Abstract:
We study the uniqueness classes of a generalized solution of the Cauchy problem
\begin{equation}
u_t=\frac12\sum_{i,j=1}^na_{ij}(x)u_{x_ix_j}+\sum_{i=1}^na_i(x)u_{x_i}\equiv Lu,\quad
u(0,x)=\varphi(x),\quad x\in\mathbf R^n,\ t\in[0,T],
\end{equation}
when the matrix $\bigl\{a_{ij}(x)\bigr\}$ is degenerate. A generalized solution is introduced with the help of an infinitesimal operator of a Markov process connected with the operator in (1). In the proof of the theorems we use probabilistic characteristics of this process.
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