Continuous Time p-Adic Random Walks and Their Path Integrals

The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in R to the groups p-nZp/pnZp give rise to probability measures on the space of Skorokhod paths on these finite groups. These measures induce probability measures on the Skorokhod space of Qp-valued paths that almost surely take values on finite grids. We study the convergence of these induced measures to their continuum limit, a p-adic Brownian motion. We additionally prove a Feynman-Kac formula for the matrix-valued propagator associated to a Schrodinger type operator acting on complex vector-valued functions on p-nZp/pnZp where the potential is a Hermitian matrix-valued multiplication operator.