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

Abstract

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.