Weak Solutions of Stochastic Differential Equations over the Field of p-Adic Numbers

TL;DR

This paper advances the theory of stochastic differential equations over p-adic numbers by refining stochastic integral definitions, establishing continuity of local times, and providing conditions for weak solutions driven by p-adic stable processes.

Contribution

It introduces an improved stochastic integral definition on p-adic fields and derives new existence conditions for weak solutions of p-adic SDEs with measurable coefficients.

Findings

Established joint continuity of local time for p-adic stable processes

Provided sufficient conditions for weak solutions of p-adic SDEs

Refined the stochastic integral framework for p-adic stochastic calculus

Abstract

Study of stochastic differential equations on the field of p-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the p-adic case, similar to the theory of ordinary stochastic integral with respect to Levy processes on the Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for p-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the p-adic stable process, with a Borel measurable coefficient.