Infinite Dimensional Ornstein-Uhlenbeck Processes Research Articles

Stationary Fluctuations of Run-and-Tumble Particles

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.

Amplitude equations for SPDEs driven by fractional additive noise with small hurst parameter

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter [Formula: see text]. Close to a change of stability measured with a small parameter [Formula: see text], we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of [Formula: see text] depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.

Markovian lifts of positive semidefinite affine Volterra-type processes

We consider stochastic partial differential equations appearing as Markovian lifts of matrix-valued (affine) Volterra-type processes from the point of view of the generalized Feller property (see, e.g., Dörsek and Teichmann in A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. arXiv:1011.2651). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein–Uhlenbeck processes whose state space is the set of matrix-valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes-type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston-type model.

Restoring uniqueness to mean-field games by randomizing the equilibria

We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external noise as an infinite dimensional Ornstein–Uhlenbeck process. We first investigate existence and uniqueness of a solution to the noisy system made of the mean-field game forced by the Ornstein–Uhlenbeck process. We also show how such a noisy system can be interpreted as the limit version of a stochastic differential game with a large number of players.

Small ball probabilities for the infinite-dimensional Ornstein–Uhlenbeck process in Sobolev spaces

While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential equations. The paper presents exact logarithmic asymptotics of the small ball probabilities in a scale of Sobolev spaces when the Gaussian measure is generated by the solution of a diagonalizable stochastic parabolic equation. Compared to the finite-dimensional case, new effects appear in a certain range of the Sobolev exponents.

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Levy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.

Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays

A class of stochastic optimal control problems of infinite dimensional Ornstein-Uhlenbeck processes of neutral type are considered. One special feature of the system under investigation is that time delays are present in the control. An equivalent formulation between an adjoint stochastic controlled delay differential equation and its lifted control system (without delays) is developed. As a consequence, the finite time quadratic regulator problem governed by this formulation is solved based on a direct solution of some associated Riccati equation.

A Class of Affine Processes Arising as Fluctuation Limits of Super-Brownian Motion in a Super-Brownian Catalytic Medium

A class of infinite dimensional Ornstein-Uhlenbeck processes that arise as solutions of stochastic partial differential equations with noise generated by measure-valued catalytic processes is investigated. It will be shown that the catalytic Ornstein-Uhlenbeck process with super-Brownian catalyst in one dimension arises as a high density fluctuation limit of a super-Brownian motion in a super-Brownian catalyst with immigration. The main tools include Laplace transformations of stochastic processes, analysis of a non-linear partial differential equation and techniques on continuity and regularity based on properties of the Sobolev spaces.

OPTIMAL POINTWISE APPROXIMATION OF INFINITE-DIMENSIONAL ORNSTEIN–UHLENBECK PROCESSES

We consider an infinite-dimensional Ornstein–Uhlenbeck process on the spatial domain ]0,1[d driven by an additive nuclear or space–time white noise, and we study the approximation of this process at a fixed point in time. We determine the order of the minimal errors as well as asymptotically optimal algorithms, both of which depend on the spatial dimension d and on the decay of the eigenvalues of the driving Wiener process W in the case of nuclear noise. In particular, the optimal order is achieved by employing drift-implicit Euler schemes with non-uniform time discretizations, while uniform time discretizations turn out to be suboptimal in general. By means of non-asymptotic error bounds and by simulation experiments, we show that the asymptotic results are predictive for the actual errors already for time discretizations with a small number of points.

A limit theorem for quadratic fluctuations in symmetric simple exclusion

We consider quadratic fluctuations V ε H ( η s ) = ε ∑ x ∈ Z H ( ε x ) η s ( x ) η s ( x + x 0 ) in the centered symmetric simple exclusion process in dimension d = 1 . Although the order of divergence of E [ ∫ 0 ε − 2 d s V ε H ( η s ) ] 2 is known to be ε − 3 / 2 if ε ↓ 0 , the corresponding limit theorem was so far not explored. We now show that ε 3 / 2 ∫ 0 t ε − 2 d s V ε H ( η s ) converges in law to a non-Gaussian singular functional of an infinite-dimensional Ornstein–Uhlenbeck process. Despite the singularity of the limiting functional we find enough structure to conclude that it is continuous but not a martingale in t . We remark that in symmetric exclusion in dimensions d ≥ 3 the corresponding functional central limit theorem is known to produce Gaussian martingales in t . The case d = 2 remains open.

Infinite dimensional stochastic differential equations of Ornstein–Uhlenbeck type

We consider the operator L f ( x ) = 1 2 ∑ i , j = 1 ∞ a ij ( x ) ∂ 2 f ∂ x i ∂ x j ( x ) - ∑ i = 1 ∞ λ i x i b i ( x ) ∂ f ∂ x i ( x ) . We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the a ij , b i , and λ i . The process corresponding to L solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein–Uhlenbeck process.

Equilibrium Fluctuations for Zero-Range-Exclusion Processes

We introduce a Markovian particle system which is a kind of lattice gas on Z consisting of particles carrying energy and whose dynamics is a combination of those of an exclusion process (for particles) and a zero-range process (for energy). It has two conserved quantities, the number of particles and the total energy. The process is reversible relative to certain product probability measures, but of non-gradient type. It is proved that under hydrodynamic scaling the equilibrium fluctuation fields of two conserved quantities converge in law to an infinite dimensional Ornstein–Uhlenbeck process.

White Noise Limits for Inertial Particles in a Random Field

In this paper we present a rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field. The mathematical model comprises Stokes's law for the particle motion and an infinite dimensional Ornstein-Uhlenbeck process for the fluid velocity field. The scaling limit studied leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term. Strong convergence methods are used to justify the limiting equations. The rigorously derived limiting equations are of physical interest for the concrete problem under investigation and facilitate the study of two-point motions in the white noise limit. Furthermore, the methodology developed may also prove useful in the study of various other asymptotic problems for stochastic differential equations in infinite dimensions.

Brownian motion on the Wiener sphere and the infinite–dimensional Ornstein–Uhlenbeck process

The infinite-dimensional Ornstein–Uhlenbeck process v is constructed from Brownian motion on the infinite-dimensional sphere S N−1 (1) (the Wiener sphere) – or equivalently, by rescaling, on S N−1( N ) – which is defined for infinite N by nonstandard analysis. This gives rigorous sense to the informal idea (due to Malliavin, Williams and others) that v can be thought of as Brownian motion on S ∞( ∞ ) . An invariance principle follows easily. The paper is a sequel to Cutland and Ng (1993) where the uniform Loeb measure on S N−1 (1) was shown to give a rigorous construction of Wiener measure.

On local continuity moduli for Gaussian processes

In this paper, we get three local continuity moduli theorems for the almost surely continuous, stationary increments Gaussian process {Y(t),t≥0}, the partial sum processesX(t,N)=Σ =1/ X k(t) of infinite dimensional Ornstein-Uhlenbeck processes {X k(t),t≥0} =1/∞ andl p-valued Gaussian processes {Y(t),t≥0}={X k(t),t≥0} =1/∞ separately. The first theorem implies the local continuity modulus theorem for the seriesX(t)=Σ =1/∞ X k(t) of infinite dimensional Ornstein-Uhlenbeck processes which has been obtained in [3].

Mehler formula and capacities for infinite-dimensional Ornstein-Uhlenbeck processes with general linear drift

Mehler formula and capacities for infinite-dimensional Ornstein-Uhlenbeck processes with general linear drift

Limit theorems for multitype epidemics

By means of the law of large numbers and the central limit theorem, we compare the spatial epidemic model proposed by Kendall with a sequence of multitype epidemics, viewed as Hilbert space-valued stochastic processes. In the limit, the fluctuation process turns out to be an infinite-dimensional Ornstein-Uhlenbeck process.

On the Law of the Iterated Logarithm for Infinite Dimensional Ornstein-Uhlenbeck Processes

AbstractLet be independent Ornstein-Uhlenbeck processes and In this paper the law of iterated logarithm for X(t, n)is considered. The results obtained improve those of Csorgő and Lin(1988) and Schmuland(1987).

On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space

The set of self-intersections of a Brownian path $b(t)$ taking values in $\mathbb{R}^3$ has Hausdorff dimension 1, for almost every such path, with respect to Wiener measure, a result due to Fristedt. Here we prove that this result (together with the corresponding result for paths in $\mathbb{R}^2$) in fact holds for quasi-every path with respect to the infinite-dimensional Ornstein-Uhlenbeck process, a diffusion process on Wiener space whose stationary measure is Wiener measure. We do this using Rosen's self-intersection local time, first proving that this exists for quasi-every path.