Green function estimates for subordinate Brownian motions : stable and beyond

A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$\phi(\lambda)=\log(1+\lambda^{\alpha/2}) (0 \alpha)$$ and $$\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m) (00, d >2) .$$Comment: We have weaken the condition (A5). References are update

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in bounded kappa-fat open set; if u is a positive harmonic function with respect to X in a bounded kappa-fat open set D and h is a positive harmonic function in D vanishing on D^c, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h.Comment: 24pages. To appear in Stochastic Processes and their Applications (http://www.journals.elsevier.com/stochastic-processes-and-their-applications

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 $Y$ is a diffusion process whose generator can be formally written as $L+\mu\cdot\nabla-\nu$ with Dirichlet boundary conditions, where $L$ is a uniformly elliptic second-order differential operator and $\mu=(\mu^1,...,\mu^d)$ is such that each component $\mu^i$, $i=1,...,d$, is a signed measure belonging to the Kato class $\mathbf{K}_{d,1}$ and $\nu$ is a (nonnegative) measure belonging to the Kato class $\mathbf{K}_{d,2}$. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for $Y$. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion $Y^D$ with measure-valued drift and potential when $D$ is one of the following types of bounded domains: twisted H\"{o}lder domains of order $\alpha\in(1/3,1]$, uniformly H\"{o}lder domains of order $\alpha\in(0,2)$ 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 $Y^D$ 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