论文标题
里曼尼亚人歧管中的Chernoff近似
Chernoff approximations of Feller semigroups in Riemannian manifolds
论文作者
论文摘要
研究了Chernoff的Chernoff近似值和Riemannian歧管中相关的扩散过程的近似值。假定这些歧管是有界的几何形状，因此包括所有紧凑的歧管和广泛的非压缩歧管。为一类二阶椭圆操作员建立了足够的条件，以在有界几何形状的（通常是非压缩的）歧管上生成一个feller semigroup。这些Feller Semigroup在轮班运营商方面为Chernoff近似建造。这为抛物线方程的初始值问题提供了解决方案的近似值，歧管上具有可变系数。它还产生一系列随机步行序列的弱收敛性，这些序列在歧管上与椭圆发生器相关的扩散过程。对于可行的歧管，该结果特别适用于在歧管上表示布朗运动作为相应随机步行的限制。
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.
