Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

We study 2d and 3d gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss Y and W freedoms and change of slicing in the solution space. For D dimensional case we find D + 1 surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the 3d case there exists an integrable slicing where charge algebra takes the form of Heisenberg ⊕ A3 where A3 is two copies of Virasoro at Brown-Henneaux central charge for AdS3 gravity and BMS3 for the 3d flat space gravity.

We discuss some applications of an intrinsic multipication in the space of simple loops in a surface.

In this paper we will show the Mumford goodness of the metrics on the logarithmic tangent bundle of the moduli spaces of curves induced by the Ricci and the perturbed Ricci metrics. It follows from these estimates that the Ricci metric can be extended to the Deligne-Mumford compactification of the moduli spaces.

Local polynomial fitting has many exciting statistical properties which where established under i.i.d. setting. However, the need for nonlinea r time series modeling, constructing predictive intervals, understanding divergence of nonlinear time series requires the development of the theory of local polynomial fitting for dependent data. In this paper, we study the problem of estimating conditional mean functions and their derivatives via a local polynomial fit. The functions include conditional moments, conditional distribution as well as conditional density functions. Joint asymptotic normality for derivative estimation is established for both strongly mixing and mixing processes.