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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 22, Fasc. 2,
pages 231 - 252
 

TIME-INHOMOGENEOUS DIFFUSIONS CORRESPONDING TO SYMMETRIC DIVERGENCE FORM OPERATORS

Andrzej Rozkosz

Abstract: We consider a time-inhomogeneous Markov family (X, P  )
     s,x corresponding to a symmetric uniformly elliptic divergence form operator. We show that for any f in the Sobolev space W 1 /~\  W 1
 p    2  with p = 2 if d = 1 and p > d if d > 1 the additive functional Xf  = (f(X )- f(X );0 < s < t)
          t       s 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 Xf  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].

2000 AMS Mathematics Subject Classification: 60J60, 60JS7.

Key words and phrases: -

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