Well-posedness and Stationary solutions of McKean-Vlasov (S)PDEs
Well-posedness and Stationary solutions of McKean-Vlasov (S)PDEs
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DOI：
10.1016/j.jmaa.2023.127301
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发表时间：
2022-11
影响因子：
1.3
通讯作者：
Letizia Angeli;Julien Barr'e;Martin Kolodziejczyk;M. Ottobre
中科院分区：
文献类型：
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作者：
Letizia Angeli;Julien Barr'e;Martin Kolodziejczyk;M. Ottobre
本文由两部分组成。在第一部分中，我们考虑McKean-Vlasov偏微分方程（PDE），作为相互作用粒子系统的热力学极限（即在极限N→∞，其中N是粒子数）。众所周知，即使当粒子系统有一个唯一的不变测度（稳态解）时，极限偏微分方程也经常会出现相变：对于某些（系数和）参数值的选择，偏微分方程有一个唯一的稳态解，但随着参数值的变化，会出现多个稳态。在本文的第一部分中，我们添加到这个流的文献，并考虑一个具体的例子的McKean-Vlasov型方程，即Kuramoto模型上的环面扰动的对称双阱势，并表明，这种PDE经历的相变类型刚才所描述的，作为扩散系数是变化的。在本文的第二部分中，我们考虑了一个相当一般的类的McKean-Vlasov偏微分方程的环面（其中包括原始的Kuramoto模型和Kuramoto模型的双阱势的第一部分）扰动（足够强）的无限维加性噪声。据我们所知，所得到的随机偏微分方程，我们称之为随机McKean-Vlasov方程，以前没有研究过，所以我们首先研究它的适定性。然后，我们表明，除了噪声的PDE具有恢复的唯一性的定态的意义上说，无论选择的系数和参数值的McKean-Vlasov PDE，随机McKean-Vlasov PDE总是承认最多一个不变的措施。
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 (ie in the limit N→∞, 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 the first part of this paper, we add to this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. In the second part of the paper, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by (strong enough) infinite-dimensional additive noise. To the best of our knowledge, the resulting Stochastic PDE, which we refer to as the Stochastic McKean-Vlasov equation, has not been studied before, so we first study its well-posedness. We then show that the addition of noise to the PDE has the effect of restoring uniqueness of the stationary state in the sense that, irrespective of the choice of coefficients and parameter values in the McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one invariant measure.