A method for solving the $p$-adic Kolmogorov–Feller equation for an ultrametric random walk in an axially symmetric external field

A method for solving the Kolmogorov–Feller equation for an ultrametric random walk in an axially symmetric external field is considered. The transition function $w(y\mid x)$, $x,y\in\mathbb Q_p$, of the process under consideration is nonsymmetric and depends on the norm of $p$-adic arguments. It is proved for the transition functions of the form $w(y\mid x)=\rho(|x-y|_p)\varphi(|x|_p)$ that solving the $p$-adic Kolmogorov–Feller equation for a random walk in a $p$-adic ball of radius $p^R$ reduces to solving a system of $R+1$ ordinary differential equations.