infinite-dimensional非线性退化扩散过程生成元的本质m-耗散性
Essential m-dissipativity for generators of infinite-dimensional non-linear degenerate diffusion processes
摘要：
本文证明了无穷维Ornstein-Uhlenbeck算子$N$在位势梯度扰动下，在一个光滑有界函数域$\mathcal{F}C_b^{\infty}$上的第一本质m-耗散性.我们的考虑允许无界扩散算子作为系数。我们得到了Kolmogorov方程解$f$，$α\in (0,\infty)$的相应二阶正则性估计，推广了Da Prato和Lunardi的一些结果.其次证明了无穷维非线性退化扩散过程的生成元$(L_Φ,\mathcal{F}C_b^{\infty})$的本质m-耗散性。我们强调，$(L_Φ,\mathcal{F}C_b^{\infty})$的本质m-耗散性对于应用Beznea，Boboc和Röckner提出的一般预解方法，构造无穷维非线性退化扩散方程的鞅/弱解是有用的.此外，$(L_Φ,\mathcal{F}C_b^{\infty})$和$(N,\mathcal{F}C_b^{\infty})$的基本m-耗散性以及正则性估计对于将Dolbeault，Mouhot，Schmeiser和Grothaus，Stilgenbauer的一般抽象Hilbert空间hypovervity方法分别应用于相应的扩散是必不可少的.
英文摘要：
First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator $N$, perturbed by the gradient of a potential, on a domain $\mathcal{F}C_b^{\infty}$ of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions $f$ of the Kolmogorov equation $αf-Nf=g$, $α\in (0,\infty)$, generalizing some results of Da Prato and Lunardi. Second we prove essential m-dissipativity for generators $(L_Φ,\mathcal{F}C_b^{\infty})$ of infinite-dimensional non-linear degenerate diffusion processes. We emphasize that the essential m-dissipativity of $(L_Φ,\mathcal{F}C_b^{\infty})$ is useful to apply general resolvent methods developed by Beznea, Boboc and Röckner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate diffusion equations. Furthermore, the essential m-dissipativity of $(L_Φ,\mathcal{F}C_b^{\infty})$ and $(N,\mathcal{F}C_b^{\infty})$, as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.
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