Class Of Superprocesses Research Articles

Superprocesses on ultradistributions

Stochastic solutions provide new rigorous results for nonlinear PDE’s and, through its local nongrid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: McKean’s and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE’s. A new class of superprocesses, namely superprocesses on ultradistributions, is proposed to extend the stochastic solution approach to a wider class of PDE’s.

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.

GENERALIZED MEHLER SEMIGROUPS AND ORNSTEIN–UHLENBECK PROCESSES ARISING FROM SUPERPROCESSES OVER THE REAL LINE

We study the fluctuation limits of a class of superprocesses with dependent spatial motion on the real line, which give rise to some new Ornstein–Uhlenbeck processes with values of Schwartz distributions.

Tanaka Formulae for ( , d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (α, d, β)-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L1+θ convergence argument (with 0<θ<β≤1) rather than L2 convergence, for the derivation of the Tanaka formulae.

State Classification for a Class of Interacting Superprocesses with Location Dependent Branching

The spatial structure of a class of superprocesses which arise as limits in distribution of a class of interacting particle systems with location dependent branching is investigated. The criterion of their state classification is obtained. Their effective state space is contained in the set of purely-atomic measures or the set of absolutely continuous measures according as one diffusive coefficient $c(x) \equiv 0$ or $|c(x)| \geq \epsilon > 0$ while another diffusive coefficient $h \in C^{2}_{b}(R)$.

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.

Dirichlet problem related to a class of superprocesses

Dirichlet problem related to a class of superprocesses

A proof of the persistence criterion for a class of superprocesses

Gorostiza and Wakolbinger (1991), and Dawson and Perkins (1991) established the same persistence criterion for a class of critical branching particle systems and for a class of superprocesses respectively. In this note we take another approach to the criterion and present a simpler proof of it.