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

L. P. Kuptsov

Matematičeskie zametki, Tome 15 (1974) no. 3, pp. 479-489.

Résumé

For an equation of the form $$ \frac{\partial u}{\partial t}-\sum_{ij=1}^n\alpha^{ij}\frac{\partial^2u}{\partial x^i\partial x^j}+\sum_{ij=1}^n\beta_j^ix^i\frac{\partial u}{\partial x^i}=0,\quad x\in R^n,\quad t\in R^1, $$ where $\alpha=(\alpha^{ij})$ is a constant nonnegative matrix and $\beta=(\beta^i_j)$ 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 $u(x_0,t_0)$ 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 $(x_0,t_0)$; finally, we prove a parabolic maximum principle.