On a smooth and nowhere equal to zero distribution density of a stochastic differential equation's solution on manifold

Background. E. Nelson [1-3] introduced derivatives on the average in the works and over time, they began to be studied as a separate class of stochastic differential equations. In the work presented in this paper, the machinery of mean derivatives is applied to finding conditions, under which the probability density functions of solutions of stochastic differential equation on simply connected manifold are $C^\infty$-smooth and nowhere equal to zero. In the paper, Einstein's summation convention with respect to shared upper and lower indices is used. The symbol $\frac{\partial}{\partial x_i}$ denotes both the partial derivative in the chart, and the vector of basis in the tangent space. Materials and methods. The study uses methods of stochastic analysis on manifolds. Results. The sufficient conditions are obtained, under which the probability density function of the solution of stochastic differential equation on simply connected manifold are $C^\infty$-smooth and nowhere equal to zero. Conclusions. Obtained results can be used for investigation of solution existence for stochastic differential equations and inclusions on manifolds.