On the Distributional Solution of the Inverse Problem Induced by the Heat Kernel Method

As the first constrained step in the investigation of the Fokker-Planck equation as a source of the probability density function of the position (or velocity) of a Brownian particle, we studied the connection between the inverse problem for the diffusion equation and the kernel methods developed in the area of statistical learning theory and its applications. The initial condition (an initial probability density function) is not known, and the solution of the diffusion equation (an output function) is presumed to be known at a finite, sufficiently large empirical set of points {(xi, yi)}—a sample. With the use of Tikhonov’s regularizing method, we reduced the problem to the minimization of the empirical functional in the reproducing kernel Hilbert space H :