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<p>We prove that a nonlinear semigroup <i>S</i>(<i>t</i>) in <i>L</i><sup>1</sup>(ℝ<sup><i>d</i></sup>)<i>, d ≥</i> 3<i>,</i> associated with the nonlinear Fokker–Planck equation <i>u</i><sub><i>t</i></sub><i>−</i>∆<i>β</i>(<i>u</i>)+div(<i>Db</i>(<i>u</i>)<i>u</i>) = 0<i>, u</i>(0) = <i>u</i><sub>0</sub><i>,</i> in (0<i>,</i>∞)× ℝ<sup><i>d</i></sup><i>,</i> with suitable conditions imposed on the coefficients <i>β</i> : ℝ → ℝ<i>, </i><i>−D</i> : ℝ<sup><i>d</i></sup> → ℝ<sup><i>d</i></sup><i>,</i> and <i>b</i> : ℝ → ℝ<i>,</i> is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu and M. Röckner, “The invariance principle for nonlinear Fokker–Planck equations,” <i>J. Different. Equat.</i>, <b>315</b>, 200–221 (2022)] on the nature of the corresponding omega-set ω (<i>u</i><sub>0</sub>) for <i>S</i>(<i>t</i>) in the case where the flow <i>S</i>(<i>t</i>) in <i>L</i><sup>1</sup>(ℝ<sup><i>d</i></sup>) does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions.</p>

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On the Ergodicity of Nonlinear Fokker–Planck Flows in L1(ℝd)

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Viorel Barbu,
  • Michael Röckner

摘要

We prove that a nonlinear semigroup S(t) in L1(ℝd), d ≥ 3, associated with the nonlinear Fokker–Planck equation ut−∆β(u)+div(Db(u)u) = 0, u(0) = u0, in (0,∞)× ℝd, with suitable conditions imposed on the coefficients β : ℝ → ℝ, −D : ℝd → ℝd, and b : ℝ → ℝ, is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu and M. Röckner, “The invariance principle for nonlinear Fokker–Planck equations,” J. Different. Equat., 315, 200–221 (2022)] on the nature of the corresponding omega-set ω (u0) for S(t) in the case where the flow S(t) in L1(ℝd) does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions.