Large deviations in total variation of occupation measures of one-dimensional diffusions

For one-dimensional diffusion processes, we find an explicit necessary and sufficient condition for the large deviation principle of the occupation measures in the total variation and of local times in L¹$L^1$.     

Sample path large deviations for super-Brownian motion

Summary We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as Kakutani-Hellinger energy. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Hölder norm of index <1/2 of the processtf, X _t possesses some finite exponential moments provided the functionf is smooth.This work was supported in part by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

Upper large deviations for the maximal flow in first-passage percolation

We consider the standard first-passage percolation in Z^d${\mathbb{Z}}^d$ for d≥2$d\ge 2$ and we denote by _?nd−1,h(n)${?}_{n^{d-1},h\left(n\right)}$ the maximal flow through the cylinder ]0,n]^d−1×]0,h(n)]${]0,n]}^{d-1}\times ]0,h\left(n\right)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension 3: under some assumptions, _?nd−1,h(n)/n^d−1${?}_{n^{d-1},h\left(n\right)}/n^{d-1}$ converges towards a constant ν$\nu$. We look now at the probability that _?nd−1,h(n)/n^d−1${?}_{n^{d-1},h\left(n\right)}/n^{d-1}$ is greater than ν+ε$\nu +\epsilon$ for some ε>0$\epsilon >0$, and we show under some assumptions that this probability decays exponentially fast with the volume n^d−1h(n)$n^{d-1}h\left(n\right)$ of the cylinder. Moreover, we prove a large deviation principle for the sequence (_?nd−1,h(n)/n^d−1,n∈N)$({?}_{n^{d-1},h\left(n\right)}/n^{d-1},n\in \mathbb{N})$.     

A PDE approach to large deviations in Hilbert spaces

We introduce a PDE approach to the large deviation principle for Hilbert space valued diffusions. It can be applied to a large class of solutions of abstract stochastic evolution equations with small noise intensities and is adaptable to some special equations, for instance to the 2D stochastic Navier–Stokes equations. Our approach uses a lot of ideas from (and in significant part follows) the program recently developed by Feng and Kurtz [J. Feng, T. Kurtz, Large Deviations for Stochastic Processes, in: Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006]. Moreover we present easy proofs of exponential moment estimates for solutions of stochastic PDE.     

Comparison of interacting diffusions and an application to their ergodic theory

Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.     

Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods. Received: 26 May 1998 / Revised version: 24 February 1999     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

Surface order large deviations for high-density percolation

Summary We derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a Bernoulli percolation process restricted to a big finite box, with sufficiently large parameter. We also establish a useful version of the isoperimetric inequality, which is the main tool of our proofs.     

Large deviations for Markov processes with mean field interaction and unbounded jumps

Summary An N-particle system with mean field interaction is considered. The large deviation estimates for the empirical distributions as N goes to infinity are obtained under conditions which are satisfied, by many interesting models including the first and the second Schlögl models.Supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University and the NSERC operating grant of D.A. Dawson     

Freidlin-Wentzell's large deviations for homeomorphism flows of non-Lipschitz SDEs

We prove a Freidlin-Wentzell large deviation principle for multi-dimensional stochastic differential equations with non-Lipschitz coefficients and apply it to the Brownian motion on the diffeomorphism group of the disc constructed recently by Airault, Malliavin and Thalmaier.     

Smoothness of the intensity measure density for interacting branching diffusions with immigrations

We consider the invariant measure for finite systems of interacting branching diffusions with immigrations. We use Malliavin calculus in order to show that the intensity measure of the invariant measure admits a density which is continuous, one times partially differentiable and bounded provided the immigration measure is absolute continuous.     

Dominating points and large deviations for random vectors

Summary We establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a Banach space. These estimates are based on the existence of dominating points in this setting.Dedicated to Peter Ney on the occasion of his 65th birthday.Supported in part by NSF Grant DMS-9503665Supported in part by NSF Grant DMS-9400024     

Finite and infinite systems of interacting diffusions

Summary We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={x _i (t),i ∈ ℤ ^d } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, , with . Our main result gives a comparison between the laws ofx(t _N ) andx ^N (t _N ) for timest _N →∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the National Science Foundation, and by the National Security Agency Research supported in part by the DFG Partly supported by S.R.63540155 of Japan Ministry of Education     

Surface order large deviations for Ising,Potts and percolation models

Summary We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq1 andd3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.     

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence (_Xi)$\left(X_i\right)$ of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional   empirical process of words, where one conditions on a typical underlying (_Xi)$\left(X_i\right)$. We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted.     

Large deviations for two scaled diffusions

Summary. We formulate large deviations principle (LDP) for diffusion pair (X ^ɛ ,ξ ^ɛ )=(X _t ^ɛ ,ξ _t ^ɛ ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X ^ɛ ,ν ^ɛ ) with ν ^ɛ (dt, dz) being an occupation type measure corresponding to ξ _t ^ɛ . In some sense we obtain a combination of Freidlin–Wentzell’s and Donsker–Varadhan’s results. Our approach relies on the concept of the exponential tightness and Puhalskii’s theorem. Received: 29 June 1995/In revised form: 14 February 1996     

Effect of truncation on large deviations for heavy-tailed random vectors

This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost.     

Large deviations for the radial processes of the Brownian motions on hyperbolic spaces

Using the heat kernel estimates by Davies (1989) and Anker et al. (1996), we show large deviations for the radial processes of the Brownian motions on hyperbolic spaces.     

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.