L. Stoica

The obstacle problem for quasilinear stochastic PDE’s

We prove an existence and uniqueness result for the obstacle problem of quasilinear parabolic stochastic PDEs. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation.

On the Construction of Hunt Processes from Resolvents

If (Kn) is an increasing sequence of compact sets such that K , c / s 1 and U K,=E, it follows from the quasileft-continuity of the process that the sen quence of hitting times TCK n converges to ~. Therefore if we denote by F x the family of all 2-excessive functions s which satisfy the inequality V~ 1 < s on the complement of some compact set K = K(s), depending on s, then from Hunt s balayage theorem one easily deduces the following relation:

On the Limits at the Martin Boundary for a Class of Functions

In this paper we prove a criterion for existence of pathwise limits at the Martin boundary for functions with gradient in Lloc2. (This implies that such functions have fine limits at almost all Martin boundary points.)

The approximation of additive functionals of diffusion processes

Let us consider a diffusion process in Rd . Around each point x one may consider a ring of size ϵ and a process which counts the crossings over the ring. Integrating with respect to a measure μ(dx) and letting ϵ→ 0 one gets an additive functional. This is a natural generalization of the approximation theorem of the local time of one dimensional Brownian motion by means of “downcrossings”. For multidimensional Brownian motion the result was established by Bally. The present paper introduces a new method which allows us to handle general diffusions