INteraction between probability and analysis on disordered media

1.在构造穿透无序介质的扩散过程问题和研究空间上存在可数无序介质时扩散过程的性质方面取得了新的成果。当介质上的狄利克雷形式域为贝索夫空间时，我们应用贝索夫空间理论，构造了穿透扩散过程。通过对能量函数变分公式的求解，得到了该过程热核的短时间渐近性质，并给出了该过程的“最可能路径”。这是与b.m.h hambly合作的作品，刊登在《Probab》杂志上。代数理论。Fields.2。我们在d-集（分形集）上构造了跳跃型过程，并得到了这些过程的详细热核估计。狄利克雷形式对应的域还是贝索夫空间，贝索夫空间的迹公式非常有用。利用概率技术获得非对角线估计。利用热核估计，我们展示了过程的瞬态/递归性，并计算了过程范围的豪斯多夫维数。这是与陈志强合作的作品，发表于Stoch。他们的应用程序。研究了自相似集上扩散过程的热核的渐近性质。我们证明了该测度的体积加倍性质与一些对角线上的热核估计之间的等价性。进一步，我们证明了一些逃逸时间加上局部纳什不等式的估计与上面的一些热核估计之间的等价性。

1.We have obtained new results for the problem to construct diffusion processes penetrating disordered media and to study properties of the processes when there are countable number of disordered media on a space. When domains of Dirichlet forms on each medium are Besov spaces, we apply the theory of Besov spaces and construct the penetrating diffusion process. We also obtain a short time asymptotic behavior of the heat kernel for the process and express the "most probable path" by means of the solution of variational formula of energy functions. This is a joint work with B.M.Hambly and appears in Probab. Theory Relat. Fields.2.We have constructed jump-type processes on d-sets (fractal sets) and obtained detailed heat kernel estimates of the processes. The corresponding domains of Dirichlet forms are again Besov spaces and the trace formula of Besov spaces are very useful. Probabilistic techniques are used to obtain the off-diagonal estimates. Using the heat kernel estimates, we show transience/recurrence of the process and computed the Hausdorff dimension of the range of the process. This is a joint work with Z.Q.Chen and appears in Stoch. Proc. Their Appl.3.We have studied the asymptotic behavior of the heat kernels for diffusion processes on self-similar sets. We prove an equivalence between the volume doubling property of the measure and some on-diagonal heat kernel estimates. Furthermore, we prove an equivalence between some estimates of escaping times plus local Nash inequalities and some type of heat kernel estimates from above.