Perturbation of Symmetric Markov Processes
### Perturbation of Symmetric Markov Processes

(To appear in * Probability Theory and Related Fields*;
Electronic copy now available here.)

We present a path-space integral representation of the semigroup
associated with the quadratic form obtained by a lower order
perturbation of the L^{2}-infinitesimal generator *L* of a general
symmetric Markov process. An illuminating concrete example for *L*
is *Δ*_{D}-(-Δ)^{s}_{D},
where *D* is a bounded Euclidean domain
in *R*^{d}, *s* ε ]0, 1[,
&Delta_{D} is the Laplace operator in
*D* with zero Dirichlet boundary condition and *-(-Δ)*^{s}_{D} is
the fractional Laplacian in *D* with zero exterior
condition. The strong Markov process corresponding to *L* is a
Lévy process that is the sum of Brownian motion in *R*^{d} and an
independent symmetric (2*s*)-stable process in *R*^{d} killed upon
exiting the domain *D*. This probabilistic representation is a
combination of Feynman-Kac and Girsanov formulas. Crucial to the
development is the use of an extension of Nakao's stochastic integral
for zero-energy additive functionals and the associated Itô
formula, both of which were recently developed in a
companion paper
of the authors.

A hard copy of this manuscript is available from the
second-named author upon request.

The manuscript can also be downloaded in
dvi form (154K)
and pdf form (293K).
(Version of February 22, 2007)

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August 21, 2006