The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, dσ)$, and show how to include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce in $\Gama(\Ha)$ operators $a(ξ)=\int_{\R^d} dx ξ(x)a(x)$, $ξ\in S$, with $a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and $\di_x$ are the creation and annihilation operators at $x$. We show that $(a(ξ))_{ξ\in S}$ is a family of commuting selfadjoint operators in $\Gama(\Ha)$ and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary $I$ between $\Gama(\Ha)$ and the $L^2$-space $L^2(S',dμ_{\mathrm G})$, where $μ_{\mathrm G}$ is the measure of Gamma white noise with intensity $σ$. The image of $a(ξ)$ under $I$ is the operator of multiplication by $\la\cdot,ξ\ra$, so that $a(ξ)$'s are Gamma field operators. The Fock structure of the Gamma space determined by $I$ coincides with that discovered in {\bf [}{\it Infinite Dimensional Analysis,
Quantum Probability and Related Topics} {\bf 1} (1998), 91--117{\bf ]}. We note that $I$ extends in a natural way the multiple stochastic integral (chaos) decomposition of the ``chaotic'' subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.
We study the growth properties of the family of i.i.d. random sequences, also known as discrete white noises, and of their continuous-domain generalization, the family of Lévy white noises. More precisely, we characterize the members of both families which are tempered in terms of their moment properties. We recover the characterization of tempered Lévy white noises obtained by Robert Dalang and Thomas Humeau and provide a new proof of there fundamental result. Our approach is based on a fruitful connection between the discrete and continuous-domain white noises.
We identify a necessary and sufficient condition for a Lévy white noise to be a tempered distribution. More precisely, we show that if the Lévy measure associated with this noise has a positive absolute moment, then the Lévy white noise almost surely takes values in the space of tempered distributions. If the Lévy measure does not have a positive absolute moment of any order, then the event on which the Lévy white noise is a tempered distribution has probability zero.
In the framework of sublinear expectation, we have introduced a new type of G-Gaussian random fields, which contain a type of spatial white noise as a special case. Based on this result, we also have introduced a spatial-temporal G-white noise. Different from the case of linear expectation, in which the probability measure need to be known, under the uncertainty of the probability measure, the spatial white noises are intrinsically different from the temporal one.
We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus $\mathbb{T}^2$, perturbed by multiplicative transport noise. The equation admits the white noise measure on $\mathbb{T}^2$ as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space-time white noise. The weak uniqueness of the latter equation is …
We identify the representation of the square of white noise obtained by L. Accardi, U. Franz and M. Skeide in [Comm. Math. Phys. 228 (2002), 123--150] with the Jacobi field of a Lévy process of Meixner's type.
This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. We first derive an existence theorem for integral transforms of Lévy white noise and prove the existence of generalized and mild solutions of second order elliptic partial differential equations. Furthermore, we discuss the generalized electric Schrödinger operator for different potential functions $V$.
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
For the white noise, the spectral density is constant, and the past (restriction to (-\infty,0)) is independent from the future (restriction to (0,+\infty)). If the spectral density is not too far from being constant, then dependence between the past and the future can be eliminated by an equivalent measure change; that is called an off-white noise. I derive from well-known results a necessary and sufficient condition for a spectral density to describe an off-white noise.
It is possible to construct Lévy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two approaches by extending the Lévy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for Lévy white noises, thereby maximally enlarging their domain of definition. Based on this connection, we provide new criteria for the practical determination of the doma…
We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-alpha-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the L…
In this article, we show that the Brownian motion on the circle constructed in Levy (1959) is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the other half-circle, and is degenerated in the sense of Minlos (1959). This raises the question of what the white noise is on the circle. We then formally define the white noise space and its associated Brownian bridge.
We obtain a representation theorem for Banach space valued Gaussian random variables as integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the existence of a white noise representation for a Gaussian random field indexed by a compact measure space. As an application we show how existing theory for integration with respect to Gaussian processes indexed by $[0,1]$ can be extended to Gaussian fields indexed by compact measure spaces.
We show that the system of point vortices, perturbed by a certain transport type noise, converges weakly to the vorticity form of 2D Navier--Stokes equations driven by the space-time white noise.
We prove pathwise uniqueness for solutions of parabolic stochastic pde's with multiplicative white noise if the coefficient is Hölder continuous of index $γ>3/4$. The method of proof is an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations.
In this paper, we established a quadratic transportation cost inequality for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise based on a new inequality we proved for the moments (under the uniform norm) of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $γ>1-\fracη{2(η+1)}$. Here $η\in(0,1)$ is a constant that defines the spatial regularity of the noise.
In this article, we give a new proof of the Itô formula for some integral processes related to the space-time Lévy white noise introduced in Balan (2015) as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the $p$-th moment of the stochastic integral, and the Itô representation theorem leading to a chaos expansion similar to the Gaussian case.
We present the expansion of the multifractional Brownian (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions.
The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smo…
In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida-Malliavin calculus, which are us…
In this paper, we prove transportation inequalities on the space of continuous paths with respect to the uniform metric, for the law of solution to a stochastic heat equation defined on $[0,T]\times [0,1]^d$. This equation is driven by the Gaussian noise, white in time and colored in space. The proof is based on a new moment inequality under the uniform metric for the stochastic convolution with respect to the time-white and space-colored noise, which is of independent interest.
In this paper we establish a framework for normal approximation for white noise functionals by Stein's method and Hida calculus. Our work is inspired by that of Nourdin and Peccati (Probab. Theory Relat. Fields 145, 75-118, 2009), who combined Stein's method and Malliavin calculus for normal approximation for functionals of Gaussian processes.
The paper is devoted to construction and investigation of some riggings of the $L^2$-space of Poisson white noise. A particular attention is paid to the existence of a continuous version of a function from a test space, and to the property of an algebraic structure under pointwise multiplication of functions from a test space.
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises in…
In this article, we consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term $σ$. We prove that under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If $σ$ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.
In this article, we consider the stochastic wave equation in dimension 1 driven by the Lévy white noise introduced in Balan (2015). Using Rosenthal's inequality, we develop a maximal inequality for the moments of order $p \geq 2$ of the integral with respect to this noise. Based on this inequality, we show that this equation has a unique solution, which is weakly intermittent in the sense of Foondun and Khoshnevisan (2009) and Khoshnevisan (2014).
We show the global-in-time well-posedness of the complex Ginzburg-Landau (CGL) equation with a space-time white noise on the 3-dimensional torus. Our method is based on [14], where Mourrat and Weber showed the global well-posedness for the dynamical $Φ_3^4$ model. We prove a priori $L^{2p}$ estimate for the paracontrolled solution as in the deterministic case [5].
Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is $\bbR^d$ and shot noise approximations available on more general spaces. Limit theorems stating the convergence of discrete random noises to stable white noise are proved. These results are then applied to study moving average spatial random fields with heavy-tailed innovations and related limit theorems. A second application deals with discrete approximation for Bro…
We consider stochastic differential equations driven by Gaussian white noise on $\R^d$. % We provide applications to models for financial %markets. Particular attention is given to the kernel $p_t,\,t\geq 0$ of the transition semigroup associated with the solution process. Under some assumptions on the coefficients, we prove that the small time asymptotic expansion of $p_t$ is Borel summable.
In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Levy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d\leq 3, then this solution can be represented as a classical random field in L2(μ), where μis the probability law of the Levy process. The starting point of our theory is a chaos expansion in terms of generalized Charlie…
We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the no…
We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution.
In this paper we consider the Anderson Hamiltonian with white noise potential on the box $[0,L]^2$ with Dirichlet boundary conditions. We show that all the eigenvalues divided by $\log L$ converge as $L\rightarrow \infty$ almost surely to the same deterministic constant, which is given by a variational formula.
Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. Especially, Neumann and Robin BVPs are considered. The main obstacle for applying the usual variational approach is that the Gaussian white noise has irregular realizations. In particular, the corresponding Neumann boundary values are not well-defined in the ordinary sense. In this work, the BVP is reformulated by replacing the continuity of the boundary trace mapping with measurability. Instead of using variational methods alone, the reformulation of t…
We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.
We introduce a discretization/approximation scheme for reflected stochastic partial differential equations driven by space-time white noise through systems of reflecting stochastic differential equations. To establish the convergence of the scheme, we study the existence and uniqueness of solutions of Skorohod-type deterministic systems on time-dependent domains. We also need to establish the convergence of an approximation scheme for deterministic parabolic obstacle problems. Both are of independent interest on their own.
In this paper we study stochastic currents of Brownian motion $ξ(x)$, $x\in\mathbb{R}^{d}$, by using white noise analysis. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and for $x=0\in\mathbb{R}$ we prove that the stochastic current $ξ(x)$ is a Hida distribution. Moreover for $x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a Hida distribution.
We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to $q$-deformed commutation relations with $q\in(-1,1)$. We construct a Gel'fand triple centered at the $q$-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a Våge-type inequality for the Wick product on the dual space.
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.
In this paper, we study the Besov regularity of d-dimensional Lévy white noises. More precisely, we describe new sample paths properties of a given white noise in terms of weighted Besov spaces. In particular, the smoothness and integrability properties of Lévy white noises are characterized using the Blumenthal-Getoor indices. Our techniques rely on wavelet methods and generalized moments estimates for Lévy white noises.
We prove, using coupling arguments, exponential convergence to equilibrium for reaction--diffusion and Burgers equations driven by space-time white noise. We use a coupling by reflection.
In this article, we improve the Strichartz estimates obtained in [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.
Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and play an important role in quantifying propagation and evolution of uncertainty. Although Fokker-Planck equations can be written explicitly for nonlinear dynamical systems excited by Gaussian white noise, they are not available in general for nonlinear dynamical systems excited by multiplicative non-Gaussian white noise. Marcus stochastic differential equations are often appropriate models in engineering and physics for stochastic dynamical systems excited by non-Gaussian white noise. In…
We consider the linear and nonlinear Schr{ö}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.
In this paper we study 3D Navier-Stokes (NS) equation driven by space-time white noise by using regularity structure theory introduced in [Hai14] and paracontrolled distribution proposed in [GIP13]. We obtain local existence and uniqueness of solutions to the 3D Navier-Stokes equation driven by space-time white noise.
We survey different approaches to study the invariance of the white noise for the periodic KdV. We mainly discuss the following two methods. First, we discuss the PDE method, following Bourgain \cite{BO4}, in a general framework. Then, we show how it can be applied to the low regularity setting of the white noise for KdV by introducing the Besov-type space \hat{b}^s_{p, \infty}, sp< -1. Secondly, we describe the probabilistic method by Quastel, Valkó, and the author \cite{OQV}. We also use this probabilistic approach to study the white noise for mKdV.
Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida's white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product.
We prove the Freidlin-Wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave (NLW) equation with white noise. We do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise. This allows to provide the first such result in the PDE setting.
Using the white noise space framework, we define a class of stochastic processes which include as a particular case the fractional Brownian motion and its derivative. The covariance functions of these processes are of a special form, studied by Schoenberg, von Neumann and Krein.
This paper calculates the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise.
The paper formulates Bayesian inverse problems for inference in a topological measure space given noisy observations. Conditions for the validity of the Bayes formula and the well-posedness of the posterior measure are studied. The abstract theory is then applied to Burgers and Hamilton-Jacobi equations on a semi-infinite time interval with forcing functions which are white noise in time. Inference is made on the white noise forcing, assuming the Wiener measure as the prior.
Aim of this work is to extend the results of Clément, Da Prato & Prüss on the fractional white noise perturbation with Hurst parameter 0<H<1. We will obtain similar results and it will turn out that the regularity of the solution u(t) of the stochastic Volterra equation increases with Hurst parameter H.
We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.
In this paper, we show existence of white noise solutions for weak formulations of modified Surface Quasi-Geostrophic (mSQG) equations. Based on previous results (\cite{FS}) on white noise solutions for mSQG equations on the torus $\mathbb{T}^{2}$, we show a similar result for the whole space $\mathbb{R}^{2}$ by letting the volume of the torus go to infinity and applying compactness methods (Skorokhod's theorem).
