There is a class of Markov processes called symmetric Markov processes whose laws do not change with time reversal. A typical example of a symmetric Markov processes is a diffusion process on a one-dimensional interval. We define a class of symmetric Markov processes similar to one-dimensional diffusion processes and study their properties (e.g., proof of the large deviation principle, existence and uniqueness of quasi-stationary distributions).

Among stochastic models describing random time evolution, there is an important class called Markov property processes with Markov property. Markov property is the property that future information does not depend on past information if present information is known. The goal of this course is to learn to use Markov property to calculate the hitting probability and the expected value of sojourn time