Nonlinear SPDEs Research Articles

Well-posedness for nonlinear SPDEs with strongly continuous perturbation - ERRATUM

Well-posedness for nonlinear SPDEs with strongly continuous perturbation - ERRATUM

Weak and strong solutions to nonlinear SPDEs with unbounded noise

We introduce an extended variational framework for nonlinear SPDEs with unbounded noise, defining three different solution types of increasing strength along with criteria to establish their existence. The three notions can be understood as probabilistically and analytically weak, probabilistically strong and analytically weak, as well as probabilistically and analytically strong. Our framework facilitates several well-posedness results for the Navier–Stokes Equation with transport noise, equipped with the no-slip and Navier boundary conditions.

Exponential convergence for nonlinear SPDEs with double reflecting walls

Exponential convergence for nonlinear SPDEs with double reflecting walls

Conditional Least Squares Estimation for Fractional Super Levy Processes in Nonlinear SPDEs

We consider infinite dimensional extension of affine models as super Levy processes satisfying a nonlinear SPDE. We obtain the asymptotics of the conditional least squares estimators. Finally we obtain the Berry-Esseen inequality.

The Neumann problem for fully nonlinear SPDE

The Neumann problem for fully nonlinear SPDE

Weak error analysis for a nonlinear SPDE approximation of the Dean–Kawasaki equation

We consider a nonlinear SPDE approximation of the Dean–Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order N-1-1/(d/2+1)logN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N^{-1-1/(d/2+1)}\\log N$$\\end{document}. Along the way we show well-posedness, a comparison principle, and an entropy estimate for a class of nonlinear regularized Dean–Kawasaki equations with Itô noise.

Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

A Graphical Representation of the Truncated Moment of the Solution of a Nonlinear SPDE

We consider a stochastic partial differential equation driven by a Lévy type noise (SPDE). Particular attention is given to the correlation function which measures the moments of the solution. Using the Feynman graph formalism, the solution of the SPDE as well as its truncated moments are given as a sum over specific graphs that are evaluated according to some rules. A remark on some applications will be given at the end of this work.

Directed mean curvature flow in noisy environment

AbstractWe consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite‐dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

Parameter estimation for semilinear SPDEs from local measurements

This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in (Ann. Appl. Probab. 31 (2021) 1–38) for linear SPDEs remains rate-optimal when applied to a large class of semilinear SPDEs. The obtained abstract results are applied to several important classes of SPDEs, including stochastic reaction-diffusion equations. Moreover, we also study the stochastic Burgers equation, as an example with first order nonlinearity, which is a borderline case of the general results. The optimal statistical results are obtained through a precise control of the spatial regularity of the solution and by using higher order fractional Lp-Sobolev type spaces. We conclude with numerical examples that validate the theoretical results.

On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment

With the recent developments on nonlinear SPDEs, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one-dimensional SPDE ∂tρ=1 2ΔΦ(ρ)−2∇(W′Φ(ρ)), where the spatial white noise W′ is replaced by a regularization Wε′, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in ε-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.

Numerical approximation of nonlinear SPDE’s

The numerical analysis of stochastic parabolic partial differential equations of the form du+A(u)dt=fdt+gdW,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} du + A(u)\\, dt = f \\,dt + g \\, dW, \\end{aligned}$$\\end{document}is surveyed, where A is a nonlinear partial operator and W a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory.

Strong convergence rates in averaging principle for slow-fast McKean-Vlasov SPDEs

In this paper, we aim to study the asymptotic behavior for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic p-Laplace type equation and also some McKean-Vlasov stochastic differential equations.

Solving some stochastic partial differential equations driven by Lévy noise using two SDEs*

ABSTRACT The method of characteristics is a powerful tool to solve some nonlinear second-order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear utility stochastic PDEs, SIAM J. Financ. Math. 4(1) (2013), pp. 737–783; A. Matoussi and M. Mrad, Dynamic utility and related nonlinear SPDE driven by Lévy noise, preprint (2020), submitted for publication. Available at https://hal.archives-ouvertes.fr/hal-03025475]. In this situation the solution is theoretically of the form where and are solutions of a system of two SDEs, is the inverse flow of and is the initial condition. Unfortunately this representation is not explicit except in simple cases where and are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations and of X and ξ. This allows us to avoid a complicated discretization in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in -norms, of the compound approximation to the compound variable , in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.

Renormalising SPDEs in regularity structures

The formalism recently introduced in [BHZ19] allows one to assign a regularity structure, as well as a corresponding “renormalisation group”, to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was shown in [CH16] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group on a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular non-linear SPDEs.

Well-posedness for nonlinear SPDEs with strongly continuous perturbation

AbstractWe consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

Modulation Equation and SPDEs on Unbounded Domains

We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation. As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are H\"older continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

Coercivity condition for higher moment a priori estimates for nonlinear SPDEs and existence of a solution under local monotonicity

Higher moment a priori estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.

An SPDE model for systemic risk with endogenous contagion

We propose a dynamic mean field model for `systemic risk' in large financial systems, which we derive from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distances-to-default of financial institutions and absorption at zero corresponds to default. As a way of modelling correlated exposures and herd behaviour, we consider a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution can cause a drop in the distances-to-default of the other institutions. In this way, we aim to capture key `system-wide' effects on risk. The resulting mean field limit is characterized uniquely by a nonlinear SPDE on the half-line with a Dirichlet boundary condition. The density of this SPDE gives the conditional law of a non-standard `conditional' McKean--Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate that is essential to the proofs. Depending on the realizations of the common noise and the rate of mean reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, the contagion mechanism can give rise to periods of significant systemic default clustering.

A new approach to the existence of invariant measures for Markovian semigroups

On etablit une approche en deux etapes pour demontrer l’existence des mesure invariantes finies pour un semigroupe de Markov donne. En fixant d’abord une mesure auxiliaire convenable, on demontre ensuite des conditions equivalentes a l’existence d’une mesure invariante finie qui est absolument continue par rapport a elle. Comme applications, on obtient une generalisation unificatrice des diverses versions du theoreme ergodique de Harris et on fournit une reponse a une question ouverte de Tweedie. On montre aussi que pour une EDP stochastique sur un triplet de Gelfand, la condition de coercivite stricte est suffisante pour garantir l’existence d’une seule mesure de probabilite pour le semigroupe associe, si une inegalite de type Harnack avec puissance est satisfaite. Un corollaire du resultat central montre que tout semigroupe uniformement borne sur $L^{p}$ possede une mesure invariante ; on donne des applications aux perturbations sectorielles des formes de Dirichlet.