Criteria for Coordinate-Homogeneity for Continuous Markov Processes

Let us consider a Markov process with an infinitesimal operator
\begin{equation} A_t=A^i(t,x)\frac{\partial}{\partial x^i } +B^{ij}(t,x)\frac{\partial^2}{\partial x^i\partial x^j} \label{eq*} \tag{*} \end{equation}
where $x=(x^1,\dots,x^n )$ is a point in Riemann space $V_n$ with a metric $g_{ij}(t,x)$.
The necessary and sufficient conditions for the existence of transformation $x^{i'}=x^{i'}(t,x^1,\dots,x^n)$ which transforms the operator \eqref{eq*} into the well-known operator
$$A_t^0=B^{i'j'}(t)\frac{\partial^2}{\partial x^{i'} \partial x^{j'}}$$
are given.
At the end of the paper an example is, given from statistics, in which these conditions are applied for establishing the density of the probabilities $f(t,x,\tau,\xi)$ of a certain Markov process.