408 research outputs found
Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials
Recently, in [Preprint (2006)], we extended the concept of intrinsic
ultracontractivity to nonsymmetric semigroups. In this paper, we study the
intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued
drifts and measure-valued potentials in bounded domains. Our process is a
diffusion process whose generator can be formally written as
with Dirichlet boundary conditions, where is a
uniformly elliptic second-order differential operator and
is such that each component , , is a
signed measure belonging to the Kato class and is a
(nonnegative) measure belonging to the Kato class . We show
that scale-invariant parabolic and elliptic Harnack inequalities are valid for
. In this paper, we prove the parabolic boundary Harnack principle and the
intrinsic ultracontractivity for the killed diffusion with measure-valued
drift and potential when is one of the following types of bounded domains:
twisted H\"{o}lder domains of order , uniformly H\"{o}lder
domains of order and domains which can be locally represented
as the region above the graph of a function. This extends the results in [J.
Funct. Anal. 100 (1991) 181--206] and [Probab. Theory Related Fields 91 (1992)
405--443]. As a consequence of the intrinsic ultracontractivity, we get that
the supremum of the expected conditional lifetimes of is finite.Comment: Published in at http://dx.doi.org/10.1214/07-AOP381 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains
For a symmetric -stable process on \RR^n with ,
and a domain D \subset \RR^n, let be the infinitesimal
generator of the subprocess of killed upon leaving . For a Kato class
function , it is shown that is intrinsic ultracontractive on a
H\"older domain of order 0. This is then used to establish the conditional
gauge theorem for on bounded Lipschitz domains in \RR^n. It is also shown
that the conditional lifetimes for symmetric stable process in a H\"older
domain of order 0 are uniformly bounded
Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes
Martin boundaries and integral representations of positive functions which
are harmonic in a bounded domain with respect to Brownian motion are well
understood. Unlike the Brownian case, there are two different kinds of
harmonicity with respect to a discontinuous symmetric stable process. One kind
are functions harmonic in with respect to the whole process , and the
other are functions harmonic in with respect to the process killed
upon leaving . In this paper we show that for bounded Lipschitz domains, the
Martin boundary with respect to the killed stable process can be
identified with the Euclidean boundary. We further give integral
representations for both kinds of positive harmonic functions. Also given is
the conditional gauge theorem conditioned according to Martin kernels and the
limiting behaviors of the -conditional stable process, where is a
positive harmonic function of . In the case when is a bounded domain, sharp estimate on the Martin kernel of is obtained
Two-sided Green function estimates for killed subordinate Brownian motions
A subordinate Brownian motion is a L\'evy process which can be obtained by
replacing the time of the Brownian motion by an independent subordinator. The
infinitesimal generator of a subordinate Brownian motion is ,
where is the Laplace exponent of the subordinator. In this paper, we
consider a large class of subordinate Brownian motions without diffusion
component and with comparable to a regularly varying function at
infinity. This class of processes includes symmetric stable processes,
relativistic stable processes, sums of independent symmetric stable processes,
sums of independent relativistic stable processes, and much more. We give sharp
two-sided estimates on the Green functions of these subordinate Brownian
motions in any bounded -fat open set . When is a bounded
open set, we establish an explicit form of the estimates in terms of
the distance to the boundary. As a consequence of such sharp Green function
estimates, we obtain a boundary Harnack principle in open sets with
explicit rate of decay.Comment: 33 page
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