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.     

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.     

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.     

Subgeometric rates of convergence of f-ergodic strong Markov processes

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f$f$-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to (f,r)$(f,r)$-regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

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.     

A remark on the uniqueness of Silverstein extensions of symmetric Dirichlet forms

We prove the uniqueness of the Silverstein extension of symmetric Dirichlet forms under some condition on intrinsic metrics. As its application, we present some non‐local Dirichlet forms which possess the uniqueness of the Silverstein extension and generate non‐conservative Hunt processes.     

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     

On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations _X0,_XΔ,…,_XnΔ$X_0,X_{\Delta },\dots ,X_{n\Delta }$, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞$n\Delta \to \infty$ and Δ→0$\Delta \to 0$) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Absolute stability of a stochastic evolution equationt†

It is shown that mean square stability of a class of nonlinear stochastic evolution equations is equivalent to that of a linear stochastic evolution equation provided noise terms in the former are dominated by that of the latter. This generalizes Morozan's result concerning the stochastic dserential equation of Lur'e type to a much wider class of stochastic evolution equations in Hilbert space. Two examples are given to illustrate the theory.     

Large deviations for Markov processes with mean field interaction and unbounded jumps

Summary An N-particle system with mean field interaction is considered. The large deviation estimates for the empirical distributions as N goes to infinity are obtained under conditions which are satisfied, by many interesting models including the first and the second Schlögl models.Supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University and the NSERC operating grant of D.A. Dawson     

On free stochastic processes and their derivatives

We study a family of free stochastic processes whose covariance kernels K$K$ may be derived as a transform of a tempered measure σ$\sigma$. These processes arise, for example, in consideration of non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal basis in the corresponding non-commutative L²$L^2$ of sample-space. We define a stochastic integral for our family of free processes.     

Diffusion approximations of age-and-position dependent branching processes

By a (G, F, h) age-and-position dependent branching process we mean a process in which individuals reproduce according to an age dependent branching process with age distribution function G(t) and offspring distribution generating function F, the individuals (located in R^N) can not move and the distance of a new individual from its parent is governed by a probability density function h(r). For each positive integer n, let Z_n(t,dx) be the number of individuals in dx at time t of the (G, F_n,h_n) age-and-position dependent branching process. It is shown that under appropriate conditions on G, F_n and h_n, the finite dimensional distribution of $\text{Z}{}_{\mathrm{n}}\text{(nt, dx)}\text{n}$ converges, as n → ∞, to the corresponding law of a diffusion continuous state branching process X(t,dx) determined by a ψ-semigroup {ψ_t: t ? 0}. The ψ-semigroup {ψ_t} is the solution of a non-linear evolution equation. A semigroup convergence theorem due to Kurtz [10], which gives conditions for convergence in distribution of a sequence of non-Markovian processes to a Markov process, provides the main tools.     

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)$(g,F)$-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)$(g,F)$-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.     

Measurable Riemannian structures associated with strong local Dirichlet forms

We introduce Riemannian‐like structures associated with strong local Dirichlet forms on general state spaces. Such structures justify the principle that the pointwise index of the Dirichlet form represents the effective dimension of the virtual tangent space at each point. The concept of differentiations of functions is studied, and an application to stochastic analysis is presented.     

Lagging and leading coupled continuous time random walks, renewal times and their joint limits

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW), which models diffusion and anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. Two different limit processes are identified when waiting times precede jumps or follow jumps, respectively, together with two limit processes corresponding to the renewal times. We calculate the joint law of all four limit processes evaluated at a fixed time t.