Brownian Motion in a Hilbert Space with a Semipermeable Membrane on a Hyperplane

Abstract

In a separable Hilbert space, we construct a continuous Markov process whose behavior coincides everywhere, except for a hyperplane S orthogonal to a given unit vector ν, with the behavior of a homogeneous Gaussian process with a given correlation operator tB, where B is a nonsingular nuclear operator. As the process hits the hyperplane, it receives an impulse infinite in modulus in the direction A such that |(A, ν)| ≤ (Bν, ν).We obtain a stochastic differential equation whose solutions are trajectories of the process constructed.