Mean value theorem and a maximum principle for Kolmogorov's equation

Abstract

For an equation of the form

$$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$

where α=(α^ij) is a constant nonnegative matrix andΒ=(Β ⁱ_i ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.