On a~class degenerative diffusion processes

I. M. Sonin

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 540-547

Résumé

It is proved that the degenerative diffusion processes with the characteristic operators reducing after a change of variables to the form \begin{gather*} Lu=\frac12\sum_{ij=1}^na_{ij}(x_1^1,\dots,x_1^n,x_2^1,\dots,x_2^n,\dots,x_N^1,\dots,x_N^n,t)\frac{\partial^2u}{\partial x_1^i\partial x_1^j}+ \\ +\sum_{i=1}^na_i(x_1^1,\dots,x_N^n,t)\frac{\partial u}{\partial x_1^i}+\sum_{i=1}^nx_1^i\frac{\partial u}{\partial x_2^i}+\sum_{i=1}^nx_2^i\frac{\partial u}{\partial x_3^i}+\dots \\ \dots+\sum_{i=1}^nx_{N-1}^i\frac{\partial u}{\partial x_N^i}+a(x_1^1,\dots,x_N^n,t)u \end{gather*} have smooth densities. The proof is carried out by constructing the fundamental solution of the corresponding parabolic equation. For this the classical method of Lévy with some modifications is used.