We consider stochastic partial differential equations on $\mathbb{R}^{d}, d\geq 1$, driven by a Gaussian noise white in time and colored in space, for which the pathwise uniqueness holds. By using the Skorokhod representation theorem we establish various strong stability results. Then, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and uniqueness of solutions as well as the continuous dependence on the coefficients.
We consider a general class of SPDEs in $\mathbb{R}^d$ driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with $d\in \{1,2,3\}$. In these pa…
In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t>0$, assuming some suitable regularity and non degeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.
In this paper, we extend Walsh's stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang's one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equati…
In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and Hölder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.
We consider the family of stochastic partial differential equations indexed by a parameter $\eps\in(0,1]$, \begin{equation*} Lu^{\eps}(t,x) = \epsσ(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} $(t,x)\in(0,T]\times\Rd$ with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $σ$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let $p^\eps_{t,x}$ denote the density of the law of $u^\eps(t,x)$ at a fixed point $(t,x)\in(0,T]\times\Rd$. We …
In this paper, we consider a system of $k$ second order non-linear stochastic partial differential equations with spatial dimension $d \geq 1$, driven by a $q$-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise, has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and Hölder's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some me…
We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.
Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (1999), and the fractional noise considered in Balan and Tudor (2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with harmonizable noise, under some mild conditions imposed on the Green function of the differential operator …
We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (2003), we prove that the solutions belong to a fractional $L^2$-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the s…
In this article, we introduce a Lévy analogue of the spatially homogeneous Gaussian noise of Dalang (1999), and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure $μ$ on $\bR^d$, whose density is given by $|h|^2$ for a complex-valued function $h$. Without assuming that the Fourier transform of $μ$ is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise.
We study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, and hence fails to be symmetric. However, we demonstrate a scenario where the failure of proposal symmetry is a higher order effect. Hence, from these microscopic dynamics we derive as a limit as the proposal size goes to zero and the number of spins to infinity, a non-local stochastic version of the harmonic map heat flow (or overdamped Landau-Lipshitz equation). The equatio…
The asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane is studied. We show that as the divergence-free advection term becomes larger and larger, the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. Firstly, we deal with the case that the stochastic perturbation is given by a singular spatially homogeneous Wiener process taking values in the space of Schwartz distributions. As in previous works, we assume here that the derivative of the period of the motion on the…
We consider the linear stochastic heat and wave equations with generalized Gaussian noise that is white in time and spatially correlated. Under the assumption that the homogeneous spatial correlation $f$ satisfies some mild conditions, we show that the solutions to the linear stochastic heat and wave equations exhibit tall peaks in macroscopic scales, which means they are macroscopically multi-fractal. We compute the macroscopic Hausdorff dimension of the peaks for Gaussian random fields with vanishing correlation and then apply this result to the solution of the linear stochastic heat and wav…
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…
In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the $L^2$-generator of a $d$-dimensional Lévy process $X=(X_t)_{t \geq 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $β$-stable process, with $β\in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov…
This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. …
The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the param…
Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience.
In this paper we study the Large Deviation Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space $\mathbb{R}^d$, with arbitrary dimension $d\geq 1$, under random influence which is a Gaussian noise, white in time and correlated in space. The differential operator is a fractional derivative operator. We prove a large deviations principle for our equation, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion. This approach reduces the proof of LDP to establishin…
We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic ge…
Stochastic partial differential equations (SPDE) on graphs were introduced by Cerrai and Freidlin [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865-899]. This class of stochastic equations in infinite dimensions provides a minimal framework for the study of the effective dynamics of much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood, partly due to complications near vertex singularities. In this work, motivated by the study of the dynamics and the genealogies of expanding populations in spatially structured environments,…
In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in $\mathbb{R}^d$ w…
Let $M$ be a compact Riemannian homogeneous space (e.g. a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation \mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot W$ in any dimension $d\ge 1$, where $f$ and $g$ are continuous multilinear mappings and $W$ is a spatially homogeneous Wiener process on $\mathbb R^d$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.
We study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. We prove that at any fixed time, a.s. the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\IR^3$, the sample paths in time are Hölder continuous functions. Hence, we obtain joint Hölder continuity in the time and space variables. Our results rely on a detailed analysis of properties…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To reproduce the fractional, and more specifically multifractal, regularity nature of fully developed turbulence, these dynamical evolutions incorporate an homogenous pseudo-differential linear operator of degree 0 that takes care of transferring energy that is injected at large scales in the system, towards smaller scales according to a cascading mechanism. In the…
The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the Navier-Stokes equation. We work in the limit of high frequenc…
We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}Δ\right) u(t,x) = ρ(u(t,x)) \:\dot{M}(t,x), \] where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $ρ$ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang's condition, namely, $\int_{\mathbb{R}^d}(1+|ξ|^2)^{-1}\hat{f}(\text{d} ξ)<\…
We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven…
We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t, x, u)$ satisfies $K^{-1}|u|^λ\leq h(t, x, u)\leq K(1+|u|)$ for $λ\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existe…
Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；National University of Defense Technology（国防科技大学）
