In this paper we introduce a new technique for depicting the phase portrait
of stochastic differential equations. Following previous work for deterministic
systems, we represent the phase space by means of a generalization of the
method of Lagrangian descriptors to stochastic differential equations.
Analogously to the deterministic differential equations setting, the Lagrangian
descriptors graphically provide the distinguished trajectories and hyperbolic
structures arising within the stochastic dynamics, such as random fixed points
and their stable and unstable manifolds. We analyze the sense in which
structures form barriers to transport in stochastic systems. We apply the
method to several benchmark examples where the deterministic phase space
structures are well-understood. In particular, we apply our method to the noisy
saddle, the stochastically forced Duffing equation, and the stochastic double
gyre model that is a benchmark for analyzing fluid transport
