BV functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space

Let (E,H,mu) be an abstract Wiener space and H be the class of functions p epsilon L-+(1) (E, mu) satisfying the ray Hamza condition in every direction l epsilon E*. For p epsilon H, the closure (delta(p), F-p) of the symmetric form delta(p)(u,v) = integral(E) [del u(z), del v(z)](H) p(z) mu(dz), u, v epsilon FC, is a quasi-regular Dirichlet form on L-2(F, p d mu) (F = Supp[p mu]), yielding an association diffusion M-p = (X-1, P-2) on F called a distorted Ornstein Uhlenbeck process. A function p on E is called a BV function (p epsilon BV(E) in notation) if p epsilon boolean ORp>1 L-p(E; mu) and V(p) = sup(G epsilon(FCb1)E..\\G\\n(z)less than or equal to 1) integral(E) del*G(z) p(Z) mu(dz) is finite. For p epsilon H boolean AND BV(E), there exist a positive finite measure \\D-p\\ on F and a weakly measurable function sigma(p): F --> H such that \\sigma(p)(z)\\(H) = 1\\Dp\\-a.e. and integral(F) del*G(z) x p(z)mu(dz) = integral(F) [G(Z), sigma(p)(z)] (H) \\Dp\\(dz), For All G epsilon (FCh1)(E)*. Further, the sample path of M-p admits an expression as a sum of E-valued CAFs, X-t - X-0 = W-t - 1/2 integral(0)(1) X-s ds + 1/2 integral(0)(t) sigma(p)(X-s) dL(3)(\\Dp\\), where W-t is an E-valued Browmian motion and L-t(\\Dp\\) is a PCAF of M-p with Revuz measure \\Dp\\. A measureable set Gamma subset of E is called Caccioppoli if I-Gamma epsilon BV(E). In this case, the support of the measure \\DIp\\ is contcentrated in partial derivative Gamma and the above equations reduce to the Gauss formula and the Skorohod equation for the modified reflecting Orstein Uhlenbeck process, respectively. A related coarea formula is also presented. (C) 2000 Academic Press.