Functional Laplace operator on a $\mathfrak p$-adic space and Feynman–Kac and Feynman formulas

Homogeneous closed PDO are constructed which are analogous to the powers of (absolute value of) infinite dimensional Laplacian and acting in Banach spaces of complex-valued functions defined on function spaces over a field of $\mathfrak p$-adic numbers. For elements of semigroups, for which these PDOs are generators, Feynman formulas and Feynman–Kac ones are obtained.