Motivated by the need of developing stochastic nonlinear model-based methods to characterize uncertainty for chemical process estimation, control, identification, and experiment design purposes, in this paper the problem of characterizing the global dynamics of single-state nonlinear stochastic system is addressed. An isothermal CSTR with Langmuir-Hinshelwood kinetics is considered as representative example with steady state multiplicity. The dynamics of the state probability distribution function (PDF) is modeled within a Fokker-Planck’s (FP) global nonlinear framework, on the basis of FP’s partial differential equation (PDE) driven by initial state and exogenous uncertainty. A correspondence between global nonlinear deterministic (stability, multiplicity and bifurcation) and stochastic (PDF stationary solution and mono/multimodality) characteristics is identified, enabling the interpretation of tunneling-like stationary-to-stationary PDF transitions, and the introduction of a bifurcation diagram with the consideration of stochastic features in the context of the CSTR case example.