Extended symmetry analysis of (1+2)-dimensional Fokker-Planck equation

Abstract

We carry out the extended symmetry analysis of an ultraparabolic degenerate Fokker-Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker-Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete and essential point symmetry groups of the Kolmogorov equation using the direct method and analyze their structure. After listing inequivalent one- and two-dimensional subalgebras of the essential Lie invariance algebra of this equation, we exhaustively classify its Lie reductions. As a result, we construct wide families of exact solutions of the Kolmogorov equation, and three of them are parameterized by single arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Kolmogorov equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.