Brownian motion in a Hilbert space with diffusion along a translucent membrane on a hyperplane. (English. Ukrainian original) Zbl 0994.60041

Zajtseva, L. L.

Theory Probab. Math. Stat. 62, 19-26 (2001); translation from Teor. Jmovirn. Mat. Stat. 62, 19-26 (2000).

The aim of this paper is to construct an infinite-dimensional analogue of generalized diffusion processes in \(d\)-dimensional Euclidean space \(R^{d}\) with diffusion matrix \(B+\beta\delta_{S}(x)\) and vector of drift \(A\delta_{S}(x),\) where \(S\) is a hyperplane in \(R^{d}\) orthogonal to a given unit vector \(\nu,\) \(B\) and \(\beta\) are constant positive symmetrical operators on \(R^{d}\) and \(S\), respectively, \(A\) is a vector in \(R^{d}\) such that \(|(A,\nu)|\leq(B\nu,\nu),\) \(\delta_{S}(x)\) is a generalized function on \(R^{d},\) action of which on a test function is reduced to the integration of the latter by \(S.\) The author begins with constructing a finite-dimensional process in the form suitable for further generalization on Hilbert space. The method of construction was proposed by B. I. Kopytko and N. I. Portenko [in: Probability theory and mathematical statistics. Lect. Notes Math. 1021, 318-326 (1983; Zbl 0531.60073)], where a particular case is considered with \(B\) as identity operator.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G20	Generalized stochastic processes
60J65	Brownian motion
58J65	Diffusion processes and stochastic analysis on manifolds
60J25	Continuous-time Markov processes on general state spaces
60J60	Diffusion processes

Keywords:

Brownian motion in Hilbert space; translucent diaphragm on hyperplane; diffusion operator of trace class

Citations:

Zbl 0531.60073