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

Two types of p-adic pseudo-differential operators (denoted, respectively, by Tf1,f2l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}$$\end{document} and Jf1,f2α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }$$\end{document}) are introduced in this article. We will show that the operator Tf1,f2l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}$$\end{document} 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.