Pascal Azerad, Mohamed Mellouk
In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a space-time white noise in dimension one. The equation we consider may also include a reaction term.
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We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the Bak-Tang-Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to the stochastic sign fast diffusion equation, previously obtained in a limiting sense only. Aside from generalizing the SVI approach to stochastic fast diffusion equations we develop a new proof of well-p…
We correct a few errors that appeared in [Convergence of invariant measures for singular stochastic diffusion equations, Stochastic Process. Appl. 122 (2012), no. 4, 1998--2017] by I. Ciotir and J.M. Tölle.
Stochastic and deterministic non-autonomous reaction-diffusion equations
本文研究了具有多项式反应项的非自治确定性与随机反应扩散方程，证明了其适定性，并得出了非自治随机卷积时空正则性的新结果。
In this short article we present new results that bring about hitherto unknown relations between certain Bernstein diffusions wandering in bounded convex domains of Euclidean space on the one hand, and processes which typically occur in forward-backward systems of stochastic differential equations on the other hand. A key point in establishing such relations lies in the fact that the Bernstein diffusions we consider are actually reversible Itô diffusions.
In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is established to describe the large time evolution, distinguishing between certain death or possible life of the population.
Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.
We study the limit of a kinetic evolution equation involving a small parameter and perturbed by a smooth random term which also involves the small parameter. Generalizing the classical method of perturbed test functions, we show the convergence to the solution of a stochastic diffusion equation.
Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely Hölder continuous in both space and time variables.
This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.
We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.
We consider stochastic reaction-diffusion equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spac…
This paper studies the limit of a kinetic evolution equation involving a small parameter and driven by a random process which also scales with the small parameter. In order to prove the convergence in distribution to the solution of a stochastic diffusion equation while removing a boundedness assumption on the driving random process, we adapt the method of perturbed test functions to work with stopped martingales problems.
We establish the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on a three dimensional periodic domain to the dynamic $Φ^4_3$ model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular $C^9$ is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their $L^p$ norms in terms of the graphs of the standard $Φ^4_3$ stochastic terms.
We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations with a polynomially growing quasi-monotone nonlinearity and multiplicative Poisson noise. We also study existence and uniqueness of invariant measures for the associated semigroup in the Markovian case. A key role is played by a new maximal inequality for stochastic convolutions in $L_p$ spaces.
Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic non- linear convection-diffusion equation which we explicitly derive in terms of appropriated functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable ge…
Martingale Solutions of Fractional Stochastic Reaction-Diffusion Equations Driven by Superlinear Noise
本文证明了一类具有任意阶多项式增长伪单调漂移和超线性扩散项的随机方程鞅解的存在性，并应用于分数阶随机反应扩散方程，利用伪单调技术和Skorokhod-Jakubowski表示定理通过Galerkin近似取极限。
A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u + f(u) + σ(u) \dot{W}$ on a bounded spatial domain never explode. We consider the case where $σ$ grows polynomially and $f$ is polynomially dissipative, meaning that $f$ strongly forces solutions toward finite values. This result demonstrates the role that the deterministic forcing term $f$ plays in preventing explosion.
Mathematical analysis of a stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises
本文对一个带有趋食性和噪声的捕食者-猎物非线性随机反应扩散系统进行数学分析，通过 Faedo-Galerkin 逼近等方法证明了弱鞅解的存在性与非负性，并利用对偶技术证明了解的唯一性。
We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.
Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: an entropy formulation
本文通过为随机变分不等式引入新熵公式，建立带保守梯度噪声的退化非线性扩散方程障碍问题的存在性、唯一性和稳定性结果，并得到带非线性反应项的确定性多孔介质方程障碍问题的新适定性结果。
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptoti…
A nonlinear stochastic diffusion-convection equation with reflection
本文研究了任意空间维度下具有反射的非线性伪单调随机扩散-对流演化问题，证明了带非负初值的方程解及防止解取负值的反射测度的存在性，并通过研究抛物容度下解的拟处处定义代表元证明了测度的极小性条件。
Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions
本文研究二氧化硫与碳酸钙石材化学反应驱动的半直线非线性反应扩散系统，边界条件为Jacobi过程，通过分裂策略证明了温和解的全局存在性与轨道唯一性。
Convergence towards the population cross-diffusion system from stochastic many-particle system
本文研究随机多粒子系统向非局部种群交叉扩散系统的严格收敛，借助代数尺度磨光的中间粒子系统，在概率意义下通过停止过程与 Gronwall 型估计完成证明。
In this Note, we present a Calderón-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.
It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.
In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions.
We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process is a reflected non-standard Brownian motion. Beyond the result, this problem is known as a prototype of non-translation invariant problem making the usual method of the "environment as seen from the particle" inefficient.
We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are Hölder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. Our proof makes use of a probabilistic interpretation of the dissipation of solutions to the advection diffusion equation.
We prove that the solutions to fast diffusion stochastic porous media equations have finite time extinction with strictly positive probability.
We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems …
Pathwise stochastic control and a class of stochastic partial differential equations
本文研究路径随机最优控制问题，证明值过程是相关Hamilton-Jacobi-Bellman随机偏微分方程在粘性意义下的唯一解，并讨论随机路径Noether定理的相应版本。
We extend the result on the stability of travelling waves for stochastic Nagumo equations in [St] to general bistable reaction-diffusion equations with both additive and multiplicative noise, using a variational approach based on functional inequalities. Our analysis yields explicit estimates on the rate of stability that can be shown in special examples to be optimal. [St] Stannat, W., Stability of travelling waves in stochastic Nagumo equations, arXiv:1301.6378, 2013.
We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$α$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic porous media equations and stochastic $p$-Laplace equations.
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with …
In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.
Macroscopic limit for stochastic chemical reactions involving diffusion and spatial heterogeneity
本文证明在高度异质环境中，一级化学反应系统的反应-扩散主方程经仔细离散化后，其宏观极限恢复为具有不均匀扩散和反应速率的宏观反应-扩散系统。
We prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem posed and left open in several works: [Barbu, Da Prato, Röckner, CMP, 2009], [Barbu, Da Prato, Röckner, CRMAS, 2009], [Barbu, Röckner, CMP, 2012], [Barbu, Da Prato, Röckner, JMAA, 2012], [Röckner, Wang, JMLS, 2013], [Barbu, MMAS, 2013]. The highly singular-degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the…
In this article, we study the unique determination of convection term and the time-dependent density coefficient appearing in a convection-diffusion equation from partial Dirichlet to Neumann map measured on boundary.
We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach spaces. The drift contains a bounded Hölder term; in spite of this, due to the space-time white noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of the associated Kolmogorov equation. The model includes examples not covered by the previous works based on Hilbert spaces or concrete SPDEs.
The aim of this paper is to give a stochastic representation for the solution to a natural extension of the Caputo-type evolution equation. The nonlocal-in-time operator is defined by a hypersingular integral with a (possibly time-dependent) kernel function, and it results in a model which serves a bridge between normal diffusion and anomalous diffusion. We derive the stochastic representation for the weak solution of the nonlocal-in-time problem in case of nonsmooth data. We do so by starting from an auxiliary Caputo-type evolution equation with a specific forcing term. Numerical simulations …
Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise
本文研究了有色噪声驱动的随机反应扩散方程，在有界一致连续函数的 Banach 空间上，针对相应转移半群的平稳与发展问题，证明了依赖于噪声颜色的 Schauder 型估计。
Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems
本文研究了 Hilbert 空间中随机时间分数阶扩散-波方程的适定性，建立了初边值问题随机弱解的存在性、唯一性与正则性估计，并将其理论应用于反源问题，证明了边界测量条件下的唯一性。
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.
Weak mean random attractors for non-local random and stochastic reaction-diffusion equations
本文研究了非局部随机和随机反应扩散方程的弱均随机吸引子存在性，证明了解的存在性与唯一性及吸引子的性质。
In this paper we establish the meta-stability of travelling waves for a class of reaction-diffusion equations forced by a multiplicative noise term. In particular, we show that the phase-tracking technique developed in [hamster2017,hamster2020] can be maintained over timescales that are exponentially long with respect to the noise intensity. This is achieved by combining the generic chaining principle with a mild version of the Burkholder-Davis-Gundy inequality to establish logarithmic supremum bounds for stochastic convolutions in the critical regularity regime.
We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains $ Ω\subset \mathbb R^d$. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of $Ω$ which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, $-\frac{1}{ρ_\infty}\,\nabla\cdot ρ_\infty\mathbb D\nabla$. The proofs are based on the method developed in [29] for quantum confinement on bounded domains in $\mathbb R^d$. In particular for stochastic confinement we combine the Liouville prope…
Stochastic solutions and singular partial differential equations
本文提出将随机解技术推广至分布值噪声驱动的偏微分方程，作为其求解方法。
We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main…
We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (…
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems.
In this paper, a pointwise weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in studying the controllability, observability and inverse problems for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce all the known Carleman estimates and observability results, for some deterministic partial differential equations, stochastic heat equations, stochastic Schrödinger equations and stochastic transport equations. Meanwhile, as its new applicati…
Fluctuation analysis for particle-based stochastic reaction-diffusion models
本文针对基于粒子的随机反应扩散模型，推导并证明了平均场极限的下一阶涨落修正，其满足带高斯噪声的随机PIDE，数值示例表明该修正可准确估计高阶统计量。
This paper is concerned with the existence of traveling wave solutions for diffusive two-species Lotka-Volterra systems with delay in both the reaction and diffusion terms without monotonicity. We extend the partial or cross monotone iteration method to systems that satisfy the partial quasi-monotone condition via construction appropriate upper and lower solutions. This is done by using Schauder's fixed point theorem.
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states (invariant measures) is considered for one example of stochastic partial differential equations.
Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems
本文利用随机冻结方法对柱形区域上反应扩散方程的行波解进行小噪声展开，严格定义了波速的随机修正，并可在噪声强度中以任意阶数计算修正量。
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