We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.
We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli' study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we i…
In this paper, we consider the approximating KPZ equation introduced by Funaki and Quastel [2], which is suitable for studying invariant measures. They showed that the stationary solution of the approximating equation converges to the Cole-Hopf solution of the KPZ equation with extra term $(1/24)t$. On the other hand, Gubinelli and Perkowski [5] gave a pathwise meaning to the KPZ equation as an application of the paracontrolled calculus. We show that Funaki and Quastel's result is extended to nonstationary solutions by using the paracontrolled calculus.
We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.
We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators. We illustrate the efficiency of this elementary approach on the examples of the 3-dimensional generalised parabolic Anderson model equation and the generalised KPZ equation driven by a (1+1)-dimensional sp…
We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.
We introduce a non-linear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two dimensional parabolic Anderson model.
Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.
We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.
We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus $\mathbb{T}^3$. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on $\mathbb{T}^3$ by Gubinelli, Koch, and the first author (2018), Okamoto, Tolomeo, and the first author (2020), and Bringmann (2020). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with t…
We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most $-1/2-κ$ for any $κ>0$. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of e…
This paper introduces the path derivatives, in the spirit of Dupire's functional Itô calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solut…
We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the…
We define the random magnetic Laplacien with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L 2. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.
This paper explores the relationship between non-Markovian fully coupled forward-backward stochastic systems and path-dependent PDEs. The definition of classical solution for the path-dependent PDE is given within the framework of functional Itô calculus. Under mild hypotheses, we prove that the forward-backward stochastic system provides the unique classical solution to the path-dependent PDE.
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: \[\partial_t μ= (Δ{+} ξ) μ{+} \sqrt{2νμ} \tildeξ\] for independent space white noise $ξ$ and space-time white noise $\tildeξ$. 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-…
In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations \begin{equation*} \partial_t u + u \cdot \nabla u = Δu - \nabla p + ζ+ ξ \;, \quad u (0, \cdot) = u_{0}(\cdot) \;, \quad \mathrm{div} (u) = 0 \;, \end{equation*} driven by additive space-time white noise $ ξ$, with perturbation $ ζ$ in the Hölder-Besov space $\mathcal{C}^{-2 + 3κ} $, periodic boundary conditions and initial condition $ u_{0} \in \mathcal{C}^{-1 + κ} $ for any $ κ>0 $. The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and …
This manuscript is a self-contained overview of essential results of stochastic calculus and stochastic differential equations, and their connection with final-value problems for second order linear PDEs.
We consider the stochastic Ginzburg-Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg-Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the low-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies t…
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic cal…
We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild …
