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 alpha-potentials. These operators are denoted and defined in the form (E-phi,E-alpha f)(x) = -F-zeta -> x(-1) ([max{1, vertical bar phi(parallel to zeta parallel to(p))vertical bar}](-alpha) (f) over cap (zeta)), x is an element of Q(p)(n), alpha is an element of R, where f is a p-adic distribution and [max{1, vertical bar phi(parallel to zeta parallel to(p))vertical bar}](-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 E-phi,E-alpha, alpha is an element of R; and demonstrate that the family (K-alpha)(alpha>0) determines a convolution semigroup on 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.