Stochastic variational inequalities on non-convex domains

The objective of this work is to prove in a first step the existence and the uniqueness of a solution of the following multivalued deterministic differential equation:{dx(t)+∂−φ(x(t))(dt)∋dm(t),t>0,x(0)=x0 , where m:R+→Rdm:R+→Rd is a continuous function and ∂−φ∂−φ is the Fréchet subdifferential of a (ρ,γ)(ρ,γ)-semiconvex function φ; the domain of φ can be non-convex, but some regularities of the boundary are required.The continuity of the map m↦x:C([0,T];Rd)→C([0,T];Rd)m↦x:C([0,T];Rd)→C([0,T];Rd) associating to the input function m the solution x of the above equation, as well as tightness criteria allows to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion:{Xt+Kt=ξ+∫0tF(s,Xs)ds+∫0tG(s,Xs)dBs,t≥0,dKt(ω)∈∂−φ(Xt(ω))(dt).