Functional limit theorems for a time-changed multidimensional Wiener process

TL;DR

This paper establishes functional limit theorems for a multidimensional Wiener process that is time-changed by an additive functional, revealing convergence to processes including multidimensional skew Brownian motion.

Contribution

It introduces new functional limit theorems for time-changed multidimensional Wiener processes with state-dependent intensities, extending classical results to more complex settings.

Findings

Convergence to multidimensional skew Brownian motion.

Limit theorems under state-dependent intensity conditions.

Asymptotic behavior characterized by different limits in each octant.

Abstract

We study the asymptotic behaviour of a properly normalized time-changed multidimensional Wiener process; the time change is given by an additive functional of the Wiener process itself. At 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. We assume 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 the possible limits there is a multidimensional analogue of skew Brownian motion.