arXiv subjects

Y. LeJan

Publications and source records attributed to Y. LeJan.

2 recordsLinked to original sources

arXiv2019

Concerning the geometry of stochastic differential equations and stochastic flows

Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $Tξ_t$, a Weitzenböck formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishing theorem. A comprehensive theory on this, its generalization to semi-elliptic case and applications is published in the book `On the geometry of diffusion operators and stochastic flows'. Related to this is also the book `The geometry of filtering'. This article is easier to read.

math.PR↗

arXiv2008

The Geometry of Filtering

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.

math.DG↗