Abstract. 
Diffusions with random switching are stochastic processes that consist of a
diffusion process $X_t$ and a continuous Markov jump process $Y_t$. Both
components interact with each other as $Y_t$ controls the dynamical regimes
of $X_t$ and $X_t$ effects the transition rate of $Y_t$. This type of
system includes a wide range of models, where the stochastic lattice model
for Madden Jullian Oscillation is one prime example. In this talk, we will
discuss when does this type of system 1) has a Lyapunov function;  2)has
geometric ergodicity, while the transition rates could be genuinely
unbounded. We will review some of the fundamental tools developed in
literature for geometric ergodicity, and apply them to diffusion processes
with random switching.