158 research outputs found
Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms
Let be a locally compact separable metric space and be a positive
Radon measure on it. Given a nonnegative function defined on
off the diagonal whose anti-symmetric part is assumed to be less singular than
the symmetric part, we construct an associated regular lower bounded
semi-Dirichlet form on producing a Hunt process on
whose jump behaviours are governed by . For an arbitrary open subset
, we also construct a Hunt process on in an analogous
manner. When is relatively compact, we show that is censored in
the sense that it admits no killing inside and killed only when the path
approaches to the boundary. When is a -dimensional Euclidean space and
is the Lebesgue measure, a typical example of is the stable-like
process that will be also identified with the solution of a martingale problem
up to an -polar set of starting points. Approachability to the boundary
in finite time of its censored process on a bounded open
subset will be examined in terms of the polarity of for the
symmetric stable processes with indices that bound the variable exponent
.Comment: Published in at http://dx.doi.org/10.1214/10-AOP633 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On unique extension of time changed reflecting Brownian motions
Let be an unbounded domain in \RR^d with . We show that if
contains an unbounded uniform domain, then the symmetric reflecting Brownian
motion (RBM) on is transient. Next assume that RBM on
is transient and let be its time change by Revuz measure
for a strictly positive continuous integrable function
on . We further show that if there is some so that
is an unbounded uniform domain, then
admits one and only one symmetric diffusion that genuinely extends it and
admits no killings. In other words, in this case (or equivalently, ) has
a unique Martin boundary point at infinity.Comment: To appear in Ann. Inst. Henri Poincare Probab. Statis
A BRIEF SURVEY ON STOCHASTIC CALCULUS IN MARKOV PROCESSES (Stochastic Analysis of Jump Processes and Related Topics)
Time changes of symmetric diffusions and Feller measures
We extend the classical Douglas integral, which expresses the Dirichlet
integral of a harmonic function on the unit disk in terms of its value on
boundary, to the case of conservative symmetric diffusion in terms of Feller
measure, by using the approach of time change of Markov processes.Comment: Published at http://dx.doi.org/10.1214/009117904000000649 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Traces of symmetric Markov processes and their characterizations
Time change is one of the most basic and very useful transformations for
Markov processes. The time changed process can also be regarded as the trace of
the original process on the support of the Revuz measure used in the time
change. In this paper we give a complete characterization of time changed
processes of an arbitrary symmetric Markov process, in terms of the
Beurling--Deny decomposition of their associated Dirichlet forms and of Feller
measures of the process. In particular, we determine the jumping and killing
measure (or, equivalently, the L\'{e}vy system) for the time-changed process.
We further discuss when the trace Dirichlet form for the time changed process
can be characterized as the space of finite Douglas integrals defined by Feller
measures. Finally, we give a probabilistic characterization of Feller measures
in terms of the excursions of the base process.Comment: Published at http://dx.doi.org/10.1214/009117905000000657 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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