1],n≥1.
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
(5.41)log⁡N(λ)≤-cexp⁡[-λa-1],λ<Λc.
Let n be large enough for λn<Λc. Denote tn=e-(λn-δ)/a-1 and write
(5.42)∫Re-λtndN(λ)=∫-∞Λce-λtnN(λ)dλ+[e-Λctn+∫Λc∞e-λtndN(λ)]=∫-∞λn-2δe-λtnN(λ)dλ+∫λn-2δλne-λtnN(λ)dλ+∫λnΛce-λtnN(λ)dλ+[e-Λctn+∫Λc∞e-λtndN(λ)]=In1+In2+In3+In4.
Let us estimate In1 – In4. Clearly,
(5.43)In4≤Ce-Λctn,
with appropriate constant C. Assumption (5.40) yields, via monotonicity,
(5.44)In2≤2δexp⁡[-(λn-2δ)tn-be-λn/(a-1)]=2δexp⁡[(δ+alog⁡tn+a)tn-be-δ/atn]=2δexp⁡[atnlog⁡tn-(be-δ/a-a-δ)tn],
here we have used the relation
(5.45)λn=-alog⁡tn-a+δ,
which comes from the definition of tn.
Consider the functionΘ:λ↦-λtn-ce-λ/a-1. Straightforward computation shows that assuming
(5.46)c>ae-δ/a,
its derivative is increasing on (-∞,λn-2δ], and
(5.47)Θn′(λn-2δ)=(caeδ/a-1)tn>1,
for n large enough. Then, with (5.41) in mind, we get
(5.48)In1≤∫-∞λn-2δeλ-λn+2δeΘn(λn-2δ)dλ=eΘn(λn-2δ)=exp⁡[atnlog⁡tn-(ceδ/a-a-δ)tn].
Similar argument leads to
(5.49)In3≤∫λn+∞e-λ+λneΘn(λn)dλ=eΘn(λn)=exp⁡[atnlog⁡tn-(δ+ceδ/a-a)tn].
Now, we can finalize the proof. Take δ>0 such that be-δ/a-a-δ>0. Note that
(5.50)aeδ/a-a-δ>0,δ+aeδ/a-a>0.
Therefore, c<a can be chosen in such a way that (5.46) holds true and
(5.51)ceδ/a-a-δ>0,δ+ceδ/a-a>0.
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,
(5.52)log⁡∫Re-λtndN(λ)≤atnlog⁡tn-ɛtn,
with some positive ɛ. This contradicts to (5.35) and proves that assumption (5.40) is impossible.Remark 5.3.
In the proof of the lower bound in Lemma5.2, 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 Theorem2.8

Our argument is based on the following version of the Gärtner-Ellis theorem.Lemma 5.4.
Consider a sequenceNn, n≥1 of nonnegative monotonous functions on ℝ, which vanish at -∞, and assume that there exist a>1, c>0, and a sequence Υn→+∞ such that
(5.53)1Υnlog⁡∫Re-μΥnxdNn(x)⟶I(μ)=cμa,n⟶∞,μ>0.
Then,
(5.54)1Υnlog⁡Nn(-x)=-I*(x)(1+o(1)),n⟶∞,
uniformly by x∈[A,B] for every [A,B]⊂(0,+∞). Here,
(5.55)I*(x)=sup⁡μ>0[μx-I(μ)]=(c(a-1))-1/(a-1)(a-1a)a/(a-1)xa/(a-1).The only difference between conditions of Lemma5.4 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∈ℝ,γ>0} be such that(a)
every functionN(·,γ), γ>0 is nonnegative and nondecreasing,(b)
for everyt>0,
(5.56)Ñ(t,γ):=∫Re-λtN(dλ,γ)<∞,(c)
for givena>1, b∈ℝ, c>0, and given sequences λn>0, γn>0, n≥1 with λnaγn-b→∞,
(5.57)log⁡Ñ(μλn1/(a-1)γn-b/(a-1),γn)=cμaλna/(a-1)γn-b/(a-1)(1+o(1)),n→∞,μ>0.
Then,(5.58)log⁡N(-λnx,γn)=(c(a-1))-1/(a-1)(a-1a)a/(a-1)λna/(a-1)γn-b/(a-1)xa/(a-1)(1+o(1)),n→∞,
uniformly by x∈[A,B] for every [A,B]⊂(0,+∞).Proof.
PutΥn=λna/(a-1)γn-b/(a-1),
(5.59)Nn(x)=N(λnx,γn),x∈R.
Because a>1 and λnaγn-b→∞, we have Υn→∞. Condition (5.57) provides (5.53). Henceforth, (5.58) follows by Lemma 5.4.Now, we can finalize the proof of Theorem2.8.Statement  (i)
Take(5.60)a=b=dp,c=νωdθd/p(pd-p)Γ(2p-dp).
For given sequences λn<0, γn>0, n≥1 and arbitrary μ>0 denote
(5.61)tn=μ(-λn)1/(a-1)γn-b/(a-1),αtn=1γn.
Condition (-λn)p/d/γn→∞ yields tn→∞, and condition (-λn)(d+2-p)/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+2-p)/2>1, and consequently (-λn)aγn-b→∞. Applying Corollary 5.5 with x=-1 and λn replaced by -λn, we obtain the required statement.Statement  (ii)
Take(5.62)a=d+4-2pd+2-2p,b=4d+2-2p,c=C2,
and keep the notation (5.58). Condition (-λn)(d+4-2p)/4/γn→∞ yields (-λn)aγn-b→∞. Condition (-λn)(d+2-p)/2/γn→0 yields tn(d+2-p)/(d+2)=o(αtn). Finally, these two conditions yield λn→0-, and consequently (-λn)(d+2-2p)/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
(5.63)lim⁡t→∞(αtt)d/plog⁡E⊗E0exp⁡[-1αt∫0tV¯(Bκs)ds]=Cψ.
Using this relation and following the proof of Statement (i) with appropriate modifications, we obtain the required statement.



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*Source: 803683-2011-06-28.xml*