Mathematical Sciences: Diffusions and Potential Theory, Diffusions on Manifolds

研究者将研究流形上的扩散以及与椭圆散度形式算子相对应的扩散的性质。扩散过程提供了广泛使用的模型，特别是在物理学中。将使用分析工具和概率工具来寻求这些重要问题的解决方案。在流形上，研究布朗运动出口分布的光滑性。在特殊结构上，研究者将研究条件布朗运动的渐近行为。对于由椭圆散度形式算子引起的扩散，将研究漂移公式、条件规范定理和样本路径性质的变换。

The investigator will examine the properties of diffusions on manifolds and diffusion corresponding to elliptic divergence form operators. Diffusion processes provide widely used models especially in Physics. Both the analytic and the probabilistic tools will be used to seek solutions to these important problems. On manifolds, the smoothness of the exit distribution of Brownian motion will be studied. On special structures, the investigator will study the asymptotic behavior of conditioned Brownian motion. For diffusions arising from elliptic divergence form operators, transformation of drift formulae, conditional gauge theorems and sample path properties will be studied.