This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the discrete or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.
By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.
In this paper, we consider stochastic homogenization of elliptic equations with unbounded and non-uniformly elliptic coefficients. Extending subadditive arguments, we get an estimate for the rate of the convergence of the solution of the Dirichlet problem under the condition that coefficients in the unit cube have a certain exponential integrability. For the coefficient field $\mathbf{a}$ in this paper, we only assume a constant decrease at a constant distance of the maximal correlation as an assumption of ergodicity, and stationarity with respect to $\mathbb{Z}^d$-translations.
The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincaré domains). We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $r\in[1,3]$) perturbed by an additive noise on Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these…
Building upon a recent work by two of the authours and J. Seidler on bw-Feller property for stochastic nonlinear beam and wave equations, we prove the existence of an invariant measure to stochastic 2-D Navier-Stokes (with multiplicative noise) equations in unbounded domains. This answers an open question left after the first authour and Y. Li proved a corresponding result in the case of an additive noise.
For a Harris ergodic Markov chain $(X_n)_{n\ge 0}$, on a general state space, started from the so called small measure or from the stationary distribution we provide optimal estimates for Orlicz norms of sums $\sum_{i=0}^τf(X_i)$, where $τ$ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function $f$ (wrt the stationary distribution) and the regeneration time $τ$ (wrt the small measure). We provide applications to tail estimates for additive functionals of the chain $(X_n)$ generated by unbounded functions as well as to classical l…
We derive the Markov-modulated generalized Ornstein-Uhlenbeck process by embedding a Markov-modulated random recurrence equation in continuous time. The obtained process turns out to be the unique solution of a certain stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its statio…
In this work, we consider the two and three-dimensional stochastic convective Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive white noise $$\mathrm{d}\boldsymbol{u}-[μΔ\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-α\boldsymbol{u}-β|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla p]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\mathrm{d}\mathrm{W},\ \nabla\cdot\boldsymbol{u}=0,$$ for $r\in[1,\infty),$ $μ,α,β>0$ in unbounded domains (like Poincaré domains) $\mathcal{O}\subset\mathbb{R}^d$ ($d=2,3$) where $\mathrm{W}(\cdot)$ is a Hilbert space valued Wiener process on…
We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation. As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, i…
We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation. As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result we sketch the proof for local existence a…
We provide an example for stabilization by noise. Our approach does not rely on monotonicity arguments due to the presence of higher order differential operators or mixing properties of the system as the noise might be highly degenerate. In the examples a scalar additive noise destroys a high-dimensional random attractor of a PDE on an unbounded domain. In the presence of small noise close to bifurcation all trajectories converge to a single stationary solution.
In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in $\mathbb{R}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the non-linear stochastic heat equation involving fractional derivative in time and fractional Laplacian in space. We give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. We also study various properties of the solution: path regularity, the behavior of second moment and the stationarity in the case of linear additi…
This paper is concerned with the following problem: Given a stochastic non-linear system controlled over a noisy channel, what is the largest class of channels for which there exist coding and control policies so that the closed loop system is stochastically stable? Stochastic stability notions considered are stationarity, ergodicity or asymptotic mean stationarity. We do not restrict the state space to be compact, for example systems considered can be driven by unbounded noise. Necessary and sufficient conditions are obtained for a large class of systems and channels. A generalization of Bode…
We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility maximization problems including the classical ones of terminal wealth or consumption, as well as the problems depending on a random time-horizon or multiple consumption instances. As an example we treat explicitly the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein-Uhle…
We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility-maximization problems including the classical ones of terminal wealth or consumption, as well as the problems that depend on a random time horizon or multiple consumption instances. As an example we explicitly treat the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein-Uh…
The strong convergence rate of the Euler scheme for SDEs driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the $2$D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order $2H$ for the case that the drift coefficient has bounded derivatives up…
We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin c…
Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains
We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=Ψ_n(X_{n-1}^x)$, $X_0^x=x$, where $Ψ_n$ are i.i.d. random continuous transformations of some unbounded closed subset of ${\mathbb{R}}$. We assume here that $Ψ_n$ behaves asymptotically like $A_nx$, for some random positive number $A_n$ [the main example is the affine stochastic recursion $Ψ_n(x)=A_nx+B_n$]. Our aim is to describe invariant Radon measures of the process $X_n^x$ in the critical case, when ${\mathbb{E}}\log A_1=0$. We prove that those measures behave at infinity like $\frac{dx}…
The asymptotically autonomous robustness of random attractors of stochastic fluid equations defined on \emph{bounded} domains has been considered in the literature. In this article, we initially consider this topic (almost surely and in probability) for a non-autonomous stochastic 2D Navier-Stokes equation driven by additive and multiplicative noise defined on some \emph{unbounded Poincaré domains}. There are two significant keys to study this topic: what is the asymptotically autonomous limiting set of the time-section of random attractors as time goes to negative infinity, and how to show th…
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.
The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the litera…
A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.
It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the ``spatial operator'' is the L^2-generator of a Lévy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly bigger than one. In addition, we prove that the solution to the SPDE is [Hölder] co…
We consider sample path properties of the solution to the stochastic heat equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a Lévy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than $-\frac d 2$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Lévy noise such that noises with a smaller ind…
