In this paper we establish a Harnack inequality for nonnegative harmonic functions of some discontinuous Markov processes with a diffusion part. AMS 2000 Mathematics Subject Classification: Primary 60J45, 60J75, Secondary 60J25.

This note aims to give a brief account on some recent progress of the simulation techniques of stochastic processes associated to divergence-form opertors with discontinuous coefficients, such as the one used in the Darcy law.

In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes. AMS 2010 Mathematics Subject Classification: Primary 60J45, Secondary 60J25, 60J50.

Xenopus oocyte maturation is an example of an all-or-none, irreversible cell fate induction process. In response to a submaximal concentration of the steroid hormone progesterone, a given oocyte may either mature or not mature, but it can exist in intermediate states only transiently. Moreover, once an oocyte has matured, it will remain arrested in the mature state even after the progesterone i...

The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description i...

The evolution of the Erdos-Rényi (ER) network by adding edges is a basis model for irreversible kinetic aggregation phenomena. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel Kij approximately ij , where ij is the product of the sizes of two merging clusters. Here we study that when the giant cluster is discoura...

We numerically solve parabolic problems in , , where is a bounded interval and is a strongly elliptic integrodifferential operator of order ! #" %$'& . A discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are used. The densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compressio...