Dimensional Ornstein-Uhlenbeck Processes Research Articles

Universal excursion and bridge shapes in ABBM/CIR/Bessel processes

Several years ago, in the context of the physics of hysteresis in magnetic materials, a simple stochastic model has been introduced: the ABBM model. Later, the ABBM model has been advocated as a paradigm for the description of a broad class of diverse phenomena, baptized ‘crackling noise phenomena’. The model reproduces many statistical features of such intermittent signals, as for instance the statistics of burst (or avalanche) durations and sizes, in particular the power law exponents that would characterize the dynamics as critical. In order to go beyond such ‘critical exponents’, the measure of the average shape of the avalanche has also been proposed. Here, the exact calculation of the average (as well as the fluctuations) of the avalanche shape for the ABBM model is presented, showing that its normalised shape does not depend on the external drive. Moreover, the average (and the fluctuations) of the multi-avalanche shape, that is the average shape of a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are exactly the same. This result is obtained using the exact solution of the ABBM model, which is obtained leveraging the equivalence with the Cox–Ingersoll–Ross process (CIR), rigorously obtained with a so called ‘time change’. A simple presentation of this and other known relevant exact results is provided: notably the correspondence of the ABBM/CIR model with the Rayleigh model and, more importantly, with the generalised Bessel process, which describes the dynamics of the modulus of the multi dimensional Ornstein–Uhlenbeck process (exactly as the Bessel process does for the Brownian process). As a consequence, our main finding, that is the correspondence between the excursion (avalanche) and bridge (multi-avalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: if we consider the distance from the origin of such diffusive particles, the (normalised) average shape of its trajectory (and the fluctuations around that) until a return in a time T is the same, whether it has returned before T or not.

Normalized Gaussian path integrals.

Path integrals play a crucial role in describing the dynamics of physical systems subject to classical or quantum noise. In fact, when correctly normalized, they express the probability of transition between two states of the system. In this work, we show a consistent approach to solve conditional and unconditional Euclidean (Wiener) Gaussian path integrals that allow us to compute transition probabilities in the semiclassical approximation from the solutions of a system of linear differential equations. Our method is particularly useful for investigating Fokker-Planck dynamics and the physics of stringlike objects such as polymers. To give some examples, we derive the time evolution of the d-dimensional Ornstein-Uhlenbeck process and of the Van der Pol oscillator driven by white noise. Moreover, we compute the end-to-end transition probability for a charged string at thermal equilibrium, when an external field is applied.

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by Levy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.

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.

The Inverse First Passage time method for a two dimensional Ornstein Uhlenbeck process with neuronal application.

The Inverse First Passage time problem seeks to determine the boundary corresponding to a given stochastic process and a fixed first passage time distribution. Here, we determine the numerical solution of this problem in the case of a two dimensional Gauss-Markov diffusion process. We investigate the boundary shape corresponding to Inverse Gaussian or Gamma first passage time distributions for different choices of the parameters, including heavy and light tails instances. Applications in neuroscience framework are illustrated.

Dynamic Looping of a Free-Draining Polymer

We revisit the celebrated Wilemski-Fixman (WF) treatment for the looping time of a free-draining polymer. The WF theory introduces a sink term into the Fokker-Planck equation for the $3(N+1)$-dimensional Ornstein-Uhlenbeck process of the polymer dynamics, which accounts for the appropriate boundary condition due to the formation of a loop. The assumption for WF theory is considerably relaxed. A perturbation method approach is developed that justifies and generalizes the previous results using either a Delta sink or a Heaviside sink. For both types of sinks, we show that under the condition of a small dimensionless $\epsilon$, the ratio of capture radius to the Kuhn length, we are able to systematically produce all known analytical and asymptotic results obtained by other methods. This includes most notably the transition regime between the $N^2$ scaling of Doi, and $N\sqrt{N}/\epsilon$ scaling of Szabo, Schulten, and Schulten. The mathematical issue at play is the non-uniform convergence of $\epsilon\to 0$ and $N\to\infty$, the latter being an inherent part of the theory of a Gaussian polymer. Our analysis yields a novel term in the analytical expression for the looping time with small $\epsilon$, which is previously unknown. Monte Carlo numerical simulations corroborate the analytical findings. The systematic method developed here can be applied to other systems modeled by multi-dimensional Smoluchowski equations.

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.

Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes

In the context of non-equilibrium statistical physics, the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state. In this paper, we establish a central limit theorem and a moderate deviation principle for the entropy production rate of $d$-dimensional Ornstein-Uhlenbeck processes, by the techniques of functional inequalities such as Poincar\'e inequality and log-Sobolev inequality. As an application, we obtain a law of iterated logarithm for the entropy production rate.

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.

A Central Limit Theorem for a sequence of Brownian motions in the unit sphere in [formula omitted

We use a Stochastic Differential Equation satisfied by Brownian motion taking values in the unit sphere S n − 1 ⊂ R n and we obtain a Central Limit Theorem for a sequence of such Brownian motions. We also generalize the results to the case of the n -dimensional Ornstein–Uhlenbeck processes.

Long-Time Average of Field Measured by an Ornstein–Uhlenbeck Wanderer

For a given field V ( x ) on R d , we shall find (i) the exponent ν such that \(X_T[\omega] = T^{-\nu}\int_0^{T} V(\omega(t))\,\mathrm{d}t\) converges as T →∞ to have a non-trivial probability distribution, and (ii) the distribution itself, where ω( t ) denotes the d -dimensional Ornstein–Uhlenbeck process starting from ω(0) = x at t = 0. The exponent and the probability distribution are different depending on whether or not a weighted average \(\int_{\mathbb{R}^d}V(x) \exp[-\beta x^2/2D]\,\mathrm{d}^{d} x\) of V vanishes, where D and β are constants of the Ornstein–Uhlenbeck process. In the non-vanishing case, they are ν=1 and a Dirac delta distribution, while in the vanishing case, ν=1/2 and the Gaussian distribution. The results have applications to the physics of chemoreception in immune systems.

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].

Testing for a Change in the Parameter Values and Order of an Autoregressive Model

The problem of testing whether or not a change has occurred in the parameter values and order of an autoregressive model is considered. It is shown that if the white noise in the AR model is weakly stationary with finite fourth moments, then under the null hypothesis of no changepoint, the normalized Gaussian likelihood ratio test statistic converges in distribution to the Gumbel extreme value distribution. An asymptotically distribution-free procedure for testing a change of either the coefficients in the AR model, the white noise variance or the order is also proposed. The asymptotic null distribution of this test is obtained under the assumption that the third moment of the noise is zero. The proofs of these results rely on Horvath's extension of Darling-Erdos' result for the maximum of the norm of a $k$-dimensional Ornstein-Uhlenbeck process and an almost sure approximation to partial sums of dependent random variables.