||1. The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry. Shen and Zheng (2010) proposed a probability method to transform the Monge-Kantorovich problem in the Euclidean plane into a Dirichlet boundary problem. Their results are original and sound. However, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results. This is a joint work with Zuo Quan Xu of Hong Kong Polytechnic University.
2. In the settings that the state space is finite or discrete, whether a transformation of a Markov chain enjoys still Markov property is known as "lampability of Markov chains". In this talk I shall describe a verifiable condition for a transformation of a continuous states valued Markov jump process enjoys still Markov property, and discuss some aspects related to this topic. Our results have applications in the study of modeling genetic coalescent processes with recombination. The talk is based on my recent joint work with Xian Chen and others.