On Dirichlet Processes Associated with Second Order Divergence Form Operators |
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Authors: | Rozkosz Andrzej |
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Institution: | (1) Faculty of Mathematics and Informatics, Nicholas Copernicus University, ul. Chopina 12/18, 87–100 Toru, Poland |
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Abstract: | Let
be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space
the process (X) admits under P
x a decomposition into a martingale additive functional (AF) M
and a continuous AF A
of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every
if is continuous, d=1 and
or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A
equals zero if q=2 and that the decomposition of (X) into the martingale AF M
and the AF of zero energy A
is strict if
for some q>d. Moreover, our decomposition provides a probabilistic representation of A
. |
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Keywords: | divergence form operator diffusion Dirichlet process additive functional |
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