Rayleigh-Stokes Equations with Space-Time White Noise: Existence, Hölder Regularity, and Parameter-Continuity

Linear and nonlinear stochastic wave equations given by a space-time Gaussian white noise are considered in a space of dimension d≥2. In the linear case the solution is a random Schwartz distribution. In the nonlinear case existence and uniqueness of solutions is proven in the framework of Colombeau random generalized functions. In the case where the nonlinear drift is given by the Fourier transform ƒof a complex measure for instance when ƒ is a trigonometric function the solution is proven to be Lp-associated with the solution of the free equation, for any p ≥ 1

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs—through their fundamental solutions—on stochastic integral equations (SIEs) driven by space–time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase in the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [Discrete Contin. Dyn. Syst. Ser. A 33 (2013) 413–463, Stoch. Dyn. 6 (2006) 521–534], where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space–time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions $d=1,2,3$. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in $d=1,2$. This gave, for the first time, an example of a space–time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space. In this paper, we introduce the $2\beta^{-1}$-order $\beta$-inverse-stable-Lévy-time Brownian motion ($\beta$-ISLTBM) SIEs, $\beta\in \{1/2^{k};k\in\mathbb{N}\}$, driven by space–time white noise. Based on the dramatic regularizing effect of the BTBM density ($\beta=1/2$), and since the kernels in these $\beta$-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs—no matter how high we take the order $1/\beta$ of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. Furthermore, we show that increasing the order of the Laplacian $\beta^{-1}$ beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches that of the Brownian sheet as $\beta\searrow0$. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and $\beta$-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we study approximations to 3D Navier–Stokes (NS) equation driven by space-time white noise by paracontrolled distribution proposed in Ref. 13. A solution theory for this equation has been developed recently in Ref. 27 based on regularity structure theory and paracontrolled distribution. In order to make the approximating equation converge to 3D NS equation driven by space-time white noise, we should subtract some drift terms in approximating equations. These drift terms, which come from renormalizations in the solution theory, converge to the solution multiplied by some constant depending on approximations.

Two-Dimensional Navier–Stokes Equations Driven by a Space–Time White Noise

In this article, we present an $L_{p}$-theory ($p\geq 2$) for the semi-linear stochastic partial differential equations (SPDEs) of type \begin{equation*}\partial^{\alpha }_{t}u=L(\omega ,t,x)u+f(u)+\partial^{\beta }_{t}\sum_{k=1}^{\infty }\int^{t}_{0}(\Lambda^{k}(\omega,t,x)u+g^{k}(u))\,dw^{k}_{t},\end{equation*} where $\alpha \in (0,2)$, $\beta <\alpha +\frac{1}{2}$ and $\partial^{\alpha }_{t}$ and $\partial^{\beta }_{t}$ denote the Caputo derivatives of order $\alpha $ and $\beta $, respectively. The processes $w^{k}_{t}$, $k\in \mathbb{N}=\{1,2,\ldots \}$, are independent one-dimensional Wiener processes, $L$ is either divergence or nondivergence-type second-order operator, and $\Lambda^{k}$ are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal $L_{p}$-regularity of the solutions. By converting SPDEs driven by $d$-dimensional space–time white noise into the equations of above type, we also obtain an $L_{p}$-theory for SPDEs driven by space–time white noise if the space dimension $d<4-2(2\beta -1)\alpha^{-1}$. In particular, if $\beta <1/2+\alpha /4$ then we can handle space–time white noise driven SPDEs with space dimension $d=1,2,3$.

Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory

The magnetohydrodynamics system forced by space–time white noise has been proposed and investigated by physicists and engineers, although without rigorous mathematical proofs of its solution’s fundamental properties. Applying the theory of regularity structures, we prove its well-posedness and strong Feller property. The proof requires a careful treatment of nonlinear terms which are sensitive to specific components of the solution that is a six-dimensional vector field, of which the first three and the last three are velocity and magnetic fields, respectively.

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math>

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.

Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space–time white noise

Three-dimensional Navier–Stokes equations driven by space–time white noise

Stefan problems for reflected SPDEs driven by space–time white noise