R. Vilela Mendes
Stochastic solutions not only provide new rigorous results for nonlinear pde's but also, through its local non-grid nature, are a natural tool for parallel computation. There are two methods to construct stochastic solutions: the McKean method and superprocesses. Here a comparison is made of these two approaches and their strenghts and limitations are discussed.
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Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: MacKean's and superprocesses. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE's. A new class of superprocesses, namely superprocesses on signed measures and on distributions, is proposed to extend the stochastic solution approach to a wider class of PDE's.
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: McKean's and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE's. A new class of superprocesses, namely superprocesses on ultradistributions, is proposed to extend the stochastic so…
Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs
本文研究与正则化非保守路径依赖非线性抛物型McKean偏微分方程相关的带消亡路径依赖McKean-Vlasov SDE，证明强解的存在唯一性及次概率律正则性，其密度为PDE弱解，并讨论关联粒子系统适定性。
This paper is composed of two parts. In the first part we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic limits of interacting particle systems (i.e. in the limit $N\to\infty$, where N is the number of particles). It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. In …
Generalized Mean-Reflected McKean--Vlasov BSDEs: Well-Posedness and a Markovian PDE Representation
本文研究了一类广义均值反射麦肯-瓦拉斯型反向随机微分方程，结合均值反射约束与广义积分，证明了解的存在性和唯一性。
McKean-Vlasov stochastic equations with Hölder coefficients
本工作针对Chaudru de Raynal确立的带Hölder系数的非退化McKean-Vlasov随机微分方程适定性问题，通过利用一致抛物型PDE的标准高斯估计给出精简直接证明，绕开测度参数求导需求并扩展至更弱假设下的亚椭圆型PDE。
Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations
本文讨论了路径依赖麦肯PDE的两种概率解释及对应的粒子近似方法，并分析了其在求解PDE中的应用。
In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which we associate a random function which is proved to converge to a solution of a regularized version of a nonlinear PDE.
Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs
本文通过将非线性PDE转化为非线性福克-平面克方程，构造了无限多个概率对应物，证明了麦肯恩意义下的非线性马尔可夫过程不唯一由一维时间边缘决定，且展示了其在二维时间边缘下的唯一性。
McKean-Vlasov PDE with Irregular Drift and Applications to Large Deviations for Conservative SPDEs
本文精细分析了具有奇异相互作用和平方根漂移项的McKean-Vlasov PDE，在随机重整化动力学解框架下建立了Dean-Kawasaki方程的小噪声大偏差，并应用于涨落Ising-Kac-Kawasaki动力学，揭示了宏观涨落理论与涨落流体力学间的一致性。
We present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. The analysis hinges on sharp gradient to time-inhomogeneous parabolic PDEs bounds. These bounds may be of independent interest. They extend the classical results of Kusuoka \& Stroock. Both algorithms are tested through two numerical examples.
This article is a continuation of our first work \cite{chaudruraynal:frikha}. We here establish some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is…
In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the results obtained for nonlinear PDE of McKean-Vlasov type generated by stable-like processes in the previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.
In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin's transformation \cite{zvonkin\_transformation\_1974} and so on the regularization properties of the associated PDE, which is stated on the space $[0,T]\times \R^d\times \mathcal{P}\_2(\R^d)$, where $T$ is a positive number, $d$ denotes the dimension equation and $\mathcal{P}\_2(\R^d)$ is the space of probability measures on $\R^d$ with finite second order moment. Especi…
Approximation of non-linear SPDEs with additive noise via weighted interacting particles systems: the stochastic McKean-Vlasov equation
本文研究通过相互作用粒子系统逼近带加性噪声的非线性SPDE问题，提出用带时变共噪权重的粒子加权经验测度得到随机McKean-Vlasov方程，补充了SPDE粒子表示与加权经验测度时变拓展相关研究。
In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time marginals of its probability distribution in a singular way. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the McKean-Vlasov equation unde…
Integrability properties and stochastic McKean-Vlasov dynamics with singular Lennard-Jones drift: a mesoscale regularization
本文研究了具有奇异Lennard-Jones漂移的随机 McKean-Vlasov 动力学，通过介观尺度正则化证明了粒子系统经验测度的收敛性及解的可积性性质。
On nonlinear Markov processes in the sense of McKean
本文研究McKean意义下的非线性时间非齐次Markov过程，针对系数仅可测且对测度变量不弱连续的McKean-Vlasov SDE，给出可检验条件使其解的路径分布构成非线性Markov过程，并建立其与一大类非线性抛物型PDE解流的一一对应。
We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed. As the number of processes in the system becomes large, the empirical measure of the population converges to the solution of a non-linear stochastic heat equation with Dirichlet boundary condition. The diffusion coefficients are allowed to have finitely many discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of solutions to the limiting stochastic PDE. As a corollary we obtain a represe…
Superpositions for General Conditional Mckean-Vlasov Stochastic Differential Equations
本文建立一般条件McKean-Vlasov随机微分方程、非线性Zakai方程与无穷维条件Fokker-Planck方程之间的叠加原理，使三者的适定性可相互蕴含。
New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes
本文证明了一类与非线性抛物方程（含p-Laplacian和分数阶p-Laplacian）相关联的带跳McKean-Vlasov过程的存在性，通过重写为FPKE并利用新的非线性叠加原理解决鞅问题，最终得到弱解。
We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.
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…
It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with some additional work, one can prove finite time die-out for two types of systems of stochastic differential equations on the lattice Z^d. Our first system involves the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When gamma = 1/2, the measure which puts mass u(t,x) at x is a super-random walk and it is well-known that the process becomes extinct i…
On gradient stability in nonlinear PDE models and inference in interacting particle systems
本文研究非线性PDE模型中的梯度稳定性问题，并通过两个例子展示了如何利用隐函数定理验证稳定性条件及统计可识别性。
Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems
研究具有非局部算子Ψ(-Δ)的非线性Fokker-Planck方程的温和解存在性与分布解唯一性，并证明对应非线性鞅问题的解存在唯一，进而构造McKean意义下的非线性马尔可夫过程。
Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
本文通过分支扩散过程表示半线性抛物型PDE，并利用自动微分技术实现非线性梯度的处理，为高维非线性PDE提供了具有误差估计的蒙特卡洛数值方法。
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