Functional limit theorems for a time-changed multidimensional Wiener process

Abstract

UDC 519.21

We study the asymptotic behavior of a properly normalized time-changed multidimensional Wiener process. The time change is described by an additive (in time) functional of the Wiener process itself. On the level of generators, the time change means that we consider the Laplace operator, which generates a multidimensional Wiener process, and multiply it by a (possibly degenerate) state-space dependent intensity. It is assumed that the intensity admits limits at infinity in each octant of the state space but the values of these limits may be different. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time-changed multidimensional Wiener process. Among possible limits, there is a multidimensional analog of the skew Brownian motion.