Abstract:In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure L t (A) = 1 t t 0 1l A (W s )ds of the d dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions M 1 (R d ) can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of R d by adding a point at ∞ that results in the compactification of M 1 (R d ) by allowing some mass to escape to the point at ∞. If one were to use only test functions that are continuous and vanish at ∞ then the compactification results in the space of sub-probability distributions M ≤1 (R d ) by ignoring the mass at ∞. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbitsThere are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.1. Motivation and Introduction 1.1 Motivation. We start with the Wiener measure P on Ω = C 0 [0, ∞); R d corresponding to the d-dimensional Brownian motion W = (W t ) t≥0 starting from the origin. Our result is motivated by the following set up. Let L t denote the normalized occupation measure of the Brownian motion until time t, i.e.,This is a random element of M 1 = M 1 (R d ), the space of probability measures on R d . We are interested in the transformed measurewith A being a measurable set in the path space of the Brownian motion andV (x − y)µ(dx)µ(dy).

We consider the smoothed multiplicative noise stochastic heat equationwhere Bε,t is a spatially smoothed (at scale ε) space-time white noise, and β > 0 is a parameter. We show the existence of aβ ∈ (0, ∞) so that the solution exhibits weak disorder when β <β and strong disorder when β >β. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.Here the white noise (the disorder) is replaced by i.i.d. random variables η = {η(n, x) : n ∈ N, x ∈ Z d }, P 0 denotes the law of a simple random walk starting at the origin corresponding to a d-dimensionalDate: January 7, 2015. 1991 Mathematics Subject Classification. 60J65, 60J55, 60F10.

We consider mean‐field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self‐attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan in terms of the Pekar variational formula, which coincides with the behavior of the partition function of the polaron problem under strong coupling. Based on this, in 1986 Spohn made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the Pekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean‐field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean‐field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence is a contribution to the understanding of the “mean‐field approximation” of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed by Mukherjee and Varadhan in 2016; its extension to the uniform strong metric for the singular Coulomb interaction was carried out by König and Mukherjee in 2015, as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by Bolthausen and Schmock.© 2017 Wiley Periodicals, Inc.

We consider p independent Brownian motions in R d . We assume that p 2 and p.d 2/ < d . Let`t denote the intersection measure of the p paths by time t, i.e., the random measure on R d that assigns to any measurable set A R d the amount of intersection local time of the motions spent in A by time t . Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass`t .R d / as t ! 1. In this paper, we derive a large-deviation principle for the normalized intersection measure t p`t on the set of positive measures on some open bounded set B R d as t ! 1 before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalized occupation times measures of the p motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure.A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set U B. This extends earlier studies on the intersection measure by König and Mörters.

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measureℙα,T=Zα,T−1exp{}α2∫−TT‍∫−TT‍normale−∣t−s∣∣ω()t−ω()s∣0.25emnormalds0.25emnormaldtnormaldℙwith respect to ℙ that governs the law of the increments of the three‐dimensional Brownian motion on a finite interval [−T, T], and Zα, T is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self‐attractive interaction. According to a conjecture of Pekar that was proved in [9], g0=limα→normal∞1α2[]limT→normal∞logZα,T2T exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit trueℙ⌢α=limT→∞trueℙ⌢α,T exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for 12T()ω()T−ω()−T under trueℙ⌢α,T and obtain an expression for the limiting variance. © 2019 Wiley Periodicals, Inc.