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

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel \(\alpha \) $\alpha$ -potentials. These operators are denoted and defined in the form \(\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}\) $$\begin{array}{c}\hfill ({\mathcal{E}}_{\mathit{\varphi },\alpha }f)(x)=-{\mathcal{F}}_{\zeta \to x}^{-1}\left({\left[{max\{1,|\mathit{\varphi }(||\zeta ||}_p,)|\}\right]}^{-\alpha },\widehat{f},(\zeta )\right),x\in {\mathbb{Q}}_p^n,\alpha \in \mathbb{R},\end{array}$$ where f is a p-adic distribution and \(\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }\) ${\left[{max\{1,|\mathit{\varphi }(||\zeta ||}_p,)|\}\right]}^{-\alpha }$ is the symbol of the operator. We will study some properties of the convolution kernel (denoted as \(K_{\alpha }\) $K_{\alpha }$ ) of the pseudo-differential operator \(\mathcal {E}_{\varvec{\phi },\alpha }\) ${\mathcal{E}}_{\mathit{\varphi },\alpha }$ , \(\alpha \in \mathbb {R}\) $\alpha \in \mathbb{R}$ ; and demonstrate that the family \((K_{\alpha })_{\alpha >0}\) ${(K_{\alpha })}_{\alpha >0}$ determines a convolution semigroup on \(\mathbb {Q}_{p}^{n}\) ${\mathbb{Q}}_p^n$ . Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.