Semi-linear Cauchy problem and Markov process associated with a $p$-adic non-local ultradiffusion operator

Abstract

This work is dedicated to study the pseudodifferential operator (𝐷𝛼𝑑1,𝑑2𝜑)(𝑥)=−∫ℚ𝑛𝑝−𝛼𝑑1,𝑑2(𝑦)[𝜑(𝑥+𝑦)−𝜑(𝑥)]𝑑𝑛𝑦, which can be seen as a generalization of Taibleson operator; here 𝛼𝑑1,𝑑2(𝑥)=max{‖𝑥‖𝑑1𝑝, ‖𝑥‖𝑑2𝑝}𝛼. We show that semi-linear Cauchy problem is well-posed in 𝔐𝜆 [a Banach space containing functions that do not belong to 𝐿1(ℚ𝑛𝑝)], assuming that semi-linear part f is a Lipschitz function. We associate to the corresponding homogeneous problem a Markov process, which is indeed a Feller process. Finally, we study the first passage time problem.

Oscar F. Casas

Associate Professor