Peter K. Friz, Paul Gassiat, Pierre-Louis Lions, Panagiotis E. Souganidis
We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.
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Pathwise regularization by noise for semilinear SPDEs driven by a multiplicative cylindrical Brownian motion
本文研究乘性圆柱布朗运动驱动的半线性SPDE，通过无穷维推广、极大正则性分析和Volterra-sewing技术，证明了奇异扩散系数下的噪声正则化现象。
We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.
This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical pr…
We prove pathwise nonuniqueness in the stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by Mueller, Mytnik and Perkins [Ann. Probab. 42 (2014) 2032-2112], the solutions of the present SPDEs are assumed to be nonnegative and have very different properties including uniqueness in law. In proving possible separation of solutions, we derive delicate properties of certain correlated approximating solutions, which is based on a novel coupling method called continuous decomposition. I…
The finite time blowup in the almost sure sense of a class of space-time fractional stochastic partial differential equations is discussed. Both the cases of white noise and colored noise are considered. The sufficient and necessary condition between the blowup and Osgood condition is obtained when the spatial domain is bounded. And the sufficient condition for the blowup is obtained when the spatial domain is the whole space. The results in this paper could be regarded as extensions to some results in Foondun and Nualart, 2021.
The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution t…
Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise
本研究借助粗糙路径理论提出通用框架，推导小乘性线性噪声扰动的随机偏微分方程的路径中心极限定理与中偏差原理，给出最优收敛速度，并应用于多类典型随机偏微分方程。
Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise
本文研究由对称 Lévy 噪声驱动的椭圆型随机偏微分方程，利用 Green 函数证明随机场解的存在唯一性，并将其推广到谱分数阶拉普拉斯算子，给出解具有连续修正的充分条件。
We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.
Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasiline…
Pathwise Holder convergence of the implicit Euler scheme for semi-linear SPDEs with multiplicative noise
本文证明在UMD Banach空间中，乘性噪声半线性抛物型随机微分方程的隐式欧拉格式具有最优路径wise Hölder收敛速率，并应用于乘性时空白噪声驱动的二阶抛物型SPDEs。
From rank-based models with common noise to pathwise entropy solutions of SPDEs
本文研究带公共噪声的秩基模型的平均场极限，证明经验累积分布函数收敛于随机偏微分方程的解，并建立鞅问题解与逐路径熵解之间的等价关系。
A pathwise approach to semilinear SPDEs with Lévy drivers
本文提出一种路径方法，将鲁棒粘性解推广到 Lévy 驱动的半线性粗糙偏微分方程，建立适定性与流性质，并给出基于粗糙 BSDE 的随机表示，为相应 SPDE 提供路径解释。
In this paper we establish the strong existence, pathwise uniqueness and a comparison theorem to a stochastic partial differential equation driven by Gaussian colored noise with non-Lipschitz drift, Hölder continuous diffusion coefficients and the spatial domain in finite interval, $[0,1]$, and with Dirichlet, Neumann or mixed nonhomogeneous random conditions imposed on the endpoints. The Hölder continuity of the solution both in time and in space variables is also studied.
This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffus…
In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self…
This paper is devoted to the study of hyperbolic systems of linear partial differential equations perturbed by a Brownian motion. The existence and uniqueness of solutions are proved by an energy method. The specific features of this class of stochastic partial differential equations are highlighted and the comparison with standard existence results for SPDEs is discussed. The small perturbations problem is studied and a large deviation principle is stated. A pathwise approximation result, similar to the stochastic differential equations case, is established, with an application to a support t…
In this paper we further study the stochastic partial differential equation first proposed by Xiong (2013). Under localized conditions on the coefficients we show that the solution is in fact distribution-function-valued and we establish the pathwise uniqueness of the solution. As applications we obtain the well-posedness of the martingale problems for two classes of measure-valued diffusions: interacting super-Brownian motions and interacting Fleming-Viot processes. Properties of the two superprocesses such as the existence of density fields and the extinction behaviors are also studied.
We consider stochastic partial differential equations on $\mathbb{R}^{d}, d\geq 1$, driven by a Gaussian noise white in time and colored in space, for which the pathwise uniqueness holds. By using the Skorokhod representation theorem we establish various strong stability results. Then, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and uniqueness of solutions as well as the continu…
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's le…
Rough Burger-like SPDEs
本文提出了一种在低正则性区域下，通过改进的粗糙路径方法，建立Burgers型随机偏微分方程路径存在性和唯一性的理论框架。
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approac…
Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x) +\bigl|u(t,x)\bigr|^γ\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<γ<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently …
Invariant Manifolds and Stability for Singular SPDEs
本文利用正则性结构理论构造奇异随机偏微分方程的随机动力系统，结合路径估计与乘性遍历定理分析Lyapunov谱，证明驻点附近存在局部稳定、不稳定和中心不变流形，并在线性化指数稳定时得到非线性方程的局部指数稳定性。
Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
研究由乘性噪声驱动、具粗糙初始数据的多维半线性随机演化方程，在系数全局Lipschitz时建解的适定性等，局部Lipschitz时证最大局部解相关性质，条件针对特定范数，可用于多种方程及处理奇异和非光滑初始数据。
Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes
本文在变分框架下利用 Kurtz 结果，将 Yamada-Watanabe 理论推广到由 Wiener 过程与 Poisson 随机测度共同驱动的非线性 SPDE，证明鞅解存在加轨道唯一性可推出唯一强解存在。
Polynomial interacting particle systems and non-linear SPDEs for market capitalization curves
该研究扩展了Fernholz等人的波动率稳定市场模型，引入共同噪声项构建多项式市场模型，证明N公司系统经验测度的极限收敛于退化非线性SPDE的唯一解，给出其McKean-Vlasov SDE条件概率表示及相关性质，并提供数值实现。
Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application
本文针对具有完全局部单调系数的随机偏微分方程，通过Faedo-Galerkin方法等建立Wong-Zakai逼近，并应用于确定解的拓扑支集，涵盖多种流体模型。
A BSDEs approach to pathwise uniqueness for stochastic evolution equations
本文利用无穷维正倒向随机微分方程系统，证明了一类Hilbert空间中漂移项为Hölder连续的随机发展方程的路径唯一性，并证明解对初值具有Lipschitz依赖性。
Controlled rough SDEs, pathwise stochastic control and dynamic programming principles
本文研究粗糙随机微分方程的最优控制，建立路径动态规划原理与HJB稳定性，无需布朗统计，统一不同控制类别。
In this paper, we consider the modified two-component Camassa-Holm System with multiplicative noise. For these SPDEs, we first establish the local existence and pathwise uniqueness of the pathwise solutions in Sobolev spaces $H^{s}\times H^{s}, s>\frac{3}{2}$. Then we show that strong enough noise can actually prevent blow-up with probability 1. Finally, we analyse the effects of weak noise and present conditions on the initial data that lead to the global existence and the blow-up in finite time of the solutions, and their associated probabilities are also obtained.
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques …
Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation
该研究以粗糙Heston模型为示例，证明局部随机粗糙波动率下条件密度SPDE的Itô-Wentzell约化仍成立，得到路径式Fokker-Planck公式，纯粗糙Heston情形下条件密度为显式对数正态形式，关联Rao-Blackwell化校准。
Nonlinear SPDE driven by Levy noise: Well-posedness, optimal control and invariant measure
针对乘性Lévy噪声驱动的非线性SPDE，论文结合紧性、Skorokhod表示与压缩法证明强解唯一存在，并通过变分法获得最优控制，同时证明无控制方程的不变测度存在。
Surface finite element approximation of parabolic SPDEs with Whittle--Matérn noise
本文提出并分析了一类带加性噪声的线性抛物型随机发展方程的全离散曲面有限元逼近方法，推导了强收敛率和路径wise收敛率，并通过数值实验验证了结果。
The Neumann problem for fully nonlinear SPDE
本文将逐路粘性解概念推广至带 Neumann 边界条件的全非线性随机偏微分方程，在区域凸性假设下证明了比较定理从而确立解的存在唯一性及对噪声的连续性，并应用于研究柱状区域中随机扰动平均曲率流的长时间行为。
Non-uniqueness for the stochastic incompressible Euler equations with a passive tracer
本文研究了随机不可压缩欧拉方程中带被动示踪剂的路径非唯一性现象，通过经典变换将SPDE转化为具有随机系数的PDE，并利用De Lellis和Székelyhidi Jr.开发的Baire类别方法构造了无限多个全局弱解，最终应用于三维随机理想磁流体动力学方程。
A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear Sensitivity
本文研究一维环面上带加性噪声与非线性敏感度的化学排斥SPDE，证明其全局唯一解、马尔可夫半群、唯一不变测度及指数收敛，并给出比高斯更重的尾界。
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spa…
Strong Stability Preservation for Stochastic Partial Differential Equations
该研究将确定性 SSP 概念推广至随机情形，以获得 SDE 与 SPDE 的无条件有界、逐路径非线性稳定数值解，并指出其在单调性数据同化中的建模价值。
In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough paths theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem, which is stable under a large class of perturbations, includ…
The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this paper), we prove a general existence and compactness theorem for $C^k$-cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with …
Motivated by the work of Musiela and Zariphopoulou \cite{zar-03}, we study the Itô random fields which are utility functions $U(t,x)$ for any $(ω,t)$. The main tool is the marginal utility $U_x(t,x)$ and its inverse expressed as the opposite of the derivative of the Fenchel conjuguate $\tU(t,y)$. Under regularity assumptions, we associate a $SDE(μ, σ)$ and its adjoint SPDE$(μ, σ)$ in divergence form whose $U_x(t,x)$ and its inverse $-\tU_y(t,y)$ are monotonic solutions. More generally, special attention is paid to rigorous justification of the dynamics of inverse flow of SDE. So that, we are a…
Strong pathwise solutions for a class of stochastic thermo-magneto-hydrodynamic-type systems with multiplicative noise
本文研究了一类带有乘性噪声的随机热磁流体动力学型系统的强解存在性和唯一性，通过Galerkin近似和紧致性论证，证明了在二维空间中解的全局性。
Milstein-type Schemes for Hyperbolic SPDEs
研究乘性高斯噪声驱动的双曲型半线性随机发展方程的时间逼近，通过Milstein型格式获得路径一致强误差的最优收敛阶，将结果从抛物型推广到双曲型SPDE。
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of t…
Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs
本文证明驯化指数欧拉格式对超线性双曲型随机偏微分方程具有1/2阶强收敛率，将结果从全局推广至局部Lipschitz非线性，并应用于多种非线性方程。
We study a generalized 1d periodic SPDE of Burgers type: $$ \partial_t u =- A^θu + \partial_x u^2 + A^{θ/2} ξ$$ where $θ> 1/2$, $-A$ is the 1d Laplacian, $ξ$ is a space-time white noise and the initial condition $u_0$ is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when $θ>1/2$. When $θ>5/4$ we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of t…
This article is a sequel to [M.Z.Z.1] aimed at completing the characterization of the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Stationary solution are viewed as random points in the infinite-dimensional state space, and the characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's …
Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
本文证明半线性SPDE时间离散格式路径一致收敛的最优速率，覆盖指数Euler、隐式Euler和Crank-Nicolson等方法，并应用于Maxwell、Schrödinger和波动方程，首次给出后两种格式的结果。
We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence.
In this article, we consider the following class of stochastic partial differential equations (SPDE): \begin{equation*} \left\{\begin{aligned}\mathrm{d} \mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d} t+\mathrm{B}(t,\mathbf{X}(t))\mathrm{d}\mathrm{W}(t)+\int_{\mathrm{Z}}γ(t,\mathbf{X}(t-),z)\widetildeπ(\mathrm{d} t,\mathrm{d} z),\; t\in[0,T],\\ \mathbf{X}(0)&=\boldsymbol{x} \in \mathbb{H},\end{aligned} \right.\end{equation*} with fully locally monotone coefficients in a Gelfand triplet $\mathbb{V}\subset \mathbb{H}\subset\mathbb{V}^*$, where the mappings \begin{align*} \mathrm{A}:[0,T]\ti…
Average dissipation for stochastic transport equations with Lévy noise
本文证明带纯跳跃 Lévy 输运噪声的 Marcus 型 SPDE 的平均解满足含分数阶 Laplace 型算子的确定性耗散方程，并给出数值模拟，将其视为迈向非局部增强耗散的第一步。
We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $(S(t,s))_{0\leq s\leq T}$ is a $C_0$-evolution family of contractions on a $2$-smooth Banach space $X$ and $(W_t)_{t\in [0,T]}$ is a cylindrical Brownian motion on a probability space $(Ω,P)$, then for every $0<p<\infty$ there exists a constant $C_{p,X}$ such that for all progressively measurable processes $g: [0,T]\times Ω\to X$ the process $(\int_0^t S(t,s)g_sdW_s)_{t\in [0,T]}$ has a continuous modification and $$E\sup_{t\in [0,T]…
In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by \begin{equation*}\partial_t h =\fracν{2}Δh + λ((\partial_1 h)^2 - (\partial_2 h)^2) + ν^\frac{1}{2}ξ,\end{equation*} where $ξ$ denotes a noise which is white in both space and time, and $λ$ and $ν$ are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill-posed. It is not possible to linearise it via the Cole-Hopf transformation and the pathwise techniques for singular SPDEs (the theory of Regularity Structures…
Global Energy Solutions for Normalized Linear Stochastic Schr{ö}dinger Equations
本文研究马尔可夫开放量子系统中的非线性薛定谔方程，在能量空间中证明线性方程的全局适定性，并在放宽单调性假设下证明非线性SPDE解的不可区分性，最终通过路径唯一性和归一化建立非线性随机方程的适定性。
We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$\mathrm{d} X= u(ω,t,X)\, \mathrm{d} t + \frac12 σ(ω,t,X)σ'(ω,t,X)\,\mathrm{d} t + σ(ω,t,X) \, \mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $σ$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(Ω; L^\infty([0,T];\dot{H}^1(\mathbb{R})))$ for any finite $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ as $t\d…
Anomalous properties of 2D active scalars perturbed by rough transport noise
研究Kraichnan型粗糙输运噪声驱动的二维活性标量SPDE，证明反常正则化与可积性，并在Euler情形建立涡度反常耗散。
Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise
本文证明了受传输噪声扰动的3D分数阶Navier--Stokes方程在无外力情况下存在无穷多个全局Leray--Hopf解，首次实现了该类方程的随机扰动非唯一性结果，其证明基于凸积分技术与流变换。
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