Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces
Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces
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DOI：
10.1016/j.jfa.2008.05.005
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发表时间：
2009-02
影响因子：
1.7
通讯作者：
V. Bogachev;G. Prato;M. Röckner
中科院分区：
文献类型：
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作者：
V. Bogachev;G. Prato;M. Röckner
我们考虑Hilbert空间H中的Kolmogorov算子L0，它与定义在适当的正则函数空间上的具有含时奇异拟耗散漂移F=F(t，⋅)：H→H的随机偏微分方程解有关。我们证明了L0在空间Lp([0，T]×H；ν)，p⩾1中本质上是m-耗散的，其中ν(dt，dx)=νt(Dx)dt，[公式：见正文]是由L0给出的福克-普朗克方程的解。因此，L0的闭包生成一个马尔可夫C0-半群。我们还证明了奇异漂移F的Fokker-Planck方程解的唯一性，并给出了它在含时变反应项反应扩散方程中的应用。这一结果推广了[V.Bogachev，G.Da Prato，M.Rökerner，弱抛物方程的解的存在性》中讨论的有限维情形，Proc.伦敦数学。SoC。(3)88(2004)753-774]，[V.Bogachev，G.Da Prato，M.Rökerner，On Parteric Eques for Measures，Comm.[V.Bogachev，G.Da Prato，M.Rökerner，W.Stannat，W.Stannat，弱抛物型方程解的唯一性，Bull.伦敦数学。SoC。39(2007)631-640]到无限维度。
We consider a Kolmogorov operator L0in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift F=F(t,⋅):H→H, defined on a suitable space of regular functions. We show that L0is essentially m-dissipative in the space Lp([0,T]×H;ν), p⩾1, where ν(dt,dx)=νt(dx)dt and the family [Formula: see text] is a solution of the Fokker–Planck equation given by L0. As a consequence, the closure of L0generates a Markov C0-semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions.