New classes of p-adic pseudo-differential operators with negative definite symbols and their applications

Abstract

This paper introduces new classes of p-adic operators representing m-dissipative pseudo-differential operators with negative definite symbols under certain conditions. We will study new types of semilinear problems and martingale problems associated with these operators, and we will prove that these pseudo-differential operators are the infinitesimal generators of strongly continuous contraction semigroups on \(L^2({\mathbb {Q}}_p^n)\). Also, this article introduces new families of measures, resolvent of measures, positive definite measures, Feller semigroups, and Markov processes.