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Time-inhomogeneous diffusions corresponding to symmetric divergence form operators

Rozkosz A.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

231--252

We consider a time-inhomogeneous Markov family (X, P3,x) corresponding to a symmetric uniformly elliptic divergence form operator. We show that for any φ in the Sobolev space W1p∩W12 with p = 2 if d = 1 and p > d > if d > 1 the additive functional Xφ = (φ (X1)- φ (Xx); 0 ≤ s < t} admits a unique strict decomposition into a martingale additive functional of finite energy and a continuous additive functional of zero energy. Moreover, we give a stochastic representation of the zero energy part and show that in case the diffusion coefficient is regular in time the functional Xφ is a Dirichlet process for each starting point (s, x). The paper contains also rectifications of incorrectly presented or incorrectly proved statements of our earlier paper [14].