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     

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.     

Classical limit theorems for measure-valued Markov processes

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.     

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.     

On regularity of superprocesses

Summary Three theorems on regularity of measure-valued processesX with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treatX as a family of random measures associated with finely open setsQ in time-space. Heuristically,X describes an evolution of a cloud of infinitesimal particles. To everyQ there corresponds a random measureX which arises if each particle is observed at its first exit time fromQ. (The stateX _t at a fixed timet is a particular case.) We consider a monotone increasing familyQ _t of finely open sets and we establish regularity properties of as a function oft. The results are used in [D6], [D7] and [D10] for investigating the relations between superprocesses and non-linear partial differential equations. Basic definitions on Markov processes and superprocesses are introduced in Sect. 1. The next three sections are devoted to proving the regularity theorems. They are applied in Sect. 5 to study parts of superprocess. The relation to the previous work is discussed in more detail in the concluding section. It may be helpful to look briefly through this section before reading Sects. 2–5.Partially supported by the National Science Foundation Grant DMS-8802667 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

Semi-stable Markov processes in rn

A Markov process in Rⁿ{x_t} with transition function P_t is called semi-stable of order α>0 if for every a>0, P_t(x, E) = P_at(a^ax, a^aE). Let ?_t(ω)=∫^t₀|x_s(ω)|^-1/α ds, T(t) be its inverse and {y_t}={x_T(t)}.Theorem 1: {Y_t} is a multiplicative invariant process; i.e., it has transition function q_t satisfying q_t(x,E)=q_t(ax,aE) for all a > 0.Theorem 2: If {x_t} is Feller, right continuous and uniformly stochastic continuous on a neighborhood of the origin, then {y_t} is Feller.     

Markov processes with free-Meixner laws

We study a time-non-homogeneous Markov process which arose from free probability, and which also appeared in the study of stochastic processes with linear regressions and quadratic conditional variances. Our main result is the explicit expression for the generator of the (non-homogeneous) transition operator acting on functions that extend analytically to complex domains.     

Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions

In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it has been noticed in previous works, there is a phase transition in the behavior of the process. Here, we examine the strongly and intermediately supercritical regimes The main result is a conditional limit theorem for the rescaled associated random walk in the intermediately case.     

One-point extensions of Markov processes by darning

This paper is a continuation of the works by Fukushima–Tanaka (Ann Inst Henri Poincaré Probab Stat 41: 419–459, 2005) and Chen–Fukushima–Ying (Stochastic Analysis and Application, p.153–196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L ²-infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L ²-infinitesimal generator of the extended process. An important role in our investigation is played by the α-order approaching probability u _α . The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206. The research of M. Fukushima is supported in part by Grant-in-Aid for Scientific Research of MEXT No.19540125.     

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α$\alpha$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α$\alpha$-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.     

Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

A limit theorem for trees of alleles in branching processes with rare neutral mutations

We are interested in the genealogical structure of alleles for a Bienaymé–Galton–Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Ji?ina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.     

Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching

In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as a special case. We prove the Feller and strong Feller continuity by means of introducing auxiliary processes and making use of the Radon-Nikodym derivatives. Then, we investigate the geometric ergodicity by the Foster-Lyapunov inequality. Moreover, we establish the V-uniform ergodicity by means of introducing additional auxiliary processes and by virtue of constructing certain order-preserving couplings of the original as well as the auxiliary processes. In addition, illustrative examples are provided for demonstration.     

Conditional law of risk processes given that ruin occurs

A risk process that can be Markovised is conditioned on ruin. We prove that the process remains a Markov process. If the risk process is a PDMP, it is shown that the conditioned process remains a PDMP. For many examples the asymptotics of the parameters in both the light-tailed case and the heavy-tailed case are discussed.     

Multifractal spectra and precise rates of decay in homogeneous fragmentations

We consider a mass-conservative fragmentation of the unit interval. Motivated by a result of Berestycki [J. Berestycki, Multifractal spectra of fragmentation processes, J. Statist. Phys. 113 (3–4) (2003) 411–430], the main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Lévy processes constrained to stay in a finite interval.     

Constructions of coupling processes for Lévy processes

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.     

Surviving particles for subcritical branching processes in random environment

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a.s$a.s$. These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.