Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

The first integral method to some complex nonlinear partial differential equations

In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.     

A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

Forward-backward stochastic differential equations and quasilinear parabolic PDEs

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate. Received: 12 May 1997 / Revised version: 10 January 1999     

A class of path-valued Markov processes and its applications to superprocesses

Summary Let ( _s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with ( _s ), which is closely related to the so-called superprocess with spatial motion ( _s ). In particular, a subsetH of the state space of ( _s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when ( _s ) is a diffusion process, we recover certain results recently obtained by Dynkin.     

Anisotropic branching random walks on homogeneous trees

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff dimension of Λ∩Ω_μ, where Ω_μ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ω_μ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ω_μ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point. Received: 30 June 1998 / Revised version: 10 March 1999     

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998     

Diffusion processes on graphs: stochastic differential equations,large deviation principle

Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices. Received: 12 February 1997 / Revised version: 3 March 1999     

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process

We consider finite-range, nonzero mean, one-dimensional exclusion processes on ℤ. We show that, if the initial configuration has ``density' α, then the process converges in distribution to the product Bernoulli measure with mean density α. From this we deduce the strong form of local equilibrium in the hydrodynamic limit for non-product initial measures.     

Trivial stationary solutions to the Kuramoto-Sivashinsky and certain nonlinear elliptic equations

We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R² are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN.     

Convergence to equilibrium for classical and quantum spin systems

Summary The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].Partially supported by the Italian National Project MURST Problemi nonlinearinell' Analisi... and by DRET under contract 901636/A000/DRET/DSISR.Partially sponsored by the KBN grant 2 2003 91 02 and by the KBN grant 2PO3A 082 08     

A new approach to the single point catalytic super-Brownian motion

Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst _c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure ^c ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX _t is determined from ^c by an explicit representation formula. Heuristically, a mass ^c (ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure ^c . In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure ^c .     

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.     

An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

We prove the existence of a weak solution of the Dirichlet problem for a class of elliptic partial differential systems in separable Orlicz–Sobolev spaces.     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

Summary. The super-Brownian motion X ^ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction. Received: 21 November 1996 / In revised form: 31 March 1997     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.