On bivariate distributions of the local time of Itô-McKean diffusions

Abstract

Denote as L the local time at 0 of an Itô-McKean diffusion X. We present a new explicit description of the distribution of $L_t$ in terms of convolution exponent and, using the excursion theory, we describe the transition density of the pair $(X,L)$. We provide a simple connection formula for the distribution of excursions of a bivariate Itô-McKean diffusion from a hyperplane. Examples involving the distribution of a local time are presented, including a formula for the distribution of $(X_t,L_{\mathrm{\infty }})$ for a transient diffusion.