ABSTRACT
This chapter summarises some analytical tools that are frequently used when studying the asymptotic behaviour of continuous time stochastic processes as the time parameter tends to infinity. It begins with the change of time technique. The time change method is used as a simple rule to transfer the invariance property of probability measures for jump diffusion processes. The chapter describes Foster-Lyapunov conditions and related Dobrushin contraction inequalities. It shows a series of application examples of Lyapunov functions for different classes of diffusion processes. The chapter discusses more advanced functional and spectral analysis techniques, including a derivation of the Poincare inequality. These techniques are applied to investigate the exponential decays to equilibrium.
