Solvability of Langevin differential inclusions with set-valued diffusion terms on Riemannian manifolds

Abstract

The existence of weak solution is proved for a Langevin type second-order stochastic differential inclusion on a complete Riemannian manifold, having both drift and diffusion terms set-valued. The construction of solution involves integral operators with Riemannian parallel translation and a special sequence of continuous ϵ-approximations for an upper semicontinuous set-valued mapping with convex bounded closed values, that is proved to converge point-wise to a Borel measurable selection.

Keywords:

• Stochastic differential inclusions
• Langevin equation
• Riemannian manifolds
• ϵ-approximations

Keywords:

• 2000 Mathematics Subject Classifications: 58C06
• 58J65
• 82C31
• 93E03