Generalized functions on a Non-Archimedean superspace

A theory is developed for superanalytic generalized functions on a superspace over a non-Archimedean Banach superalgebra with trivial annihilator of the odd part. A Gaussian distribution and the Volkenborn distribution are introduced on the non-Archimedean superspace. Existence and uniqueness theorems are proved for the Cauchy problem for linear differential equations with variable coefficients. The Cauchy problem for non-Archimedean superdiffusion, the Schrödinger equation, and the Schrodinger equation for supersymmetric quantum mechanics on a non-Archimedean Riemann surface are considered as applications.