New classes of parabolic pseudo-differential equations, Feller semigroups, contraction semigroups and stochastic process on the p-adic numbers

Abstract

Abstract Two types of p -adic pseudo-differential operators (denoted, respectively, by $${\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}$$ T f 1 , f 2 l and $${\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }$$ J f 1 , f 2 α ) are introduced in this article. We will show that the operator $${\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}$$ T f 1 , f 2 l determines certain Feller semigroups and stochastic processes with state space the p -adic numbers. The second type of these operators (defined on a new class of p -adic Sobolev space) are connected with contraction semigroups and parabolic pseudo-differential equations.