论文标题
非线性fokker-planck流动$ l^1（\ mathbb r^d）$
The ergodicity of nonlinear Fokker-Planck flows in $L^1(\mathbb R^d)$
论文作者
论文摘要
在这项工作中，一个人证明了$ l^1（\ Mathbb r^d）$，$ d \ geq 3 $与非线性fokker-planck方程$ _t-u_t-Δβ（u） $（0，\ infty）\ times \ times \ mathbb r^d $，在适当条件下$β：\ mathbb r \ to \ mathbb r $，$ d：\ mathbb r^d \ to \ mathbb r^d $ and $ b：\ mathbb r^d $ and $ b：\ mathbb r \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ to \ mathbb rgodic。特别是，这意味着对相应的McKean-Vlasov随机微分方程的解决方案的时间边际定律的平均成分性。这完成了[7]中关于$ s（t）$的相应omega-set $ω（u_0）$的性质的结果，在$ l^1中的流$ s（t）$（\ mathbb r^d）中的情况下，$ s（t）$没有固定点，因此通用的固定fokker-planck方程没有解决方案。
One proves in this work that the nonlinear semigroup $S(t)$ in $L^1(\mathbb R^d)$, $d\geq 3$, associated with the nonlinear Fokker-Planck equation $u_t-Δβ(u)+\text{div}(Db(u)u){=}0$, $u(0)=u_0$ in $(0,\infty)\times\mathbb R^d$, under suitable conditions on the coefficients $β:\mathbb R\to\mathbb R$, $D:\mathbb R^d\to\mathbb R^d$ and $b:\mathbb R\to\mathbb R$, is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws of the solutions to the corresponding McKean-Vlasov stochastic differential equation. This completes the results established in [7] on the nature of the corresponding omega-set $ω(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb R^d)$ has not a fixed point and so the corresponding stationary Fokker-Planck equation has no solutions.
