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arXiv 2210.13624math.PRmath.AP
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$L^1(\mathbb R^d)$中非线性Fokker-Planck流的遍历性

The ergodicity of nonlinear Fokker-Planck flows in $L^1(\mathbb R^d)$

Viorel Barbu, Michael Röckner

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摘要：

本文证明了与非线性Fokker-Planck方程$u_t-Δβ(u)+\text{div}(Db(u)u){=}0$，$u(0)=u_0$在$(0,\infty)\times\mathbb R^d$中相关联的非线性半群$L^1(\mathbb R^d)$中的$S(t)$，$d\geq 3$，在关于系数$β:\mathbb R\to\mathbb R$，$D:\mathbb R^d\to\mathbb R^d$和$b:\mathbb R\to\mathbb R$的适当条件下，是平均遍历的.特别是，这意味着平均遍历相应的McKean-Vlasov随机微分方程的解的时间边际律。这完成了在[7]中建立的关于在$L^1(\mathbb R^d)$中的流动$S(t)$没有固定点并且因此相应的定态Fokker-Planck方程没有解的情况下$S(t)$的相应ω-集$ω(u_0)$的性质的结果。

英文摘要：

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.

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