Another view on the local time of self-intersections for a function of the Wiener process

The article is devoted to the local time of self-intersections for the process $F(w),$ where $F: {\mathbb R}^2\to{\mathbb R}^2$ is smooth function, and $w$ is the standard planar Brownian motion. We present the local time of self-intersections for the process $F(w)$ in terms of a manifold.