Markov properties of multiparameter processes and capacities

Summary We study a class of multiparameter symmetric Markov processes. We prove that this class is stable by subordination in Bochner's sense. We show then that for these processes, a probabilistic and an analytic potential theory correspond to each other. In particular, additive functionals are associated with finite energy measures, hitting probabilities are estimated by capacities, quasicontinuity corresponds to path-continuity. In the last section, examples show that many earlier results, as well as new ones, in this domain can be obtained by our method.     

On conservativeness and recurrence criteria for Markov processes

In the present paper, we give general criteria of conservativeness and recurrence for Markov processes associated with, not necessarily symmetric, Dirichlet spaces. The conservativeness criterion is applied to discuss a comparison theorem of conservativeness for diffusion processes. Also some sufficient conditions of conservativeness and recurrence for diffusion processes.are given.     

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.     

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 associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Potential theory related to some multiparameter processes

In a first part, we present a potential theory constructed form a continuous kernel on a locally compact space. The notions of capacity, quasi-continuity, equilibrium measures and potentials are specially studied. In a second part, we particularize the framework, and, in the third part, we give probabilistic interpretations in this particular case. The process then involved is a sum of independent symmetric Levy processes in ^d , viewed as a multiparameter process. For instance, hitting probabilities for the process are estimated in terms of capacity.     

Crossing estimates for symmetric Markov processes

A crossing estimate is established for symmetric Markov processes on general state spaces. Received: 20 May 1999 / Revised version: 30 August 2000 / Published online: 9 March 2001     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Construction of Markov processes and associated multiplicative functionals from given harmonic measures

Summary LetE be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures onE that behave like harmonic measures associated with all relatively compact open sets inE (i.e. that satisfy a certain consistency condition), one can construct a Markov process onE and a multiplicative functional with values in [0, ) such that the hitting distributions of the process inflated by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the spaceE equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).     

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.     

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.     

Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces

Summary In the work of Donsker and Varadhan, Fukushima and Takeda and that of Deuschel and Stroock it has been shown, that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric Markov process is given in terms of its Dirichlet form. We give a short proof generalizing this principle to general state spaces that include, in particular, infinite dimensional and non0metrizable examples. Our result holds w.r.t. quasi-every starting point of the Markov process. Moreover we show the corresponding weak upper bound w.r.t. quasi-every starting point.This research was supported by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

How many diffusions exist on the Vicsek snowflake?

We consider Kusuoka's construction of Dirichlet forms on the Vicsek snowflake. a nested fractal. His method is generalized and all irreducible, local Dirichlet forms which can be constructed in this way are characterized. We end up with a one-parameter family of different possible forms. This proves that Brownian motion on fractals is not unique if the isometry group of the fractal is too small.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

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.     

Nonlinear potential theory and PDEs

We consider equations like -div(|u| ^p–2u)=, where is a nonnegative Radon measure and 1u and the measure are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.     

Minorations pour les chaînes de Markov unidimensionnelles

Summary We prove a space-time lower estimate for reversible random walks on graphs having linear growth. This estimate complements a similar upper bound that follows from a work of Hebisch and Saloff-Coste.

     

Some Markov processes with Brownian exit distributions

Summary LetD be a bounded domain inR ^d with regular boundary. LetX=(X_t, P^x) be a standard Markov process inD with continuous paths up to its lifetime. IfX satisfies some weak conditions, then it is possible to add a non-local part to its generator, and construct the corresponding standard Markov process inD with Brownian exit distributions fromD.This work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany