Valued Ornstein-Uhlenbeck Processes Research Articles

Hydrodynamics of a class of [formula omitted]-urn linear systems

In this paper we are concerned with hydrodynamics of a class of N-urn linear systems, which include voter models, pair-symmetric exclusion processes and binary contact path processes on N urns as special cases. We show that the hydrodynamic limit of our process is driven by a C[0,1)′-valued linear ordinary differential equation and the fluctuation of our process, i.e, central limit theorem from the hydrodynamic limit, is driven by a C[0,1)′-valued Ornstein–Uhlenbeck process. To derive above main results, we need several replacement lemmas. An extension in linear systems of Chapman–Kolmogorov equation plays key role in proofs of these replacement lemmas.

SIMPLE SIMULATION SCHEMES FOR CIR AND WISHART PROCESSES

We develop some simple simulation algorithms for CIR and Wishart processes. We investigate rigorously the square of a matrix valued Ornstein–Uhlenbeck process, the main idea being to split the generator and to reduce the problem to the simulation of the square of a matrix valued Ornstein–Uhlenbeck process to be added to a deterministic process. In this way, we provide a weak second-order scheme that requires only the simulation of i.i.d. Gaussian r.v.'s and simple matrix manipulations.

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C 0 semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.

Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dW H (t), t≥0, where A is the generator of a C 0-semigroup S on a separable real Banach space E and W H (t) t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S H .

Self-Intersection Local Time for ′(ℝd)-Ornstein-Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure-valued process yield different types of 𝒮′(ℝd)-valued Ornstein-Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α-stable process. In this paper we prove existence and path continuity results for the self-intersection local time of these Ornstein-Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.

Self-Intersection Local Time for SomeS′(ℝd)-Ornstein-Uhlenbeck Processes Related to Inhomogeneous Fields

We study existence and path continuity of the self-intersection local time (SILT) for some S′(ℝd)-valued Ornstein-Uhlenbeck processes. Examples of such processes arise in particular as fluctuation limits ofparticle systems. We analyze the effect that different types ofmeasures involved in the covariances of the processes have on existence and continuity of SILT. These measures do not necessarily have the regularity and homogeneity properties of those that have been considered before; this precludes some key ingredients of the previous techniques. We develop new technical tools and prove sufficient conditions and some necessary conditions for existence of SILT, and sufficient conditions for path continuity of SILT. The questions of existence and continuity involve problems of existence of integrals and singular integrals. We give examples which illustrate how different types of measures e.g., atomic or with L2-density) may produce different critical dimensions for existence of SILT. One of our motivations is the desire for a better understanding ofwhat SILT represents for S′(ℝd)-valued processes.

SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

Existence and continuity results are obtained for self-intersection local time of [Formula: see text]-valued Ornstein–Uhlenbeck processes [Formula: see text], where X0 is Gaussian, Wt is an [Formula: see text]-Wiener process (independent of X0), and T't is the adjoint of a semigroup Tt on [Formula: see text]. Two types of covariance kernels for X0 and for W are considered: square tempered kernels and homogeneous random field kernels. The case where Tt corresponds to the spherically symmetric α-stable process in ℝd, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: Wt, T't X0 and [Formula: see text], from which the results for Xt are derived. As a by-product, a class of non-finite tempered measures on ℝd whose Fourier transforms are functions is identified. The tools are mostly analytical.

Self-similarity of Brownian motion and a large deviation principle for random fields on a binary tree

Using self-similarity of Brownian motion and its representation as a product measure on a binary tree, we construct a random sequence of probability measures which converges to the distribution of the Brownian bridge. We establish a large deviation principle for random fields on a binary tree. This leads to a class of probability measures with a certain self-similarity property. The same construction can be carried out forC[0, 1]-valued processes and we can describe, for instance, aC[0, 1]-valued Ornstein-Uhlenbeck process as a large deviation of Brownian sheet.

Path properties for 𝑙^{∞}-valued Gaussian processes

We prove moduli of continuity results for l ∞ {l^\infty } -valued Gaussian processes in general, as well as for l ∞ {l^\infty } -valued Ornstein-Uhlenbeck processes in particular.

Gaussian and Non–Gaussian Distribution–Valued Ornstein–Uhlenbeck Processes

Generalized (distribution-valued) Ornstein-Uhlenbeck processes, which by definition are solutions of generalized Langevin equations, arise in many investigations on fluctuation limits of particle systems (eg. Bojdecki and Gorostiza [1], Dawson, Fleischmann and Gorostiza [5], Fernández [7], Gorostiza [8,9], Holley and Stroock [10], Itô [12], Kallianpur and Pérez-Abreu [16], Kallianpur and Wolpert [14], Kotelenez [17], Martin-Löf [19], Mitoma [22], Uchiyama [25]). The state space for such a process is the strong dual Φ′ of a nuclear space Φ. A generalized Langevin equation for a Φ′-valued process X ≡ {Xt} is a stochastic evolution equation of the formwhere {At} is a family of linear operators on Φ and Z ≡ {Zt} is a Φ'-valued semimartingale (in some sense) with independent increments. Equations of the type (1.1) where Z does not have independent increments also arise in applications (eg. [9,14,20]) but here we are interested precisely in the case when Z has independent increments (we restrict the term generalized Langevin equation to this case in accordance with the classical Langevin equation).

Sample Path Properties of lp -Valued Ornstein-Uhlenbeck Processes

AbstractWe give conditions under which a vector valued Ornstein Uhlenbeck process has continuous sample paths in lp for 1 ≦ p < ∞. We also show when the space lp is not entered at all, i.e., when it has zero capacity.

Continuity of $l^2$-Valued Ornstein-Uhlenbeck Processes

A stationary $l^2$-valued Ornstein-Uhlenbeck process is considered which is given formally by $dX_t = -AX_t dt + \sqrt 2a dB_t$, where $A$ is a positive self-adjoint operator on $l^2, B_t$ is a cylindrical Brownian motion on $l^2$ and $a$ is a positive diagonal operator on $l^2$. A simple criterion is given for the almost-sure continuity of $X_t$ in $l^2$ which is shown to be quite sharp. Furthermore, in certain special cases, we obtain simple necessary and sufficient conditions for the almost-sure continuity of $X_t$ in $l^2$.

The existence of invariant measures forC[0,1]-valued diffusions

We prove the existence of an invariant measure for processes arising from a perturbation of theC[0,1]-valued Ornstein-Uhlenbeck process with a drift taking values in the Cameron-Martin space. We study the infinitesimal generator, and a partial integration onC[0,1] will yield conditions on the drift which enable us to use arguments of perturbation theory to prove the existence of an invariant measure which is absolutely continuous with respect to the Wiener measure.

Large Deviations for $l^2$-Valued Ornstein-Uhlenbeck Processes

A stationary $l^2$-valued Ornstein-Uhlenbeck process given formally by $dX(t) = - AX(t) dt + \sqrt{2a} dB(t)$, where $A$ is a positive self-adjoint constant operator on $l^2$ and $B(t)$ is a cylindrical Brownian motion on $l^2$, is considered. An upper bound on $P(\sup_{t \in \lbrack 0, T \rbrack}\|X(t)\| > x)$ is established and the asymptotics for the given bound, as $x \rightarrow \infty$, is derived.