Zaitseva L. Diffusion processes with membranes in Hilbert space

The wide class of generalized diffusion processes in finite-dimensional space is constructed in the form that is invariant of the dimension of the phase space. This construction is extended to the arbitrary separable Hilbert space. We consider two approaches to construction of such processes: analytical and probabilistic. Using analytical approach we construct transition probability of diffusion process with generalized drift vector and diffusion matrix as the solution of initial-boundary problem for partial differential equation of parabolic type in such a way that this result has analogue in Hilbert space. Also we find stochastic differential equation such that constructed process is the weak solution to this equation.Using probabilistic approach we construct stochastic process, which drift vector and diffusion matrix include delta-function concentrated on the given hyperplane, as the strong solution to the stochastic differential equation. This approach with some improvements also takes place in infinite-dimensional case.