Hypoellipticity of Ito’s Second Order Parabolic Equations

Abstract

0.1. Smoothness of solutions of deterministic parabolic equations increases as the smoothness assumptions on their coefficients increase. This is a typical feature of parabolic equations. Moreover, under wide assumptions, the smoothness of solutions for t>0 depends only on the smoothness of coefficients and does not depend on the smoothness of the initial functions. This is important for example, in the study of the fundamental solution of a parabolic equation, since we can consider this solution as a solution of the corresponding Cauchy problem where the initial function is the Dirac-function. Hypoellipticity is a particular case of the growth of smoothness property mentioned above.