Harnack Inequalities for Stochastic Partial Differential Equations
Harnack Inequalities for Stochastic Partial Differential Equations
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DOI：
10.1007/978-1-4614-7934-5
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发表时间：
2013-08
影响因子：
5.6
通讯作者：
Feng-Yu Wang
中科院分区：
文献类型：
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作者：
Feng-Yu Wang
Harnack不等式的关键是比较偏微分方程正解在两个不同点的值。这个不等式由Harnack[21]于1887年在欧氏空间上的调和函数中引入，并由Serrin[46]于1955年和Moser[34]于1961年推广到椭圆型或抛物型偏微分方程解。在许多其他应用中，Harnack不等式在1986年被Li和Yau[26]用来推导显式热核估计，在1993年被Hamilton[20]用来研究Ricci流的正则性，然后在Perelman对Poincaré猜想的证明中使用了它。然而，所有这些Harnack不等式都是依赖于维度的，因此对于无限维空间上的方程是无效的。在这本书中，我们旨在给出一个完备的Harnack不等式的描述和随机泛函/偏微分方程解的半群的应用。由于相关的Fokker-Planck方程是无穷维空间上的偏微分方程组，所以我们将要研究的Harnack不等式是无量纲的。这与上述经典的Harnack不等式有本质的不同。此外，我们研究的主要工具是一种新的耦合方法(即通过改变测量进行耦合)，而不是偏微分方程组和几何分析文献中通常的最大值原理。
The key point of Harnack’s inequality is to compare values at two different points for positive solutions of a partial differential equation. This inequality was introduced by Harnack [21] in 1887 for harmonic functions on a Euclidean space, and was generalized by Serrin [46] in 1955 and Moser [34] in 1961 to solutions of elliptic or parabolic partial differential equations. Among many other applications, Harnack’s inequality was used by Li and Yau [26] in 1986 to derive explicit heat kernel estimates, and by Hamilton [20] in 1993 to investigate the regularity of Ricci flows, which was then used in Perelman’s proof of the Poincaré conjecture. All these Harnack inequalities are, however, dimension-dependent and thus invalid for equations on infinite-dimensional spaces.In this book we aim to present a self-contained account of Harnack inequalities and applications for the semigroup of solutions to stochastic functional/partial differential equations. Since the associated Fokker–Planck equations are partial differential equations on infinite-dimensional spaces, the Harnack inequalities we are going to investigate are dimension-free. This is essentially different from the abovementioned classical Harnack inequalities. Moreover, the main tool in our study is a new coupling method (ie, coupling by change of measure) rather than the usual maximum principle in the literature of partial differential equations and geometric analysis.