Non-Archimedean Parabolic-Type Equations with Variable Coefficients

Abstract

This chapter is devoted to the study of a class of parabolic-type equations with variable coefficients, which generalizes the parabolic-type equations attached to the \(\boldsymbol{W}\) operators. The theory of these equations was initiated by Kochubei in [81], see also [80, Chapter 4]. He studied, in dimension one, p-adic parabolic-type equations with variable coefficients and their associated Markov processes. Later Rodriguez-Vega in [97] extended some of the results of Kochubei to the n-dimensional case by using the Taibleson operator instead of the Vladimirov operator. Building up on [25] and [80], Chacon-Cortes and Zuniga-Galindo developed a theory of p-adic parabolic-type equations with variable coefficients attached to operators \(\boldsymbol{W}\), which contains the one-dimensional p-adic heat equation of [111], the equations studied by Kochubei in [80], and the equations studied by Rodriguez-Vega in [97]. In this chapter, we establish the existence and uniqueness of solutions for the Cauchy problem for these equations. We show that the fundamental solutions of these equations are transition density functions of Markov processes, and finally, we study the well-possednes of the Cauchy problem.