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 measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of SPDEs. In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span $\R^k$. We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span $\R^k$ which contains this point has a non void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension $d\in \{1,2,3\}$.
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.
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 continu…
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 …
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.
We prove a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs, with non-homogeneous second order operator on a bounded domain and driven by a space-time white noise. Our method based on an approximation of the domain and the coefficients of the operator, does not require regularity assumptions. As in previous works, the results are consequences of Itô's formula and estimates for the positive part of local solutions which are non-positive on the lateral boundary.
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 exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas f…
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…
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…
Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert--Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have imp…
We consider a model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.
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…
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 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…
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725--2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite-dimensional equations are adopted. For such F…
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+ξ|u|^{1+λ}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $λ\geq 0$ and the coefficients depend on $(ω,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $λ$, a sufficient condition for the unique solvabili…
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,…
We prove a quantitative propagation of chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard \cite{FonGueMe} and Fournier \cite{Fournier} where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions
We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, ener…
We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of Boltzmann's and Landau's equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.
For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the diffusion coefficient of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six…
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present paper aims to extend that result to more-general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their…
We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela SPDE on a homogeneous weighted Sobolev space.
We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decrease for any initial condition. We overcome this difficulty thanks to a functional which is decreasing in time in a neighborhood of any Dirac mass (in the sense of the Wasserstein distance). The results are…
