Diffusions and Their Applications

我们将研究概率论中的几个问题。第一个是分形上扩散的唯一性问题。对于无限分叉的分形，如Sierpinski地毯，尚不知道是否所有状态空间为分形的布朗运动的可能结构都会导致相同的过程。这个问题适用于数学物理。另一个问题是将概率技术应用于热点问题，这是分析中的一个难题。热点问题是确定具有反射边界的热方程的解在何处取最大值。第三个问题是热核的估计，这是一个与偏微分方程有关的问题。散度形式的椭圆算子的热方程有一个与拉普拉斯方程相当的基本解。当状态空间不是欧几里得空间，而是更一般的流形时，这是真的吗？我们将研究概率论中的几个问题。它们都与分析和偏微分方程有关，特别是与热方程有关。热方程支配着热流和许多其他物理量。有趣的是，对随机过程的研究，如布朗运动，可以对各种介质中热方程的解提供大量的启示。

Several problems in probability theory will be studied. The first is the question of uniqueness for diffusions on fractals. For infinitely ramified fractals such as the Sierpinski carpet it is not yet known whether all possible constructions of Brownian motion whose state space is the fractal lead to the same process. This problem has applications to mathematical physics. Another concerns applying probabilistic techniques to the hot spots problem, a difficult problem in analysis. The hot spots problem is to determine where the solution to the heat equation with reflecting boundaries takes its maximum. A third problem is the estimation of the heat kernel, a problem that is related to partial differential equations. The heat equation for elliptic operators in divergence form has a fundamental solution that is comparable to that for the Laplacian. Is this true when the state space is not Euclidean space, but rather a more general manifold? Several problems in probability theory will be studied. They are all related to analysis and partial differential equations, in particular, to the heat equation. The heat equation governs the flow of heat as well as many other physical quantities. Interestingly enough, the study of random processes such as Brownian motion, can shed a great deal of light on the solutions of the heat equation in various media.