Joint continuity of the intersection local times of Markov processes
This paper gives simple conditions on the transition densities of two independent Markov processes X and Y in \({\mathbb{R}}^ d\), which guarantee the existence of the intersection local time of X and Y, i.e. of a continuous density for the measure: \[ f\to \iint_{H}f(X_ s-Y_ t)ds dt, \] where H is any bounded Borel subset of \(({\mathbb{R}}_+)^ 2\). These conditions are satisfied in the case of smooth elliptic diffusions in \({\mathbb{R}}^ d\), for \(d=2\) or 3. Under similar assumptions, the author also proves the existence of a local time of self-intersections for the single process X, continuous except at \(x=0\). In the case of smooth elliptic diffusions, he establishes a Tanaka-like formula for this local time, which in the two-dimensional case can be used to study the singularity at \(x=0\).
existence of the intersection local time
scientific article; zbMATH DE number 4009481