Uniqueness and representation theorems for solutions of Kolmogorov-Fokker-Planck equations

Abstract

We consider a class of ultraparabolic operators of the following type I= div(A(x,t)D) +(x,BD)- д, where B is a constant matrix, A(z)= AT(z) > 0. We show that fi u si a solution of Lu = 0 on R\×jO,TI and w(×, 0) = 0, then each of the following conditions: |2(x, t)| can be bounded (in some sense) by e c|x|2, or u > 0, implies u= 0. We use a technique which is well known in the classic parabolic case and which relies on some pointwise estimates of the fundamental solution of L Next, we prove a representation theorem and a Fatou type theorem for non-negative solutions of Lu = 0 in R^ ×10, TI.

Keywords

• ultraparabolic equation
• cauchy problem
• fatou theorem