Geometric Meanings of Curvatures in Finsler Geometry

In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.Comment: 17 pages, This article is for the 20th Winter School on Geometry and Physics at Srni in Czech Republic. It was written during my visit at Institute of Mathematics and Informatics (IMI) at University of Debrecen in Hungary. The author would like to thank Dr. S. B\'{a}cs\'{o} and Dr. L. Kozma for their great help and hospitalit

Markov semi-groups generated by elliptic operators with divergence-free drift

In this paper we construct a conservative Markov semi-group with generator $L=\Delta+b\cdot\nabla$ on $\mathbb{R}^n$, where $b$ is a divergence-free vector field which belongs to $L^{2}\cap L^{p}$ with $\frac{n}{2}<p$. The research is motivated by the question of understanding the blow-up solutions of the fluid dynamic equations, which attracts a lot of attention in recent years.Comment: 12 page

Parabolic equations with divergence-free drift in space LtlLxqL_{t}^{l}L_{x}^{q}

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $\gamma=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $l=\infty,q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}\left(0,T;L^{n}\left(\mathbb{R}^{n}\right)\right)$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Semenov, 2006]. One of the results obtained in the present paper establishes the Aronson type estimate for critical and supercritical cases and for vector fields $b$ which are divergence-free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition enters the study of parabolic equations rather recently, mainly due to the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_{t}$ appears naturally.Comment: 31 page

Martingale representations for diffusion processes and backward stochastic differential equations

In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of continuous Hunt processes under certain technical conditions. In particular we establish the martingale representation theorem for the martingale parts of (reflecting) symmetric diffusions in a bounded domain with a continuous boundary. Together with an approach put forward in Lyons et al(2009), our martingale representation theorem is then applied to the study of initial and boundary problems for quasi-linear parabolic equations by using solutions to backward stochastic differential equations over the filtered probability space determined by reflecting diffusions in a bounded domain with only continuous boundary.Comment: 28 page