Abstract
Stochastic processes on totally disconnected topological groups are investigated. In particular, they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudodifferentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including the irreducible ones of these groups. In addition, stochastic processes on general Banach-Lie groups, loop monoids, loop spaces, and path spaces of manifolds on Banach spaces over non-Archimedean local fields are also investigated.