Mathematical Sciences: Stochastic Processes

研究者将研究分形上的扩散和欧几里德空间中的扩散。分形扩散领域的问题包括在高维欧几里德空间中构造分形的布朗运动、估计谱维数、估计过渡密度和证明律的唯一性。欧几里得空间上的扩散问题包括证明分段连续系数扩散的唯一性和研究相应的狄利克雷问题。研究者正在研究概率、数学物理（即无序物质的物理）和几何之间的界面问题。他将在分形上构造布朗运动（粒子位移的概率模型）。这种构造的一个结果是分形的光谱维数的估计，分形的研究经常显示出它是物理学中的重要维数。

The investigator will study diffusion on fractals and diffusion in Euclidean spaces. Problems in the area of diffusions on fractals include constructing Brownian motions on fractals imbedded in higher dimensional Euclidean spaces, estimating spectral dimension, estimating transition densities, and proving uniqueness in law. Problems in the area of diffusions on Euclidean spaces include proving uniqueness in law for diffusions with piecewise continuous coefficients and investigating the corresponding Dirichlet problems. The investigator is working on problems at the interface between probability, mathematical physics (i.e. the physics of disordered material), and geometry. He will construct Brownian motions (a probabilistic model for the displacement of a particle) on fractals. One result from this construction will be the estimation of the spectral dimension of the fractal, which the study of fractals has shown often to be the important dimension in physics.