We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost…

If $\mu$ is a probability measure on the set of suitable weak solutions of the 3D Navier-Stokes equations, invariant for the time-shift, with finite mean dissipation rate, then at every time $t$ the set of singular points is empty $\mu$-a.s.…

We study the asymptotics as $t \to 0$ of the transition density of a class of $\mu$-symmetric diffusions in the case when the measure $\mu$ has a multi-fractal structure. These diffusions include singular time changes of Brownian motion on the…

We extend Krylov's $L_p$-solvability theory to the Cauchy problem for systems of parabolic stochastic partial differential equations. Some additional integrability and regularity properties are also presented.

We are interested in the component intervals of the complements of a monotone sequence $R_n \subseteq \dots \subseteq R_1$ of regenerative sets, for two natural embeddings. One is based on Bochner's subordination, and one on the intersection of independent regenerative…

The small diffusion and fast dying out asymptotics is calculated for nonlinear equations of a class of superprocesses on manifolds, and the corresponding logarithmic limit of the solution is shown to be given by a solution of a certain problem…

This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role and influence on the connection between the sub-Gaussian estimates, parabolic…

Equivalent upper and lower bounds for the $L^1$ norm of Hilbert space valued infinitely divisible random variables are obtained and used to find bounds for different types of stochastic integrals.

We consider ballistic diffusion limited aggregation on a finite strip $[0, L-1]$ times $\mathbb{Z}_+$ in $\mathbb{Z}^2$ for some $L$ in $\mathbb{Z}_+$. We provide numerical bounds on the growth in the height process.

We give a new relatively compact proof of the famous identity for the distribution of the first hitting time of a linear boundary by a skip-free process with stationary independent increments. The proof uses martingale identities and change of measure.