S-harmonic Function and RieszDecomposition Theorem of Superprocesses

RecentlyItheremanyprogressinthetheoryofsuperprocesses(seeLZ'4--gi).Inthispaper,wewillestablishtheRicszdecompositiontheoremofexcessivefunctionsofaclassofsuperprocesses,andgeneralizetheclassicalRicszdecompositiontheoremofHuntprocess.FortheconstructionofsupcrprocesscswecanseeL2,4,sj.ConsiderthefollowingrealfunctionPX,9(x,z)=a(x)z b(x)zZ J,(e":--I "z)n.(du),(1)Pcowheren.(du)isakernelfromR'lto(0,~)and'a(x),b(x)andA(x)~I(uZAu)n.(du)areJ0positiveboundedBorclfunctions.AssumingthatE~{Et,11.}is…     

Central Limit Theorems for Supercritical Superprocesses with Immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.     

L log L criterion for a class of multitype superdiffusions with non-local branching mechanisms

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with nonlocal branching mechanisms under a martingale change of measure. As an application of this decomposition,we obtain a necessary and sufficient condition(called the L log L criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results obtained in Kyprianou et al.(2012),Kyprianou and Murillo-Salas(2013) and Liu et al.(2009) for superprocesses with purely local branching mechanisms and in Kyprianou and Palau(2018) for super Markov chains.     

Backbone Decomposition of Multitype Superprocesses

Journal of Theoretical Probability - In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are...     

A Generalization of Lazard's Elimination Theorem

Abstract

Using the classical Lazard's elimination theorem, we obtain a decomposition theorem for Lie algebras defined by generators and relations of a certain type.     

无限状态空间上的测度和鞅

建立无限状态空间上的Radon－Nikodym定理和Riesz表示定理，引进测度值鞅并应用于超过程．     

The prolific backbone for supercritical superprocesses

We develop an idea of Evans and O’Connell (1994) [13], Engländer and Pinsky (1999) [10] and Duquesne and Winkel (2007) [4] by giving a pathwise construction of the so-called ‘backbone’ decomposition for supercritical superprocesses. Our results also complement a related result for critical (1+β)-superprocesses given in Etheridge and Williams (2003) [11]. Our approach relies heavily on the use of Dynkin-Kuznetsov N-measures.     

Markov snakes and superprocesses

Summary We suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v–v ². We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.Partially supported by National Science Foundation Grant DMS-9301315 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University     

A Reflection Principle and an Orthogonal Decomposition in Clifford Algebras

In this article we investigate spaces of functions defined in a domain Ω ? R with values in the Clifford algebra R _n. According to an inner product an orthogonal decomposition is proved. By this decomposition, we obtain a subspace A ₂(Ω) of regular functions with respect to the Dirac operator. In the orthogonal complement the Dirac equation with homogeneous boundary values is solvable. The decomposition can be proved in two ways: by a reflection principle and by Sobolev's regularity theorem. It will turn out, that the existence of the orthogonal decomposition and Sobolev's theorem is equivalent. So also a reflection principle will be proved, which describes the jump behavior of a Cauchy type integral. By the reflection principle, a countable dense subset of A ₂(Ω) can be obtained. Further considerations lead to a minimal generating system, by which the Bergman kernel function can be obtained. As a conclusion we also obtain Runge's theorem.     

Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [https://doi.org/10.1007/s00013-018-1244-3]. This article is the final peer reviewed version.

     

<Emphasis Type="Italic">L</Emphasis>log <Emphasis Type="Italic">L</Emphasis> Condition for Supercritical Branching Hunt Processes

In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum Llog L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrscheinlichkeitstheor. Verw. Geb. 36:195–212, 1976) and Hering (Branching Processes, pp. 177–217, 1978) for branching diffusions.     

Superprocesses arising from interactive stochastic flows

In this paper, we give a unified construction for superprocesses with dependent spatial motion constructed by Dawson, Li, Wang and superprocesses of stochastic flows constructed by Ma and Xiang. Furthermore, we also give some examples and rescaled limits of the new class of superprocesses.     

Law of large numbers for supercritical superprocesses with non-local branching

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.     

Decomposition theorem for the cd-index of Gorenstein<Superscript>*</Superscript> posets

We prove a decomposition theorem for the cd-index of a Gorenstein^* poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.     

New Linking Theorem and Sign-Changing Solutions

Abstract

In this paper, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A general liking type theorem is given with the location of the critical point produced specified in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities, and may or may not be resonant with respect to Fu?ík spectrum.     

The complete characterization of a general class of superprocesses

In an important paper, [7], Dynkin, Kuznetsov and Skorohod showed that, under mild conditions, the log-Laplace functional of every branching measure-valued process is the solution of an evolution equation determined by three parameters, ξQ and ℓ. This paper essentially deals with the converse of this result. We consider a general class ℋ of BMV (subject to some mild conditions). First, we derive from [7] that every process X∈ℋ is a (ξΦ, K)-superprocesses, where the triples (ξ, Φ, K) are subject to some conditions ANS1-ANS3. Conversely, we show that, for each of these triples satisfying ANS1-ANS3, there exists a version X of the (ξ, Φ, K)-superprocess which belongs to ℋ. Consequently, ANS1-ANS3 fully characterizes ℋ, and subject to mild conditions, BMV and superprocesses are equivalent concepts. This requires to prove a general existence theorem for superprocesses, the existence of a regular version of these processes and that for processes in ℋ, branching characteristics Q and ℓ are continuous. Received: 11 April 1996 / Revised version: 25 May 1999     

随机流的超过程的状态分解

In this paper, we reconstruct the superprocesses of stochastic flows by martingale method, and prove that if and only if the infinitesimal particles never hit each other, then atomic part and diffuse part of this kind of superprocesses will be also superprocesses of stochastic flows. This result completely answers the open problem in [1].     

Local boundary controllability for the semilinear plate equation

weijiu_liu@uow.edu.

The problem of local controllability for the semilinear plate equation with Dirichlet boundary conditions is studied. By making use of Schauder's fixed point theorem and the inverse function theorem, we prove that this system is locally controllable under a super-linear assumption on the nonlinearity, that is, the initial states in a small neighborhood of 0 in a certain function space can be driven to rest by Dirichlet boundary controls. Our super-linear assumption includes the critical exponent.     

Regularities of multifractal measures

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in ” ^d . This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.     

一般介质下的超过程的研究

本文考虑一般介质作用下的分枝粒子系统，进而建立了一类比较广泛分枝机制下的超布朗运动．同时本文研究了这类超过程的轨道性质．亦即证明了其轨道连续性和局部绝对连续性