asymptotics of negative exponential moments for annealed brownian motion in a renormalized poisson potential

Hindawi Publishing Corporation International Journal of Stochastic Analysis Volume 2011, Article ID 803683, 43 pages doi:10.1155/2011/803683 Research Article Asymptotics of Negative Exponential Moments for Annealed Brownian Motion in a Renormalized Poisson Potential Xia Chen1 and Alexey Kulik2 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv 01601, Ukraine Correspondence should be addressed to Alexey Kulik, kulik@imath.kiev.ua Received 24 December 2010; Accepted April 2011 Academic Editor: Nikolai Leonenko Copyright q 2011 X Chen and A Kulik This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited 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 Schrodinger ă operators with their potential terms represented by renormalized Poisson potentials 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 for a systematic representation, to for a survey, and to 3–6 for specific topics and for recent development on this subject Let ω dx be a Poisson field in 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 International Journal of Stochastic Analysis function K on Rd known as a shape function , define the respective random function known as a Poisson potential V x Rd K y − x ω dy , 1.1 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 dμt,ω dP0 t exp − Zt,ω V Bκs ds , Zt,ω E0 exp − t V Bκs ds 1.2 Here, κ is a positive parameter, responsible for the time scaling s → κs, introduced here for further references convenience In the annealed setting, the model averages on both the Brownian motion and the environment, and respective Gibbs measure μt is defined by dμt d P ⊗ P0 exp − Zt t V Bκs ds , E ⊗ E0 exp − Zt t V Bκs ds 1.3 Heuristically, the integral t V Bκs ds 1.4 measures the total net attraction to which the Brownian particle is subject up to the time t, and henceforth, under the law μt,ω or μt , the Brownian paths heavily impacted by the Poisson obstacles are penalized and become less likely In the Sznitman’s model of “soft obstacles,” the shape function K is assumed to be locally bounded and compactly supported However, these limitations may appear to be too restrictive in certain cases Important particular choice of a shape function, physically motivated by the Newton’s law of universal attraction, is K x θ|x|−p , x ∈ Rd , 1.5 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 , it was proposed to consider, apart with a Poisson potential 1.1 , a renormalized Poisson potential V x Rd K y − x ω dy − νdy 1.6 International Journal of Stochastic Analysis Assume for a while that K is locally bounded and compactly supported Then, V x Rd Rd K y − x ω dy − νdy K y − x ω dy − ν V x −ν Rd Rd K y − x dy 1.7 K 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 this is where the word “renormalization” comes from On the other hand, for unbounded and not locally supported K, the renormalized potential 1.6 may be well defined, while the potential 1.1 blows up The most important example here is the shape function 1.5 under the assumption d/2 < p < d In that case, V is well defined as well as the Gibbs measures dμt,ω dP0 Zt,ω dμt d P ⊗ P0 Zt exp − exp − t V Bκs ds , Zt,ω E0 exp − t V Bκs ds , 1.8 V Bκs ds , 1.9 t V Bκs ds , Zt E ⊗ E0 exp − t see 7, Corollary 1.3 We use separate notation μt,ω , μt because the Gibbs measures 1.2 and 1.3 are not well defined now The above exposition shows that using the notion of the renormalized Poisson potential, one can extend the class of the shape functions significantly Note that in general, the domain of definition for 1.6 does not include the one for 1.1 For instance, for the shape function 1.5 , the potential V , and the renormalized potential V are well defined under the mutually excluding assumptions p > d and d/2 < p < d, respectively This, in particular, does not give one a possibility to define respective Gibbs measures in a uniform way This inconvenience is resolved in the terms of the Poisson potential V h , partially renormalized at the level h; see 7, Chapter By definition, Vh x Rd K y − x − h ω dy Rd K y − x ∧ h ω dy − νdy , 1.10 V, V ∞ V It is known see 7, where h ∈ 0, ∞ is a renormalization level Clearly, V h Chapter that V is well defined for every h ∈ 0, ∞ as soon as V h is well defined for some h ∈ 0, ∞ , and in that case, there exists a constant CK,h,h such that V h − V h 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 V h with any h ∈ 0, ∞ In addition, such a definition extends the class of shape functions: for K given by 1.5 , V h with h ∈ 0, ∞ is well defined for p > d/2 International Journal of Stochastic Analysis The main objective of this paper is to study the asymptotic behavior, as t → the annealed exponential moments E ⊗ E0 exp − αt ∞, of t V Bκs ds 1.11 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 Zt in the formula 1.3 for the annealed Gibbs measure In the quenched setting, similar problem was studied in the recent paper In some cases, we also consider 1.11 with a renormalized Poisson potential V replaced by either a Poisson potential V or a partially renormalized potential V h with h ∈ 0, ∞ The function αt in 1.11 appears, on one hand, because of our further intent to study in further publications the a.s behavior t V Bκs ds, t −→ ∞ 1.12 On the other hand, this function can be naturally included into the initial model One can think about making penalty 1.4 to be additionally dependent on the length of the time interval by dividing the total net attraction for the Brownian particle by some scaling parameter Because of this interpretation, further on, we call the function 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 κΔu t, x ± Q x u t, x , ∂t u t, x u 0, x 1, x ∈ Rd 1.13 This problem appears in the context of chemical kinetics and population dynamics Its name goes back to the work by Anderson 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 International Journal of Stochastic Analysis It is well known that 1.13 is solved by the following Feynman-Kac representation u t, x Ex exp ± t Q B2κs ds , 1.14 when Q is Holder continuous and satisfies proper growth bounds When Q V with K ă from 1.5 , local unboundedness of K induces local irregularity of Q Proposition 2.9 in , 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 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 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 < 2, d Theorem 1.4 and Theorem 1.5 in In view of 1.14 , our main problem relates immediately to the asymptotic behavior of the moments of the solution to the parabolic Anderson problem 1.13 with the sign “−” Here, we cite 10–20 as a partial list of the publications that deal with various asymptotic topics related to the parabolic Anderson model Another problem related to our main one is the so called Lifshitz tails asymptotic behavior of the integrated density of states function N of a random Schrodinger operator of ă the type H Δ Q 1.15 This function, written IDS in the sequel, is a deterministic spectral mean-field characteristic of H Under quite general assumptions on the random potential Q, it is well defined as N λ lim U↑Rd |U| 1λk,U ≤λ , 1.16 k 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 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 log N 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 International Journal of Stochastic Analysis Connection between the Lifshitz tails asymptotics for the IDS function N, and the problem discussed above is provided by the representation for the Laplace transform of N R e−λt dN λ 2πκt −d/2 E ⊗ Eκt 0,0 exp − t Q Bκs ds , t ≥ 1.17 Here, Eκt 0,0 denotes the distribution of the Brownian bridge, that is, the Brownian motion Our estimates for 1.11 appear to be process insensitive to some conditioned by Bκt extent and remain true with E0 in 1.11 replaced by Eκt 0,0 This, via appropriate Tauberian theorem, provides information on Lifshitz tail asymptotics for the respective IDS function N Note that in this case, the asymptotic behavior of the log N λ as λ → −∞ should be studied, because the bottom of the spectrum is equal λ0 −∞, unlike the usual Poisson case, where λ0 This difference is caused by the renormalization procedure, which brings the negative part to the potential We now outline the rest of the paper The main results about negative exponential moments for annealed Brownian motion in a renormalized Poisson potential are collected in Theorem 2.1 They are formulated for the shape function defined by 1.5 Depending on p in this definition, we separate three cases o td αt td αt ∼ αt d 2−p / d 2−p / d 2−p / d , o αt , with some t −→ ∞, 1.18 t −→ ∞, 1.19 α > 0, t −→ ∞, 1.20 calling them a “light-scale,” a “heavy-scale,” and a “critical” case, respectively There is a close analogy between our “light” versus “heavy” scale classification for a renormalized Poisson potential and the well-known “classic” versus “quantum” regime classification for a usual Poisson potential; see detailed discussion in Section In all three cases listed above, our approach relies on the identity E ⊗ E0 exp − αt t E0 exp ν V Bκs ds Rd ψ ξ t, x αt dx , 1.21 with e−u − ψ u t ξ t, x u, K Bκs − x ds, 1.22 1.23 see Proposition 2.7 and Proposition 3.1 in 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 International Journal of Stochastic Analysis process Then, we choose a ball in the Wiener space, which simultaneously is “sufficiently heavy” in probability and “sufficiently small” in size This smallness allows one to transform the integral in the r.h.s of 1.21 into ν Rd ψ αt t K −x ds dx ν Rd ψ t K −x αt dx, 1.24 which after a straightforward transformation gives a lower bound that coincides with the universal upper bound obtained before We call this approach the “small heavy ball method” It is quite flexible, and by means of this method, we also give a complete description of the light-scale asymptotic behavior for a Poisson potential V and a partially renormalized Poisson potential V h Theorem 2.4 This method differs from the functional methods, typical in the field, which go back to the paper 41 by Pastur It gives a new and transparent principle explaining the transition from quantum to classical regime; note that the phenomenology of such a transition is a problem discussed in the literature intensively; see 32, Section 3.5 for a detailed overview In the context of the small heavy ball method, we can identify the classic regime with the situation where a sufficient amount of Brownian paths stay in a suitable neighborhood So, the relation V Bκt ≈ V 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 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 Schrodinger operators, ă with their potential terms represented by either renormalized Poisson potential or partially renormalized Poisson potential Main Results Throughout the paper, ωd denotes the volume of the d-dimensional unit ball We denote Fd g ∈ W21 Rd : Rd g x dx , 2.1 where W21 Rd is used for the Sobolev space of functions that belong to L2 together with their first order derivatives We also denote ϕ u − e−u , Ξ u, v ψ u − e−u ϕ v e−u−v − u, u, v ∈ R, 2.2 ψ is introduced in 1.22 Clearly, the functions ψ, −ϕ, and Ξ are convex; this simple observation is crucial for the most constructions below International Journal of Stochastic Analysis Our main results about the asymptotics of negative exponential moments for annealed Brownian motion in a renormalized Poisson potential are represented by the following theorem Theorem 2.1 Let p ∈ d/2, d i In the “light-scale” case, lim t→∞ αt t d/p ν Rd log E ⊗ E0 exp − αt t V Bκs ds ψ θ|x|−p dx p d−p νωd θd/p 2.3 2p − d p Γ −νωd θd/p Γ p−d p ii In the “critical” case, lim t−d/ d t→∞ log E ⊗ E0 exp − sup ν Rd g∈Fd θ α ψ αt t V Bκs ds 2.4 g2 y κ p dy dx − x−y Rd ∇g y Rd iii In the “heavy-scale” case, under additional assumption p < d 4/ d 2−2p − d 4−2p / d 2−2p lim αt t t→∞ sup ⎧ ⎨ νθ2 g∈Fd ⎩ log E ⊗ E0 exp − g2 y Rd Rd x−y p dy αt κ dx − 2 dy /2, t V Bκs ds Rd ∇g y ⎫ ⎬ dy 2.5 ⎭ Remark 2.2 The additional assumption p < d /2 in Statement iii is exactly the condition for ξ t, x to be square integrable see 45 , and henceforth, for respective central limit theorem to hold true, see Proposition 4.4 and discussion in Section 4.4 below Let us discuss this theorem in comparison with the following, well-known in the field, results for annealed Brownian motion in a Poisson potential Theorem 2.3 Let K be bounded and satisfy K x ∼ θ|x|−p , with p > d |x| −→ ∞, 2.6 International Journal of Stochastic Analysis i (see [41]) If p ∈ d, d 2, lim t−d/p log E ⊗ E0 exp − t→∞ t −νωd θd/p Γ V Bκs ds ii (see [40]) If p lim t−d/ d d t→∞ p−d p 2.7 2, log E ⊗ E0 exp − t V Bκs ds − inf g∈Fd iii (see [46]) If p > d ν Rd Rd x−y 2.8 p dy dx κ Rd ∇g y dy 2, lim t−d/ d t→∞ ϕ θ g2 y log E ⊗ E0 exp − t V Bκs ds − inf ν g∈Fd Rd 1g x >0 dx 2.9 κ Rd ∇g y dy 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 and p ∈ d, d , 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 and “quantum regime” p ∈ d, d is frequently used Theorem 2.1 shows that the dichotomy “classic versus quantum regimes” is still in force for the model with a renormalized Poisson potential, with conditions on the shape function K to be either heavy or light tailed replaced by conditions on the scale αt to be, respectively, light or heavy Note that for αt ≡ 1, 1.18 and 1.19 transform exactly to p < d 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 Theorem 2.1 can be extended to other types of potentials, like a Poisson potential V or a partially renormalized Poisson potential V h We strongly believe that such an extension is possible in a whole generality; however, we cannot give such an extension in the quantum regime i.e., critical and heavy-scaled cases so far, because we 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 10 International Journal of Stochastic Analysis In the classic regime i.e., light scale case , such an extension can be made efficiently Moreover, in this case, the assumptions on the shape function K can be made very mild: instead of 1.5 , we assume 2.6 with p > d/2 and, when p < d, Rd ψ K x dx < ∞, 2.10 which is just the assumption for V to be well defined Theorem 2.4 Let the shape function K satisfy 2.6 and scale function αt satisfy 1.18 i Statement (i) of Theorem 2.1 holds true assuming K satisfies 2.10 ii For p > d, αt t lim t→∞ d/p −ν iii For p −p Rd ϕ θ|x| dx t αt V Bκs ds −νωd θ d/p 2.11 p−d Γ p d and h > 0, log E ⊗ E0 exp − ν log E ⊗ E0 exp − Rd where Eu 1.5 , αt t V h Bκs ds 2.12 αt K y , h − t −Γ dy ωd θ Eu t αt t o , αt t −→ ∞, 0, 57721 · · · is the Euler constant In particular, when K has the form log E ⊗ E0 exp − αt t νωd θ log αt t V h Bκs ds 2.13 log h Eu t αt t o , αt t −→ ∞ The following theorem shows that statements of Theorems 2.1 and 2.4 are process insensitive to some extent Theorem 2.5 Relations 4.4 , 2.3 – 2.5 , 2.11 , 2.12 , and 2.13 hold true with E0 replaced by Eκt 0,0 , that is, the expectation w.r.t the law of the Brownian bridge This theorem makes it possible to investigate the Lifshitz tails asymptotics for the integrated density of states of the random Schrodinger operators with partially ă renormalized Poisson potentials Let us outline the construction of respective objects 30 International Journal of Stochastic Analysis see 45 , we have by the dominated convergence theorem that the characteristic functions 4.55 converge pointwise to E0 exp − z2 Rd ξ2 t, x dx , 4.59 which is just the characteristic function of Ut The integrated Density of States 5.1 Proof of Proposition 2.6 Statement A We proceed in two steps First, we show that under conditions of the proposition, almost all realizations of Q are bounded from below on a given cube We V ; the case of a partially renormalized potential is quite analogous consider the case Q To simplify notation, we assume in the sequel ν Write V x Rd K g x − y ω dy − dy Rd Kg x − y ω dy − dy 5.1 The function K g is supported by some ball {x : |x| ≤ R}, which brings the lower bound −ωd Rd 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 function L : Rd → R, every realization of respective renormalized Poisson potential belongs to W21 Rd , and ∇ Rd L x − y ω dx − dy Rd ∇L x − y ω dy − dy 5.2 This, by usual approximation argument, provides that almost all realizations of the renormalized Poisson potential with kernel K belong to W21 Rd , and 5.2 holds true in Sobolev sense for K L Since K is Lipschitz, ∇K is bounded; the function K is bounded, as well Then, Proposition 2.7 in provides E exp V ∇V < ∞ 5.3 exp V x ∇V x dx < ∞ 5.4 By shift invariance, this gives E U International Journal of Stochastic Analysis 31 1 U Therefore, almost all realizations of V belong to W∞ p>1 Wp U and henceforth are continuous by Sobolev’s inclusion theorem e.g., 55 In particular, these realizations are bounded The second part of the proof is represented by the following deterministic lemma Lemma 5.1 Let Q : Rd → R be a function bounded from below and U a given cube Denote QN Q ∧ N, 5.5 Q and consider the Schrăodinger operators HU N , N ≥ with the Dirichlet boundary conditions, defined by 2.14 with Q replaced by QN Then, QN Q Q e−tHU , t ≥ converge strongly as N → ∞ to a continuous semigroup Rt,U , t ≥ i Rt,UN Q of self-adjoint operators in L2 U, dx Rt,U , t ≥ admits the Feynman-Kac representation 2.15 , Q ii every operator Rt,U , t ≥ is of trace class, Q Q iii for the generator HU of the semigroup Rt,U , t ≥ 0, the function 2.17 is well defined, and its Laplace transform admits the representation e−λt dNU λ Q R 2πκt −d/2 |U| U Eκt 0,0 exp − t Q Bκs dx 5.6 e−tHU , t ≥ admits the Feynman-Kac repre- sentation 2.15 An alternative form of this representation integral operator with the kernel Q x QN Q Proof For every N, the semigroup Rt,UN rt,UN x, y x ds χU,t B· pt x, y Etx,y exp − Q is that Rt,UN is an 1, page 13 t QN Bκs ds χU,t B· , 5.7 where pt x, y 2πκt −d/2 −|x−y|2 /2κt 5.8 e is the transition probability density for the process Bκs , s ≥ 0, and Etx,y denotes the expectation w.r.t law of the Brownian bridge which takes values x and y at s and s t, respectively By Q Propositions 3.2 and 3.5 in , for a given N, the function rt,UN is continuous and symmetric Then, by the monotone convergence theorem, there exists a monotonous limit Q rt,U x, y Q lim rt,UN x, y N →∞ 2πκt −d/2 −|x−y|2 / 2κt e Etx,y exp − t Q Bκs ds χU,t B· , 5.9 32 International Journal of Stochastic Analysis Q Q which is bounded and symmetric Integral operators Rt,UN converge to the operator Rt,U with Q kernel rt,U in the Hilbert-Schmidt norm, because respective kernels converge in L2 U×U, dx This proves Statement i immediately Since Q Rt,U Q Q 5.10 Rt/2,U Rt/2,U , Q Q and every Rt/2,U is a Hilbert-Schmidt operator, the operator Rt,U is of trace class This proves Statement ii Q We have just proved that Rt,U is of trace class, which means that it has a purely point spectrum and Q λk,Q,t < ∞, Trace Rt,U 5.11 k Q where {λk,Q,t } denote eigenvalues of Rt,U , counted with their multiplicities By the spectral Q Q decomposition theorem, this yields that the generator HU of the semigroup Rt,U , t ≥ has a purely point spectrum locally finite on every interval −∞, λ In addition, λk,Q,t e−λk,Q t , Q where {λk,Q } are respective eigenvalues of HU , counted with their multiplicities Therefore, the function 2.17 is well defined, and its Laplace transform has the form e−λt dNU λ Q R |U| Q Trace Rt,U |U| e−tλk,U k |U| Q U×U Q rt/2,U x, y rt/2,U y, x dxdy, 5.12 in the last equality, we have used the standard relation e.g., 56, Chapter III, Section Trace A∗ A A HS , 5.13 with the Hilbert-Shmidt norm of the operator A in the right hand side By the Feynman-Kac representation 2.15 and the Markov property of the Brownian bridge, the last integral can be written as 2πκt − d/2 |U| U Eκt x,x exp − t Q Bκs ds χU,t B· dx 5.14 which completes the proof of the lemma Statement B sketch of the proof In the second part of the proposition, the classic argument which goes back to 21 is applicable In order to keep the exposition self-sufficient, we give a brief sketch of this argument here The random fields Q V and Q V h are ergodic or metrically transitive in the sense that the σ-algebra generated by functionals, invariant w.r.t the transformations Sh : Q · −→ Q · h , h ∈ Rd 5.15 International Journal of Stochastic Analysis 33 is degenerate The argument here is a straightforward modification of the classic one for onedimensional moving average integrals; see Theorem 1.1 and Example 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, lim f Q · U↑Rd |U| x dx Ef Q , 5.16 U both almost surely and in mean sense By Proposition 2.7 in , Ee−cQ < ∞ for every c > This, by the Jensen inequality, provides E ⊗ Eκt 0,0 exp − t Q Bκs ds < ∞, t ≥ 0, 5.17 the argument here is the same as at the beginning of Section 3.1 Then, 5.16 applied to the function f : q −→ Eκt 0,0 exp − t q Bκs ds 5.18 yields d U↑R |U| lim U Eκt 0,0 exp − t Q Bκs x ds E ⊗ Eκt 0,0 exp − dx t Q Bκs ds 5.19 both in mean and in a.s sense Straightforward calculation shows that with probability 1, lim U↑Rd |U| χU,t B· x dx 5.20 U Together with mean convergence 5.19 , this provides d U↑R |U| lim U Eκt 0,0 exp − t Q Bκs x ds − χU,t B· x dx 0, 5.21 which completes the proof 5.2 Proof of Theorem 2.5 5.2.1 Classic Regime Arguments in Section 3.1 are process insensitive Henceforth, the upper bounds in 2.3 , 2.11 , and 3.4 hold true with E0 replaced by Eκt 0,0 34 International Journal of Stochastic Analysis On the other hand, Brownian bridge measures enjoy the scaling property similar to the Brownian one 1, page 140 : its law Pt0,0 is the image of measure of P10,0 under the map w · −→ √ · t tw 5.22 In addition, the Brownian bridge measure P10,0 has the small balls asymptotics similar to the Brownian one 59 ε2 log P10,0 sup |B s | ≤ ε s∈ 0,1 −→ − j 2d−2 /2 , 5.23 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 Eκt 0,0 instead of E0 The argument which deduce 2.12 from 3.4 is process insensitive 5.2.2 Quantum Regime The upper bound in 4.4 with Eκt 0,0 instead of E0 can be deduced from the same upper bound in its original form In the proof, we combine the standard trick based on the Markov property of the Brownian bridge e.g., Lemma in 27 with the “universal” upper bound provided by the convexity; see the end of Section 4.1 Write η t, x η t − 1, x t ζ t, x , ζ t, x L Bκs − x ds 5.24 t−1 For every γ ∈ 0, , we have by convexity υ η t, x t ≤ γυ η t − 1, x γt 1−γ υ ζ t, x 1−γ t 5.25 Analogously to 4.6 , we have Rd υ ζ t, x 1−γ t dx ≤ t Rd t−1 υ L Bκs − x 1−γ t dxds 5.26 Rd υ Lx 1−γ t dx International Journal of Stochastic Analysis 35 The last term vanishes when t → ∞ Therefore, for every γ ∈ 0, , lim sup Eκt 0,0 exp t t→∞ t Rd η t, x t υ ≤ lim sup Eκt 0,0 exp γt t→∞ t Rd υ dx 5.27 η t − 1, x γt dx On the other hand, applying the Markov property at time t − 1, we arrive at Eκt 0,0 exp γt Rd υ η t − 1, x γt E0 exp γt dx ≤ E0 exp γt Rd Rd υ η t − 1, x γt υ η t, x γt dx p1 Bκ t−1 , dx p1 Bκ t−1 , 5.28 Because p1 x, y ≤ 2πκ −d/2 , x, y ∈ Rd , 5.29 we get from the upper bound in 4.4 that for every γ ∈ 0, , lim sup Eκt 0,0 exp t t→∞ t Rd ≤ sup γ υ γ d R g∈Fd η t, x t υ dx κ L x − y g y dy dx − d R 5.30 Rd ∇g y dy Passing to the limit γ → completes the proof of the upper bound The lower bound in 4.4 with Eκt 0,0 instead of E0 can be obtained by almost the same argument, as it was used to prove lower bound in 4.4 in its initial form Only minor changes of the argument are required; let us discuss these changes Consider the large deviation result by Kac, which was the basic point in the proof of the lower bound log E0 exp t→∞ t t f Bκs ds lim ≥ CΥ,f,R sup g∈Fd f x g x dx − Rd 5.31 Rd ∇g x dx Note that under P0 , Bκs − s Bκt κt 5.32 36 International Journal of Stochastic Analysis is a Gaussian process with the covariance κ s∧s −κ ss /t and, therefore, has the distribution Pt0,0 1, page 140 Then, for any Lipschiz continuous bounded function f, 5.31 holds true with Eκt 0,0 instead of E0 Note that the statement of the Lemma 4.2 still holds true when BR is replaced with any class of functions 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 f y −R,R d f x L y − x dx 5.33 is Lipschitz continuous and bounded Applying the modified 5.31 and proceeding literally as in the proof of the lower bound in Theorem 4.1, we get the required lower bound Once we have proved the modified large deviation asymptotics 4.4 , we can repeat the arguments from Sections 4.2 and 4.3 with the the scaling property of the Brownian bridge used instead of the same property of the Brownian motion and deduce 2.4 , 2.5 with E0 replaced by Eκt 0,0 5.3 Proof of Theorem 2.7 Theorem 9.7 in 42 , Chapter IV gives 2.20 as a straightforward corollary of 2.3 Because t → t log t is a regularly varying function of the order 1, this theorem is not applicable when 2.21 is considered From 2.13 with Eκt 0,0 instead of E0 , we have log R e− λ νωd θ log h Eu t h dN V λ νωd θt log t ot , t −→ ∞ 5.34 Henceforth, the proof of 2.21 by elementary transformations can be reduced to the proof of the following lemma Lemma 5.2 Let, for a nonnegative and nondecreasing function N λ , λ ∈ R, log R e−λt dN λ at log t ot , t −→ ∞, 5.35 with some a > Then, log N λ −a exp − λ −1 a o1 , λ −→ −∞ 5.36 International Journal of Stochastic Analysis 37 Proof The upper bound, in a standard way, is provided by the Chebyshev inequality: for every t > 0, λ < 0, log N λ ≤ λt log R e−χt dN χ 5.37 exp −λ/a − , the solution of the minimization problem Take tλ tλ Clearly, tλ → arg λt at log t 5.38 ∞, λ → −∞ By 5.35 and 5.37 , log N λ ≤ −atλ λ −→ −∞, o tλ , 5.39 which gives the upper bound in 5.36 Assume that the lower bound in 5.36 fails; that is, there exist b > a and a sequence λn → −∞ such that log N λn ≤ −b exp − λn −1 , a n ≥ 5.40 Fix c < a and δ > 0, which will be specified below Since the upper bound in 5.36 is already proved, there exists Λc < such that log N λ ≤ −c exp − Let n be large enough for λn < Λc Denote tn R e−λtn dN λ Λc −∞ e−λtn N λ dλ λn −2δ −∞ e−Λc tn In1 In2 In3 Λc e− λn −δ e−Λc tn e−λtn N λ dλ ∞ λ −1 , a λn λn −2δ /a−1 ∞ Λc λ < Λc 5.41 and write e−λtn dN λ e−λtn N λ dλ Λc λn e−λtn N λ dλ 5.42 e−λtn dN λ In4 Let us estimate In1 – In4 Clearly, In4 ≤ Ce−Λc tn , 5.43 38 International Journal of Stochastic Analysis with appropriate constant C Assumption 5.40 yields, via monotonicity, In2 ≤ 2δ exp − λn − 2δ tn − be−λn / a−1 2δ exp δ a log tn a tn − be−δ/a tn 5.44 2δ exp atn log tn − be−δ/a − a − δ tn , here we have used the relation −a log tn − a λn δ, 5.45 which comes from the definition of tn Consider the function Θ : λ → −λtn − ce−λ/a−1 Straightforward computation shows that assuming c > ae−δ/a , 5.46 its derivative is increasing on −∞, λn − 2δ , and c δ/a e − tn > 1, a Θn λn − 2δ 5.47 for n large enough Then, with 5.41 in mind, we get In1 ≤ λn −2δ −∞ eλ−λn 2δ Θn λn −2δ e dλ eΘn λn −2δ exp atn log tn − ceδ/a − a − δ tn 5.48 Similar argument leads to In3 ≤ ∞ e−λ λn Θn λn e dλ eΘn λn exp atn log tn − δ ceδ/a − a tn 5.49 λn Now, we can finalize the proof Take δ > such that be−δ/a − a − δ > Note that aeδ/a − a − δ > 0, δ aeδ/a − a > 5.50 Therefore, c < a can be chosen in such a way that 5.46 holds true and ceδ/a − a − δ > 0, δ ceδ/a − a > 5.51 International Journal of Stochastic Analysis 39 Under such a choice of the constants δ and c, 5.43 , 5.44 n large enough, log R 5.48 , and 5.49 provide that for e−λtn dN λ ≤ atn log tn − εtn , 5.52 with some positive ε This contradicts to 5.35 and proves that assumption 5.40 is impossible Remark 5.3 In the proof of the lower bound in Lemma 5.2, we have combined the upper bound from the same lemma with the estimates, typical for the Laplace method e.g., 60 Note that such a structure of the proof is similar to the one for the Găartner-Ellis theorem see Section 1.1 in 51 although we cannot deduce the statement of the lemma from the Găartner-Ellis theorem directly 5.4 Proof of Theorem 2.8 Our argument is based on the following version of the Găartner-Ellis theorem Lemma 5.4 Consider a sequence Nn , n ≥ of nonnegative monotonous functions on R, which vanish at −∞, and assume that there exist a > 1, c > 0, and a sequence Υn → ∞ such that log Υn R e−μΥn x dNn x −→ I μ cμa , n −→ ∞, μ > 5.53 n −→ ∞, 5.54 Then, log Nn −x Υn −I ∗ x o1 , uniformly by x ∈ A, B for every A, B ⊂ 0, ∞ Here, I∗ x sup μx − I μ μ>0 c a−1 −1/ a−1 a−1 a a/ a−1 xa/ a−1 5.55 The only difference between conditions of Lemma 5.4 and standard assumptions of the Găartner-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 Găartner-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 function N ·, γ , γ > is nonnegative and nondecreasing, 40 International Journal of Stochastic Analysis b for every t > 0, N t, γ : R e−λt N dλ, γ < ∞, 5.56 c for given a > 1, b ∈ R, c > 0, and given sequences λn > 0, γn > 0, n ≥ with λan γn−b → ∞, 1/ a−1 log N μλn a/ a−1 cμa λn −b/ a−1 γn , γn 5.57 −b/ a−1 γn o1 , n → ∞, μ > Then, log N −λn x, γn −1/ a−1 c a−1 a/ a−1 a−1 a a/ a−1 λn −b/ a−1 γn xa/ a−1 o1 , n → ∞, 5.58 uniformly by x ∈ A, B for every A, B ⊂ 0, ∞ Proof Put Υn a/ a−1 λn −b/ a−1 γn , Nn x x ∈ R N λn x, γn , 5.59 Because a > and λan γn−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 Theorem 2.8 Statement (i) Take a d , p b c νωd θd/p 2p − d p Γ d−p p 5.60 For given sequences λn < 0, γn > 0, n ≥ and arbitrary μ > denote tn μ −λn 1/ a−1 −b/ a−1 γn , αtn γn 5.61 Condition −λn p/d /γn → ∞ yields tn → ∞, and condition −λn d 2−p /2 /γn → ∞ yields o tn Therefore, 5.57 is provided by 2.3 In addition, we have −λn /γn → ∞ αtn because p/d < 1, d − p /2 > 1, and consequently −λn a γn−b → ∞ Applying Corollary 5.5 with x −1 and λn replaced by −λn , we obtain the required statement International Journal of Stochastic Analysis 41 Statement (ii) Take a d d − 2p , − 2p b d , − 2p c C2 , 5.62 and keep the notation 5.58 Condition −λn d 4−2p /4 /γn → ∞ yields −λn a γn−b → ∞ d 2−p / d o αtn Finally, these two conditions Condition −λn d 2−p /2 /γn → yields tn d 2−2p /4 /γn → ∞ which is equivalent to tn → ∞ yield λn → 0−, and consequently −λn 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 lim t→∞ αt t d/p log E ⊗ E0 exp − αt t V Bκs ds Cψ 5.63 Using this relation and following the proof of Statement i with appropriate modifications, we obtain the required statement Acknowledgments The authors are grateful for the referee for useful comments 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sup-norms, with an application to the supremum of Bessel local times,” Journal of Theoretical Probability, vol 9, no 4, pp 915–929, 1996 60 E T Copson, Asymptotic Expansions, Cambridge Tracts in Mathematics and Mathematical Physics, No 55, Cambridge University Press, New York, NY, USA, 1965 Copyright of International Journal of Stochastic Analysis is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use ... model in a renormalized Poisson potential, ” Annales de l’Institut Henry Poincare Accepted X Chen, “Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic... 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 ; calculation of the integral... the potential We now outline the rest of the paper The main results about negative exponential moments for annealed Brownian motion in a renormalized Poisson potential are collected in Theorem