论文标题
Liouville定理用于一类傅立叶乘数及其与耦合的联系
The Liouville theorem for a class of Fourier multipliers and its connection to coupling
论文作者
论文摘要
经典的liouville属性说，$ \ mathbb {r}^n $中的所有有限的谐波函数，即满足$Δf= 0 $的所有有限函数都是恒定的。在本文中，我们以傅立叶乘数运算符$ m（d）$的象征获得了必要和足够的条件，因此解决方案$ f $ to $ m（d）f = 0 $是lebesgue a.e. \ constant（如果$ f $有限）或偶然的lebesgue lebesgue a.e.傅立叶乘数类别包括莱维过程的（一般非本地）发电机。对于Lévy流程的发电机，我们获得了强大的Liouville定理的必要条件，在该定理中，$ f $是积极的，最多可以迅速增长。作为上述结果的应用，我们证明了时空lévy过程的耦合结果。
The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes.
