Heat Kernels for Differential Operators with Radical Function Coefficients

Abstract

The first part of the paper deals with finding the heat kernel by probabilistic methods for the 1-dimensional elliptic differential operator $\frac{1}{2} (1+x^2) \frac{d^2}{dx^2} + (\sqrt{1+x^2}+\frac{x}{2}) \frac{d}{dx}$. In the second part we apply the same method to the 2-dimensional operator $\frac{1}{2} (\frac{\partial^2}{\partial x_1^2} + 2\sqrt{1+x_2^2} \frac{\partial ^2}{\partial x_1 \partial x_2} + (1+x_2^2) \frac{\partial^2}{\partial x_2^2}) + \frac{1}{2} x_2 \partial x_2$ and provide explicit formulas for its heat kernel.