On bivariate distributions of the local time of It\^o-McKean diffusions

Denote as $L$ the local time at $0$ of an It\^o-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 It\^o-McKean diffusion from a hyperplane. Examples involving the distribution of a local time are presented, including a formula for the distribution of $(X_t,L_{\infty})$ for a transient diffusion. Based on joint work with Jacek Jakubowski.