Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator

Feynman and Feynman-Kac formulas for solutions to the Cauchy problems for the heat equation with respect to complex-valued functions on the product of the real half-line and the p-adic line are obtained. Four theorems are presented that define notation, terminology, and preliminaries, pseudodifferential operators and Chernoff's theorem, representation of solutions to the cauchy problem in terms of generalized Poisson measures on function spaces, representation of solutions to the cauchy problem in terms of an analogue of the Wiener measure. The role of the Laplace operator in these equations is played by the Vladimirov operator and similar formulas can be obtained for Schrödinger-type equations and for the case of a multidimensional space. Such equations may be useful in constructing mathematical models of processes on scales characterized by Planck length and time and phenomenological models in chemistry, continuum mechanics, and psychology.