p-adic Bessel α-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 α α -potentials. These operators are denoted and defined in the form $${aligned} (E_{{φ },α }f)(x)=-F⁻¹ζ → x( [ max \{1,|{φ }(||ζ ||ₚ)|\} ] -α{f}(ζ )) , x∈ {Q}ₚⁿ, \ \ α ∈ R, {aligned}( E ϕ , α f ) ( x ) = - F ζ → x - 1 max 1 , | ϕ ( | | ζ | | p ) | - α f ^ ( ζ ) , x ∈ Q p n , α ∈ R , where f is a p -adic distribution and[ max \{1,|{φ }(||ζ ||ₚ)|\}] -αmax 1 , | ϕ ( | | ζ | | p ) | - α is the symbol of the operator. We will study some properties of the convolution kernel (denoted asKα$$ K α ) of the pseudo-differential operator $$E_{{φ },α }$$ E ϕ , α , $$α ∈ R$$ α ∈ R ; and demonstrate that the family $$(Kα)α >0$$ ( K α ) α > 0 determines a convolution semigroup on $$Qₚⁿ$$ 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.