Mathematical Sciences: Stochastic Processes

研究人员将研究分形上的扩散， 欧氏空间中的扩散 领域问题 分形上的扩散包括在 分形嵌入在高维欧几里德空间， 估计光谱维数，估计跃迁密度， 并证明法律上的唯一性。扩散领域的问题 在欧几里得空间中的唯一性定理包括： 具有分段连续系数的扩散， 研究相应的Dirichlet问题。 调查员正在解决接口处的问题 概率，数学物理（即物理学） 无序材料）和几何形状。他将构造布朗 运动（一个概率模型的位移， 粒子）上的分形。 这种结构的一个结果是 分形谱维数的估计， 分形的研究往往被证明是 物理学中的维度

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.