过滤几何
The Geometry of Filtering
摘要：
描述了两个扩散算子（光滑半椭圆型，二阶微分算子）在不同空间上但被光滑映射交织在一起所产生的几何。特殊情况下出现的黎曼淹没时的运营商是拉普拉斯-贝尔特拉米运营商，从等变运营商的全空间的主丛，并为运营商的同胚群所产生的随机流。经典的非线性滤波问题也会导致这种配置。一个基本的工具是，可能的，非线性的“半连接”引起的这一设置，导致一个典型的分解域空间上的运营商。讨论的主题包括：广义Wietzenbock曲率中产生的等变的情况下，斜积分解的扩散过程，条件过程，经典过滤，分解的随机流，和连接所确定的随机微分方程。
英文摘要：
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space of a principal bundle, and for the operators on the diffeomorphism group arising from stochastic flows. Classical non-linear filtering problems also lead to such conffigurations. A basic tool is the, possibly, non-linear "semi-connection" induced by this set up, leading to a canonical decomposition of the operator on the domain space. Topics discussed include: generalised Wietzenbock curvatures arising in the equivariant case, skew -product decompositions of diffusion processes, conditioned processes, classical filtering, decomposition of stochastic flows, and connections determined by stochastic differential equations.
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