  We consider the problem of navigation with safety constraints. The safety
constraints are probabilistic, where a given set is assigned a degree of
safety, a number between zero and one, with zero being safe and one being
unsafe. The deterministic unsafe set will arise as a particular case of the
proposed probabilistic description of safety. We provide a convex formulation
to the navigation problem with probabilistic safety constraints. The convex
formulation is made possible by lifting the navigation problem in the dual
space of density using linear transfer operator theory methods involving
Perron-Frobenius and Koopman operators. The convex formulation leads to an
infinite-dimensional feasibility problem for probabilistic safety verification
and control design. The finite-dimensional approximation of the optimization
problem relies on the data-driven approximation of the linear transfer
operator.
