Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Probabilistic approach to the Dirichlet problem of second order elliptic PDE

In this paper we provide a probabilistic approach to the following Dirichlet Problem{(∑x~4(α~(ij) x~j) ∑b~ix~i ξ)u=0, iD u=g, on D,without assuming that the eigenvalues of the operator∑x~i(α~(ij)x~j) ∑b~ix~i ξwith Dirichlet boundary conditions are all strictly negative. The results of this paper generalizedthose of Ma.     

The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that X = {X_t, t≥0; P_μ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ_0 of the generator of the mean semigroup of X. Then Mt := e~(-λ_0t)〈φ0,X_t〉 is a positive martingale. Let M_∞ be the limit of M_t. It is known(see Liu et al.(2009)) that M_∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γ_t on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,lim/t→∞γ_t〈φ0,X_t〉=W, a.s.-P_μ~.We also give the almost sure limit of γ_t〈f, X_t〉for a class of general test functions f.     

氧杂环丁烷系列含能粘合剂的合成研究进展

新型含能粘合剂的出现进一步提高了固体推进剂的能量和性能。其中,在3-位取代含能基团实现氧杂环丁烷的含能化并以此类单体合成的含能粘合剂,由于其具有相对分子量和官能度可控性好、多分散性和玻璃化转变温度低而受到重视,引起广泛关注。     

Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

Scale Invariant Boundary Harnack Principle at Infinity for Feller Processes

In this paper we prove a uniform and scale invariant boundary Harnack principle at infinity for a large class of purely discontinuous Feller processes in metric measurespaces.     

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L+μ⋅∇+ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1,1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.