Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain

Let D be a bounded homogeneous domain in ℂ ⁿ . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ^∞(D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ^∞(D) to the Bloch space of D are the operators whose symbol is bounded.     

Properties of Super-Poisson Processes and Super-Random Walks with Spatially Dependent Branching Rates

The global supports of super-Poisson processes and super-random walks with a branching mechanism ψ（z）=z^2 and constant branching rate are known to be noncompact. It turns out that, for any spatially dependent branching rate, this property remains true. However, the asymptotic extinction property for these two kinds of superprocesses depends on the decay rate of the branching-rate function at infinity.     

A weighted occupation time for a class of measured-valued branching processes

Summary A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on R ^d. Considered as a process in its own right, the first and second order asymptotics are found as time t. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.Research supported in part by Natural Sciences and Engineering Research Council of Canada     

On States of Total Weighted Occupation Times for Superdiffusions

Suppose X is a superdiffusion in R ^d with general branching mechanism ψ, and Y _{r D} denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that Y _{r D} has absolutely continuous states. And for particular ψ₍ z) = z ^1+β, 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, Y _{r D} is absolutely continuous with respect to the Lebesgue measure in , whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of Y _{r D} have not been discussed before. Received February 1, 1999, Revised February 25, 2000, Accepted March 9, 2000     

Asymptotic properties of a stochastic chemostat including species death rate

This paper deals with a stochastic system which models the population dynamics of a chemostat including species death rate. On the basis of the theory on Markov semigroup, we demonstrate that the probability densities of the distributions for the solutions are absolutely continuous. The densities will convergence in L¹ to an invariant density or weakly convergence to a singular measure under appropriate conditions. We also give the sufficient criteria for extinction exponentially of the species. To be specific, when D₁>D and the strength of perturbation is relatively small, we derive a precise threshold for the species survival.     

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     

Existence and Regularity for a Class of Infinite-Measure (ξ, ψ, K)-Superprocesses

We extend the class of (, , K)-superprocesses known so far by applying a simple transformation induced by a weight function for the one-particle motion. These transformed superprocesses may exist under weak conditions on the branching parameters, and their state space automatically extends to a certain space of possibly infinite Radon measures. It turns out that a number of superprocesses which were so far not included in the general theory fall into this class. For instance, the hyperbolic branching catalyst of Fleischmann and Mueller⁽¹²⁾ is included and we are able to extend it to the case of -branching. In the second part of this paper, we discuss regularity properties of our processes. Under the assumption that the one-particle motion is a Hunt process, we show that our superprocesses possess right versions having càdlàg paths with respect to a natural topology on the state space. The proof uses an approximation with branching particle systems on Skorohod space.     

On the spectrum of a class of differential operators and embedding theorems

The author considers the embedding problem of weighted Sobolev spacesH _p ⁿ in weightedL _s spacesL _s,r , and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions. The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator, and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete. Supported by the National Natural Science Fundation of China and the Natural Science Foundation of Inner Mongolia.     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

Hardy Inequality and Weighted Heat Trace

Upper bounds are obtained for the heat trace weighted by a negative power of the distance to the boundary of an open set D in a complete Riemannian manifold, provided the Dirichlet-Laplace-Beltrami operator acting in L ²(D) satisfies a strong Hardy inequality.      

The automorphism group of the derived category for a weighted projective line

We show that up to a translation each automorphism of the derived category D ^b X of coherent sheaves on a weighted projective line X, equiv-alently of the derived category D ^b A of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X. In the case of genus one this implies that the automorphism group is a semi-direct product of the braid group on three strands by a finite group.

Moreover we prove that most automorphisms lift from the Grothendieck group to the derived category.     

Elementary divisors of induced transformations on tensors via the representation theory of sl 2

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.     

A Regularly Preemptive Model and its Approximations

A regularly preemptive model D,MAP/D ₁,D ₂/1 is studied. Priority customers have constant inter-arrival times and constant service times. On the other hand, ordinary customers' arrivals follow a Markovian Arrival Process (MAP) with constant service times. Although this model can be formulated by using the piecewise Markov process, there remain some difficult problems on numerical calculations. In order to solve these problems, a novel approximation model MAP/MR/1 with Markov renewal services is proposed. These two queueing processes become different due to the existence of idle periods. Thus, a MAP/MR/1 queue with a general boundary condition is introduced. It is a model with the exceptional first service in each busy period. In particular, two special models are studied: one is a warm-up queue and the other is a cool-down queue. It can be proved that the waiting time of ordinary customers for the regular preemption model is stochastically smaller than the waiting time of the former model. On the other hand, it is stochastically larger than the waiting time of the latter model.     

Dirichlet型空间上的紧加权复合算子的谱

该文研究了单位多圆柱情形Dirichlet型空间上的加权复合算子的谱.对一类紧加权复合算子,给出了其谱的完全刻画,推广了已有结果.     

S-polar sets of super-brownian motions and solutions of nonlinear differential equations

This paper gives probabilistic expressions of the minimal and maximal positive solutions of the partial differential equation -1/2△v(x) γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 < α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.     

The lifetime of diffusion processes with application to conditioned diffusions

Let τ _D . denote the lifetime of a diffusion process on domainD ⊂R ^d. This paper presents a sufficient condition for the exponential moment of τ _D to be finite. Here, both of the domain and the diffusion operator are general. As an application, the main result of Gao (1995) for conditioned diffusions is improved on. Research supported in part by NNSFC (19631060) and Beijing Normal University     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Boundedness, Compactness and Schatten-class Membership of Weighted Composition Operators

The boundedness and compactness of weighted composition operators on the Hardy space H²{{\mathcal H}^2} of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces A²_a(\mathbbD){\mathcal{A}^2_\alpha(\mathbb{D})} are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.