On the number of crossings of an elastic absorbing membrane on a hyperplane by a Brownian motion. (English. Ukrainian original) Zbl 0928.60060

Durdyev, A. K.; Portenko, M. I.

Theory Probab. Math. Stat. 56, 67-80 (1998); translation from Teor. Jmovirn. Mat. Stat. 56, 67-80 (1997).

Let \(S\) be a hyperplane in \(R^{n}\) which is orthogonal to a fixed basis vector \(\nu\in R^{n}\). Let \(q(x)\) be a given continuous function on \(S\) with values in \([-1,1]\). Let \(r(x)\) be a given continuous bounded function on \(S\) with nonnegative values and let \(\sigma _{S}(x)\) be a generalized function on \(R^{n}\), the action of which on test function is reduced to the integration along the hyperplane \(S\). Then for the generalized diffusion process in the \(d\)-dimensional Euclidean space \(R^{n}\) with a constant diffusion operator \(B\), the drift vector \(q(x)\sigma _{S}(x)B\nu\) and the killing coefficient \(r(x)\sigma _{S}(x)\), the limit theorem for the number of intersections of hyperplane by a discrete approximation of this process is proved under the condition that the value of discretization tends to zero.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60J60	Diffusion processes
60J65	Brownian motion

Keywords:

generalized diffusion process; multidimensional Brownian motion; limit theorem; number of intersections