  It is a challenging issue to analyze complex dynamics from observed and
simulated data. An advantage of extracting dynamic behaviors from data is that
this approach enables the investigation of nonlinear phenomena whose
mathematical models are unavailable. The purpose of this present work is to
extract information about transition phenomena (e.g., mean exit time and escape
probability), from data of stochastic differential equations with non-Gaussian
L\'evy noise. As a tool in describing dynamical systems, the Koopman semigroup
transforms a nonlinear system into a linear system, but at the cost of
elevating a finite dimensional problem into an infinite dimensional one. In
spite of this, using the relation between the stochastic Koopman semigroup and
the infinitesimal generator of a stochastic differential equation, we learn the
mean exit time and escape probability from data. Specifically, we first obtain
a finite dimensional approximation of the infinitesimal generator by an
extended dynamic mode decomposition algorithm. Then we identify the drift
coefficient, diffusion coefficient and anomalous diffusion coefficient for the
stochastic differential equation. Finally, we compute the mean exit time and
escape probability by finite difference discretization of the associated
nonlocal partial differential equations. This approach is applicable in
extracting transition information from data of stochastic differential
equations with either (Gaussian) Brownian motion or (non-Gaussian) L\'evy
motion. We present one - and two-dimensional examples to demonstrate the
effectiveness of our approach.
