Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $\alpha\ge1$, we consider the set $\Theta_\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\alpha$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $\Theta_\alpha$, will never leave $\Theta_\alpha$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics

Invariant, super and quasi-martingale functions of a Markov process

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures

Dimension-independent Harnack inequalities for subordinated semigroups

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $\alpha$ satisfies a dimension-free Harnack inequality provided $\alpha \in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $\alpha \in (0,1).$ Some infinite-dimensional examples are also presented

Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.Comment: Published in at http://dx.doi.org/10.1214/12-AOP763 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc

Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients

In this paper we give an affirmative answer to an open question mentioned in [Le Bris and Lions, Comm. Partial Differential Equations 33 (2008), 1272--1317], that is, we prove the well-posedness of the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.Comment: 11 pages. The proof has been modifie

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.Comment: 31 pages, published in Appl. Math. Opti

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution