Annie Millet, Pierre-Luc Morien
We study the speed of convergence of the explicit and implicit space-time discretization schemes of the solution $u(t,x)$ to a parabolic partial differential equation in any dimension perturbed by a space-correlated Gaussian noise. The coefficients only depend on $u(t,x)$ and the influence of the correlation on the speed is observed.
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In this paper, we focus on the existence of the density for the law of the solutions to parabolic stochastic partial differential equations with two reflecting walls. The main tool is Malliavin calculus.
In this paper we construct the stationary weak solutions of parabolic SPDEs by a general infinite horizon backward doubly stochastic differential equations (BDSDEs for short) with non-degenerate terminal functions. For this, we first study the existence, uniqueness and stationarity of solutions of such kind of BDSDEs in weighted $L^p(dx)\bigotimes L^2(dx)$ space ($p>2$). Then the corresponding stationary solutions of parabolic SPDEs can be obtained by the connection between the solutions of BDSDEs in weighted $L^p(dx)\bigotimes L^2(dx)$ space and the weak solutions of parabolic SPDEs. This res…
We consider a general class of $L^2$-valued stochastic processes that arise primarily as solutions of parabolic SPDEs on p.c.f. fractals. Using a Kolmogorov-type continuity theorem, conditions are found under which these processes admit versions which are function-valued and jointly continuous in space and time, and the associated Hölder exponents are computed. We apply this theorem to the solutions of SPDEs in the theories of both da Prato--Zabczyk and Walsh. We conclude by discussing a version of the parabolic Anderson model on these fractals and demonstrate a weak form of intermittency.
Sample path properties of parabolic SPDEs with non constant coefficients
研究由非恒定系数抛物型算子及高斯白噪声驱动的SPDE，证明解的存在唯一性，并刻画其时空样本路径正则性。
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.
In this paper we show that the Cahn-Hilliard stochastic SPDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. This is done proving general results on SPDEs with globally Lipschitz coefficients associated with operators on smooth domains of $\mathbb{R}^d$ which are parabolic in the sense of Petrovskii}, and do not necessarily define a semi-group of operators. We study the regularity of the trajectories of the solutions and the absolute continuity of the law at some given time and position.
The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of $L^{p}(Ω\times(0,T),W^{2,p}(G))$-norms of solutions. The Hölder continuity of solutions and their derivatives is also obtained by embedding.
We prove uniqueness in law for a class of parabolic stochastic partial differential equations in an interval driven by a functional A(u) of the temperature u times a space-time white noise. The functional A(u) is Hölder continuous in u of order greater than 1/2. Our method involves looking at an associated system of infinite-dimensional stochastic differential equations and we obtain a uniqueness result for such systems.
We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the "islands," the regions of large values. These results are obtained by analyzing the correlation length of the solution.
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.
The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type $$ \partial_t u=\frac12Δu +σ(u)η\qquad\text{on $(0\,,\infty)\times\mathbb{R}^3$}$$ such that the solution exists and is unique as a random field in the sense of Dalang and Walsh, yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below $3$. En route, it will be proved that there exist a large family of parabolic SPDEs whose moment Lyapunov…
Well-posedness and invariant measure for quasilinear parabolic SPDE on a bounded domain
本文研究了有界区域上带一般乘性噪声的拟线性抛物型随机偏微分方程，在 $L^{1}$ 框架下证明了解的适定性、比较结果与收缩性质，并在特定扩散与噪声条件下证明了不变测度的存在性、唯一性与遍历性。
In this paper, we show that the exponential integrator scheme both in spatial discretization and time discretization for a class of stochastic partial differential equations has a unique stationary distribution whenever the stepsize is sufficiently small, and reveal that the weak limit of the law for the exponential integrator scheme is in fact the counterpart for the stochastic partial differential equation considered.
New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs
本文研究涉及状态及其梯度的抛物型SPDE的零能控性，通过推导新的全局Carleman估计和加权能量型估计，利用不动点论证将线性结果扩展到半线性正/反向抛物型SPDE。
We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.
The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fa…
On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation
研究有界区间上非线性抛物型SPDE解的时空增量，确定了精确的局部和一致模连续性，并推广到开放KPZ方程。
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.
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 …
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving path properties of the solution to a parabolic Cauchy problem in evolution form.
This paper considers second-order stochastic partial differential equations with additive noise given in a bounded domain of $\mathbb R^n$. We suppose that the coefficients of the noise are $L^p$-functions with sufficiently large $p$. We prove that the solutions are Hölder-continuous functions almost surely (a.s.) and that the respective Hölder norms have finite momenta of any order.
In this paper estimates for the uniform norm of solutions of parabolic SPDEs are derived. The result is obtained through iteration techniques, motivated by the work of Moser in deterministic settings. As an application of the main result, solvability of a class of semilinear SPDEs is derived.
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
Almost Sure Asymptotic Stability of Parabolic SPDEs with Small Multiplicative Noise
本文研究确定性Hopf分岔点附近带小乘性噪声的抛物型半线性SPDE平凡解的几乎必然指数渐近稳定性，证明不变测度的存在唯一性，并通过渐近展开近似指数增长率，为稳定性变化提供稳健指标。
The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in \cite{[Ghidaglia-1986]} for deterministic PDEs.
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.
We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + σ(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $σ:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function and $\dot{W}$ denotes space-time white noise. If additionally $σ(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probabili…
In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial d…
In this paper, we address the question of the discretization of Stochastic Partial Differential Equations (SPDE's) for excitable media. Working with SPDE's driven by colored noise, we consider a numerical scheme based on finite differences in time (Euler-Maruyama) and finite elements in space. Motivated by biological considerations, we study numerically the emergence of reentrant patterns in excitable systems such as the Barkley or Mitchell-Schaeffer models.
Consider an infinite system \[\partial_tu_t(x)=(\mathscr{L}u_t)(x)+ σ\bigl(u_t(x)\bigr)\partial_tB_t(x)\] of interacting Itô diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solution under standard regularity assumptions on the nonlinearity $σ$. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the $k$th moment Lyapunov exponent is frequently of sharp order $k^2$, in contrast to the continuous-space stochastic heat equation whose $k$th moment Lyapunov expone…
We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatia…
We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal mono…
We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $ε$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $Δt$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $ε$, in terms of $Δt$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in …
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。
Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise
本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。
Points of slow growth for parabolic SPDEs
本文研究了抛物型随机偏微分方程在时间零附近的慢增长点，证明了存在随机点使得解的波动具有$t^{1/4}$阶量级。
Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs
本文提出了一种预条件技术，用于高阶采样抛物型半线性随机偏微分方程的不变分布，通过设计高阶积分器提高了计算效率。
We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.
In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the $L^2$-generator of a $d$-dimensional Lévy process $X=(X_t)_{t \geq 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $β$-stable process, with $β\in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov…
Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme
本文研究了加性空间时间白噪声下全离散非线性随机偏微分方程的长期行为，证明了其不变测度的近似值与真实测度的收敛率。
A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise
针对带输运噪声的抛物型SPDEs正则性理论中Hölder连续性这一开放问题，本工作在噪声系数空间足够正则的假设下，基于Kunita随机特征线方法结合新先验估计与适配公式，解决该问题并确立拟线性SPDEs全局正则解的存在性。
Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise
针对乘性和加性噪声驱动的半线性抛物型随机偏微分方程，采用有限元空间离散和指数积分器时间离散，在迹类噪声假设下证明了均方L2范数收敛性，并通过二维线性反应扩散方程和非线性随机对流扩散反应方程验证了方法。
We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
In this paper, we aim to study the asymptotic behaviour 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 e…
Efficient simulation of nonlinear parabolic SPDEs with additive noise
本文提出并分析了一种基于线性噪声泛函的简化数值格式，用于高效模拟加性噪声非线性抛物型SPDEs，实现了更高的收敛阶。
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …
Error bounds for full space-time splitting discretizations of semi-linear SPDEs -- with a focus on dG domain decompositions
研究具有乘性噪声的抛物型随机偏微分方程全离散数值格式，用Douglas-Rachford分裂格式整合区域分解并结合间断伽辽金方法进行空间离散，给出强时空收敛结果并经数值实验验证。
Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs
本文研究了SPDEs的数值方案，提出了一种非渐近性一致时间误差界证明方法，分析了非全局Lipschitz非线性SPDEs的数值方案，证明了全隐式和两种驯化方案的误差界，并提供了数值比较。
This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. …
Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs
研究SPDEs弱误差中测试函数正则性的影响，发现即使噪声非退化，控制测试函数导数对收敛速度至关重要，且弱收敛阶可能低于强收敛阶。
Asymptotic error distribution of numerical methods for parabolic SPDEs with multiplicative noise
本文研究带乘性噪声抛物型SPDE数值方法的渐近误差分布，揭示指数Euler方法的极限误差由无穷多个独立Q-Wiener过程驱动的线性SPDE支配，并将其结果推广到全离散格式与空间半离散谱Galerkin方法。
Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime
本文提出一种基于生成元方程的代数方法，证明 Da Prato–Debussche 机制下抛物型 SPDE 不变测度的非高斯性，并覆盖 δ<14/5 的 Φ^4_δ 测度情形。
We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the second power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variable.
We study linear backward stochastic partial differential equations of parabolic type with special boundary condition that connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results for the solutions are obtained.
We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.
Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergod…
Non-local quasilinear singular SPDEs
本文研究二维环面上系数为非局部泛函的拟线性广义抛物型安德森模型，在扩散系数正性假设下，利用抛物控制微分框架建立了局部时间解理论。
Asymptotic error distribution of accelerated exponential Euler method for parabolic SPDEs
针对随机偏微分方程数值方法归一化误差过程是否存在非平凡极限分布的未解问题，本文推导了抛物型方程加速指数Euler方法的渐近误差分布，并建立一致逼近定理克服无穷维困难。
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