Differential equations perturbed by multiplicative fractional Brownian motions are considered. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ODE, YDE, or RDE. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness with irregular drift: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case [CG16, Ger22]. In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.
We construct solutions to Burgers type equations perturbed by a multiplicative space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $σ$. When $σ$ is constant and for every $H\in (0,1)$, it was proved in [19] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-α}$ where $α\in (0,1)$ (depending on $H$). In [11], this result has been extended to the multiplicative case when $H\textgreater{}1/2$. In this paper, we obtain these types of results in t…
We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarily geometric) rough path of Hölder regularity $γ>1/3.$ Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity …
In this article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac14, \frac12)$ in space. We prove the existence and uniqueness of the mild Skorohod solution, establish lower and upper bounds for the $p$-th moment of the solution for all $p\ge2$, and obtain the Hölder continuity in time and space variables for the solution.
We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in (\frac14,\frac12)$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than $H$, and Hölder continuous in time of order smaller than $γ$, where $γ=H$ for the wave equation and $γ=H/2$ for the heat equation.
We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) $L^1$-contraction. Our approach is motivated by t…
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced…
This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. The…
Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst index $α$ (in particular, for $α\le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of $B$ defined in \cite{Unt08}. The regularization procedure is applied to 'Fourier normal ordered' iterated i…
Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in $L^1(\mathcal {O})$ on bounded domains $\mathcal {O}$. The generation of a continuous, order-preserving random dynamical system on $L^1(\mathcal {O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal {O})$ norm. Uniform $L^{\infty}$ bounds and uniform space-time…
In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index $H\in (\frac14,\frac12)$. Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the $p$-th moment of the solution, for any $p\geq 2$. Condition $H>\frac14$ turns out to be necessary for the existence of solution. Secondly, we show that this so…
In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the well-posedness of the problem, the linear operator of the equations is only need to be a generator of a $\CC_0$-semigroup and the proposed noises are quite general, which include space-time white noise and rougher noises. When the linear operator generates an analytic $\CC_0$-semigroup, we derive the optimal trajectory regularity of the solution through a generalize…
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the ki…
Nonlinear PDEs with modulated dispersion III: multiplicative noises
CNRS, Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay（法国国家科学研究中心、凡尔赛数学实验室、巴黎西岱大学、巴黎萨克雷大学）；School of Mathematics, Monash University（蒙纳士大学数学学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）
The purpose of this article is threefold. First, we introduce a new type of boundary condition for the multiplicative-noise stochastic heat equation on the half space. This is essentially a Dirichlet boundary condition but with a nontrivial normalization near the boundary which leads to inhomogeneous transition densities (roughly, those of a Brownian \textit{meander}) within the associated chaos series. Secondly, we prove a new convergence result of the directed-polymer partition function in an octant to the multiplicative stochastic heat equation with this type of boundary condition, which in…
A remark on pathwise well-posedness of the 1-$d$ stochastic heat equation
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；The University of Edinburgh（爱丁堡大学）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）
We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a $γ$-Hölder rough path, for $γ\in(1/3,1/2]$. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical sy…
We study a class of semi-implicit Taylor-type numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T.~Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a one-sided Lipschitz condition only. We prove well-posedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments sho…
In this paper, we study the stochastic heat equation with a general multiplicative Gaussian noise that is white in time and colored in space. Both regularity and strict positivity of the densities of the solution have been established. The difficulty, and hence the contribution, of the paper lie in three aspects, which include rough initial conditions, degenerate diffusion coefficient, and weakest possible assumptions on the correlation function of the noise. In particular, our results cover the parabolic Anderson model starting from a Dirac delta initial measure.
We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques are used to provide an almost sure continuity result for the solution with respect to the solution of the 'smooth' equation obtained by replacing the noise with an absolutely continuous process. This continuity result is then exploited to prove weak convergence results based on Donsker and Kac-Stroock type approximations of the noise.
We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/α$-variation for any $α\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf…
This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product be…
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [GVR].
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 …
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla + \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces $W^{k,p}.$ We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e.\ when $Y=0$), we use this framework…
In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by \begin{equation*}\partial_t h =\fracν{2}Δh + λ((\partial_1 h)^2 - (\partial_2 h)^2) + ν^\frac{1}{2}ξ,\end{equation*} where $ξ$ denotes a noise which is white in both space and time, and $λ$ and $ν$ are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill-posed. It is not possible to linearise it via the Cole-Hopf transformation and the pathwise techniques for singular SPDEs (the theory of Regularity Structures…
The University of Edinburgh（爱丁堡大学）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；CNRS, Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay（法国国家科学研究中心，凡尔赛数学实验室，凡尔赛大学，巴黎萨克雷大学）；Monash University（莫纳什大学）；Beijing Institute of Technology（北京理工大学）
We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic ge…
A cylindrical neural approximation theorem for conditional laws of McKean-Vlasov equations with common noise
KTH Royal Institute of Technology（KTH皇家理工学院）；Digital Futures（数字未来中心）；Grenoble INP – Ensimag（格勒诺布尔国立理工学院-格勒诺布尔国立高等计算机与应用数学学院）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；University of Ljubljana（卢布尔雅那大学）
