论文标题
关于麦基恩的非线性马尔可夫进程
On nonlinear Markov processes in the sense of McKean
论文作者
论文摘要
我们从麦基恩的开创性工作中研究了非线性时间抗均匀的马尔可夫进程[32]。这些被作为法律家族$ \ mathbb {p} _ {s，ζ} $，$ s \ geq 0 $，在路径空间上，其中$ζ$通过$ \ mathbb {r}^d $的一组可允许的初始概率指标运行。在本文中，我们专注于每一个$ \ mathbb {p} _ {s，ζ} $作为麦基恩 - 维拉索夫·SDE解决方案的路径定律，后者仅允许其仅具有可测量的系数，而后者不一定是弱持续的。我们的主要结果是在这种一般的McKean-Vlasov SDES上鉴定了一般和可检查条件，这意味着其解决方案的路径定律构成了非线性马尔可夫过程。我们对非线性马尔可夫财产的概念是麦基恩的精神，但更笼统的是要包含一维时间边际密度的过程，更准确地说，更确切地说，非线性的抛物线PDE，一个非线性的fokker-planck-kolmogorov方程，例如最近在其中的burgers equient和vargers'等方程，以及在其中的杂物，以及杂物的层面层面，并散发出杂物的层面，这是我们的杂物层面和杂物。 $ 2D $涡旋navier-stokes方程和$ p $ laplace方程。在所有这些情况下，相关的McKean-Vlasov SDE都使它们的扩散和漂移系数都奇异地依赖于其解决方案的一维时间边缘。我们强调的是，为了我们的主要结果，非线性的Fokker-Planck-Kolmogorov方程不必得到充分的体积。因此，我们在大型非线性抛物线PDE和非线性马尔可夫过程的解决方案流之间建立一对一的对应关系。
We study nonlinear time-inhomogeneous Markov processes in the sense of McKean's seminal work [32]. These are given as families of laws $\mathbb{P}_{s,ζ}$, $s\geq 0$, on path space, where $ζ$ runs through a set of admissible initial probability measures on $\mathbb{R}^d$. In this paper, we concentrate on the case where every $\mathbb{P}_{s,ζ}$ is given as the path law of a solution to a McKean-Vlasov SDE, where the latter is allowed to have merely measurable coefficients, which in particular are not necessarily weakly continuous in the measure variable. Our main result is the identification of general and checkable conditions on such general McKean-Vlasov SDEs, which imply that the path laws of their solutions form a nonlinear Markov process. Our notion of nonlinear Markov property is in McKean's spirit, but more general in order to include processes whose one-dimensional time marginal densities solve a nonlinear parabolic PDE, more precisely, a nonlinear Fokker-Planck-Kolmogorov equation, such as Burgers' equation, the porous media equation and variants thereof with transport-type drift, and also the very recently studied $2D$ vorticity Navier-Stokes equation and the $p$-Laplace equation. In all these cases, the associated McKean-Vlasov SDEs are such that both their diffusion and drift coefficients singularly depend (i.e. Nemytskii-type) on the one-dimensional time marginals of their solutions. We stress that for our main result the nonlinear Fokker-Planck-Kolmogorov equations do not have to be well-posed. Thus, we establish a one-to-one correspondence between solution flows of a large class of nonlinear parabolic PDEs and nonlinear Markov processes.
