Equations for probability distributions of local time on surface for diffusion processes and control problems

The paper studies local time on surface for general $n$-dimensional diffusion process. Analogs of Kolmogorov–Fokker–Planck equations for the characteristic function and probability distributions of local time are derived and investigated for a wide class of $(n-1)$-dimensional surfaces. A general explicit formula which is an analog of the Tanaka formula is presented. Optimal control problems with functionals which depend on local time are investigated.