Space-time White Noise Research Articles

Stochastic Quantization of an Abelian Gauge Theory

We study the Langevin dynamics of a U(1) lattice gauge theory on the two-dimensional torus, and prove that they converge for short time in a suitable gauge to a system of stochastic PDEs driven by space-time white noises. We fix gauge via a DeTurck trick. This also yields convergence of some gauge invariant observables on a short time interval. The proof relies on a discrete version of the theory of regularity structures.

Pathwise analysis and parameter estimation for the stochastic Burgers equation

We analyze the solution to the stochastic Burgers equation with additive space-time white noise. We show that this solution can be expressed as a sum of a random field that solves the stochastic heat equation with additive space-time white noise and a more regular random field. We apply these results in order to estimate the drift parameter of solution to the Burgers equation by exploiting the behavior of the p-variations of the solution in time and in space.

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability results. We show the existence of a unique global mild solution to such equation using a fixed point method and stopping time arguments. The existence of a local mild solution (up to a stopping time) is proved via contraction mapping principle. Then, establishing a uniform bound for the solution, we show the existence and uniqueness of global mild solution to the stochastic generalized Burgers–Huxley equation. Finally, we discuss the inviscid limit of the stochastic Burgers–Huxley equation to the stochastic Burgers as well as Huxley equations.

Modelling Lévy space‐time white noises

Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. Similar to existing definitions, we introduce Levy-valued additive sheets, and show that integrating a Levy-valued random measure in space defines a Levy-valued additive sheet. This relation is manifested by the result, that a Levy-valued random measure can be viewed as the weak derivative of a Levy-valued additive sheet in the space of distributions.

The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise

We study the asymptotic speed of a random front for solutions $$u_t(x)$$ to stochastic reaction–diffusion equations of the form $$\begin{aligned} \partial _tu=\frac{1}{2}\partial _x^2u+f(u)+\sigma \sqrt{u(1-u)}{\dot{W}}(t,x),~t\ge 0,~x\in {\mathbb {R}}, \end{aligned}$$ arising in population genetics. Here, f is a continuous function with $$f(0)=f(1)=0$$ , and such that $$|f(u)|\le K|u(1-u)|^\gamma $$ with $$\gamma \ge 1/2$$ , and $${\dot{W}}(t,x)$$ is a space-time Gaussian white noise. We assume that the initial condition $$u_0(x)$$ satisfies $$0\le u_0(x)\le 1$$ for all $$x\in {\mathbb {R}}$$ , $$u_0(x)=1$$ for $$x<L_0$$ and $$ u_0(x)=0$$ for $$x>R_0$$ . We show that when $$\sigma >0$$ , for each $$t>0$$ there exist $$R(u_t)<+\infty $$ and $$L(u_t)<-\infty $$ such that $$u_t(x)=0$$ for $$x>R(u_t)$$ and $$u_t(x)=1$$ for $$x<L(u_t)$$ even if f is not Lipschitz. We also show that for all $$\sigma >0$$ there exists a finite deterministic speed $$V(\sigma )\in {\mathbb {R}}$$ so that $$R(u_t)/t\rightarrow V(\sigma )$$ as $$t\rightarrow +\infty $$ , almost surely. This is in dramatic contrast with the deterministic case $$\sigma =0$$ for nonlinearities of the type $$f(u)=u^m(1-u)$$ with $$0<m<1$$ when solutions converge to 1 uniformly on $${\mathbb {R}}$$ as $$t\rightarrow +\infty $$ . Finally, we prove that when $$\gamma >1/2$$ there exists $$c_f\in {\mathbb {R}}$$ , so that $$\sigma ^2V(\sigma )\rightarrow c_f$$ as $$\sigma \rightarrow +\infty $$ and give a characterization of $$c_f$$ . The last result complements a lower bound obtained by Conlon and Doering (J Stat Phys 120(3–4):421–477, 2005) for the special case of $$f(u)=u(1-u)$$ where a duality argument is available.

Geometric stochastic heat equations

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\mathbf{R}^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties: - For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using It\^o calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation). - Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid. - Every solution theory satisfies an analogue of It\^o's isometry. - The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation. In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and It\^o interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension 1+1, where the disorder is given by the occupation field of a Poisson system of independent random walks on Z. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel.In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an L1-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.

On a Monte Carlo scheme for some linear stochastic partial differential equations

Abstract The aim of this paper is to study the simulation of the expectation for the solution of linear stochastic partial differential equation driven by the space-time white noise with the bounded measurable coefficient and different boundary conditions. We first propose a Monte Carlo type method for the expectation of the solution of a linear stochastic partial differential equation and prove an upper bound for its weak rate error. In addition, we prove the central limit theorem for the proposed method in order to obtain confidence intervals for it. As an application, the Monte Carlo scheme applies to the stochastic heat equation with various boundary conditions, and we provide the result of numerical experiments which confirm the theoretical results in this paper.

Global well posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain

We study the two-dimensional wave equation with cubic nonlinearity posed on R2 with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for initial data in Hs, s>45.

A rough super-Brownian motion

We study the scaling limit of a branching random walk in static random environment in dimension d=1,2 and show that it is given by a super-Brownian motion in a white noise potential. In dimension 1 we characterize the limit as the unique weak solution to the stochastic PDE ∂tμ=(Δ+ξ)μ+ 2νμξ˜ for independent space white noise ξ and space-time white noise ξ˜. In dimension 2 the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-Brownian motion.

A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

Chaotic characterization of one dimensional stochastic fractional heat equation

We study the Cauchy problem of the nonlinear stochastic fractional heat equation ∂tu=−ν2(−∂xx)α2u+σ(u)W˙(t,x) on real line R driven by space-time white noise with bounded initial data. We analyze the large-|x| fixed-t behavior of the solution ut(x) for Lipschitz continuous function σ:R→R under three cases. (1) σ is bounded below away from 0. (2) σ is uniformly bounded away from 0 and ∞. (3) σ(x)=cx (the parabolic Anderson model). From the sensitivity to the initial data of stochastic fractional heat equation, we describe that the solution to the Cauchy problem of stochastic fractional heat equation exhibits chaotic behavior at fixed time before the onset of intermittency.

Stochastic Lotka-Volterra competitive reaction-diffusion systems perturbed by space-time white noise: Modeling and analysis

Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation enables spatial dependence of the species. In addition, the noise process is allowed to be space-time white noise. In this work, well-posedness, regularity of solutions, existence of density, and existence of an invariant measure for stochastic reaction-diffusion systems with non-Lipschitz and non-linear growth coefficients and multiplicative noise are considered. By combining the random field approach and infinite integration theory approach in SPDEs for mild solutions, analysis is carried out. Then this paper develops a Lotka-Volterra competitive system under general setting; longtime properties are studied with the help of newly developed tools in stochastic calculus.

The Osgood condition for stochastic partial differential equations

We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ is a space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman in (Phys. D 238 (2009) 209–215), our first result shows that the solution blows up in finite time if and only if for some $a>0$, \begin{equation*}\int _{a}^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,\end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbf{R}^{d}$.

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

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.

Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha > 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha $. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$. These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

Higher order pathwise approximation for the stochastic Burgers' equation with additive noise

This article aims to investigate the pathwise convergence of the higher order scheme, introduced by Jentzen (2011) [9], for the stochastic Burgers' equation (SBE) driven by space-time white noise. In particular, first and second order derivatives of the non-linear drift term of the SBE are assumed to be defined and bounded in Sobolev spaces using the definition of distribution derivative i.e. Lemma 4.7 in Blömker and Jentzen (2013) [2] is extended. Based on this extension, temporal convergence analysis of the higher order scheme is carried out for the SBE with additive noise. As a result, minimum temporal convergence order is improved from θ (Theorem 4.1 in Blömker et al. (2013) [3]) to 2θ, where every θ∈(0,12)). Numerical experiments are performed to validate the theoretical findings.

On the Small Time Asymptotics of the Dynamical Φ41 Model

In this paper, we establish a small time large deviation principle (small time asymptotics) for the dynamical Φ14 model, which not only involves study of the space-time white noise with intensity $$\sqrt \varepsilon $$ , but also the investigation of the effect of the small (with e) nonlinear drift.

Stochastic nonlinear Schrödinger equation on an upper-right quarter plane with Dirichlet random boundary

In this paper, we study the nonhomogeneous stochastic initial-boundary value problem for the nonlinear Schrödinger equation on an upper-right quarter plane with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the Wiener additive noise on the boundary. Our approach allows us to show the local existence and uniqueness of solutions in the space H2. The basic properties of the solutions such as the continuity and the boundary-layer behavior are also studied using the Itô calculus. Despite several technical difficulties, we believe that the approach developed in this paper can be applied to the case of a large class of noise including fractional Wiener space time white noise, homogeneous noise, and Levy noise.