BV Functions and Distorted Ornstein Uhlenbeck Processes over the Abstract Wiener Space

Abstract

Let (E, H, μ) be an abstract Wiener space and H be the class of functions ρ∈L1+(E; μ) satisfying the ray Hamza condition in every direction ℓ∈E*. For ρ∈H, the closure (Eρ, Fρ) of the symmetric form Eρ(u, v)=∫E〈∇u(z), ∇v(z)〉Hρ(z)μ(dz), u, v∈FC1b, is a quasi-regular Dirichlet form on L2(F, ρdμ) (F=Supp[ρμ]), yielding an associated diffusion Mρ=(Xt, Pz) on F called a distorted Ornstein Uhlenbeck process. A function ρ on E is called a BV function (ρ∈BV(E) in notation) if ρ∈∪p>1Lp(E; μ) and V(ρ)=supG∈(FC1b)E*, ‖G‖H(z)⩽1∫E∇*G(z)ρ(z)μ(dz) is finite. For ρ∈H∩BV(E), there exist a positive finite measure ‖Dρ‖ on F and a weakly measurable function σρ:F→H such that ‖σρ(z)‖H=1‖Dρ‖-a.e. and ∫F∇*G(z)×ρ(z)μ(dz)=∫F〈G(z), σρ(z)〉H‖Dρ‖(dz), ∀G∈(FC1b)E*. Further, the sample path of Mρ admits an expression as a sum of E-valued CAFs, Xt−X0=Wt−12∫t0Xsds+12∫t0σρ(Xs)dL‖Dρ‖s, where Wt is an E-valued Brownian motion and L‖Dρ‖t is a PCAF of Mρ with Revuz measure ‖Dρ‖. A measurable set Γ⊂E is called Caccioppoli if IΓ∈BV(E). In this case, the support of the measure ‖DIΓ‖ is concentrated in ∂Γ and the above equations reduce to the Gauss formula and the Skorohod equation for the modified reflecting Ornstein Uhlenbeck process, respectively. A related coarea formula is also presented.