Convergence rate for spherical processes with shifted centres.

The paper is devoted to study of a modification of the random walks on spheres in a finite domain G ⊂ Rm, m ≥ 2. It is proved that the considered spherical process with shifted centres converges to the boundary of G very rapidly. Namely, the average number of steps before fitting the ε-neighborhood of the boundary has the order of ln |ln ε| as ε → 0 instead of the standard order of |ln ε|. Thus, the spherical process with shifted centres can be effectively used for Monte Carlo solution of different problems of the mathematical physics related to the Laplace operator. [ABSTRACT FROM AUTHOR]