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 L2-infinitesimal generator L of a general
symmetric Markov process. An illuminating concrete example for L
is ΔD-(-Δ)sD,
where D is a bounded Euclidean domain
in Rd, s ε ]0, 1[,
&DeltaD is the Laplace operator in
D with zero Dirichlet boundary condition and -(-Δ)sD 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 Rd and an
independent symmetric (2s)-stable process in Rd 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
