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Short Communications

On a class degenerative diffusion processes

I. M. Sonin

Moscow

Abstract: 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.

Received: 05.05.1966

 English version: Theory of Probability and its Applications, 1967, 12:3, 490–496

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