We consider a class of random perturbations of Hamiltonian systems with many degrees of freedom. We assume that the perturbations consist of two components: a larger one which preserves the energy and destroys all other first integrals, and a smaller one which is a non-degenerate white noise...

Bernstein-type inequalities for local martingales are derived. The results extend a number of well-known exponential inequalities and yield an asymptotic inequality for a sequence of asymptotically continuous martingales.

We consider the Lyapunov exponents of flows generated by a class of Markovian velocity fields. The existence of the exponents is obtained for flows on a compact set, but with the most general form of the velocity field. As a particular class, we study the homogeneous and incompressible flows. In...

We deal with the following general kind of stochastic partial differential equations:with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density...

We consider financial market models based on Wiener space with two agents on different information levels: a regular agent whose information is contained in the natural filtration of the Wiener process W, and an insider who possesses some extra information from the beginning of the trading...

We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in or regular fractals we show that there are unbounded fluctuations in the...

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205-228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its...

It is proved that under suitable conditions the probability laws of two arbitrary solutions of the infinite dimensional stochastic equationdXt=AXt dt+f(t,Xt) dt+Q1/2 dWtconverge to each other, as time goes to infinity, in the strong (variational) topology. To this end, some lower estimates on...

We establish several methods for constructing stationary self-similar random fields (ssf's) on the integer lattice by "random wavelet expansion", which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random...

Let a Brownian motion in the unit ball be absorbed if it hits a set generated by a radially symmetric Poisson point process. The point set is "fattened" by putting a ball with a constant hyperbolic radius on each point. When is the probability non-zero that the Brownian motion hits the boundary...