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 $$\alpha $$ α -potentials. These operators are denoted and defined in the form $$\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}$$ ( E ϕ , α f ) ( x ) = - F ζ → x - 1 max { 1 , | ϕ ( | | ζ | | p ) | } - α f ^ ( ζ ) , x ∈ Q p n , α ∈ R , where f is a p-adic distribution and $$\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }$$ max { 1 , | ϕ ( | | ζ | | p ) | } - α is the symbol of the operator. We will study some properties of the convolution kernel (denoted as $$K_{\alpha }$$ K α ) of the pseudo-differential operator $$\mathcal {E}_{\varvec{\phi },\alpha }$$ E ϕ , α , $$\alpha \in \mathbb {R}$$ α ∈ R ; and demonstrate that the family $$(K_{\alpha })_{\alpha >0}$$ ( K α ) α > 0 determines a convolution semigroup on $$\mathbb {Q}_{p}^{n}$$ 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.