p-adic Bessel $$\alpha $$ -potentials and some of their applications

Abstract

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel \(\alpha \)-potentials. These operators are denoted and defined in the form

$$\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}$$

where f is a p-adic distribution and \(\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }\) is the symbol of the operator. We will study some properties of the convolution kernel (denoted as \(K_{\alpha }\)) of the pseudo-differential operator \(\mathcal {E}_{\varvec{\phi },\alpha }\), \(\alpha \in \mathbb {R}\); and demonstrate that the family \((K_{\alpha })_{\alpha >0}\) determines a convolution semigroup on \(\mathbb {Q}_{p}^{n}\). Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.