Almost Sure Reachability in Continuous-time Stochastic Systems

A team from Max Planck Institute for Software Systems and Newcastle University has introduced rigorous mathematical certificates for proving almost sure reachability in continuous-time stochastic systems governed by stochastic differential equations (SDEs). The authors first demonstrate a critical gap: the standard Euler-Maruyama discretization commonly used in simulations can fail to preserve the almost sure reachability property. Using a double-well Langevin system as a counterexample, they show that a target set reachable in continuous time may appear unreachable under time discretization, highlighting risks in relying solely on numerical approximations.

To address this, they propose a pair of certificates — a drift function and a variant function — that provide necessary and sufficient conditions for almost sure reachability of an open bounded target set. For linear SDEs, they characterize reachability via the spectral structure of system matrices. For polynomial SDEs, they fix polynomial templates for the drift function and use exponential-of-polynomial templates for the variant function, converting conditions into sum-of-squares (SOS) constraints solvable via semidefinite programming. An alternating scheme resolves bilinearities. The SOS approach successfully recovers the reachability property lost under discretization on the double-well Langevin example and is verified on a polynomial system. This work offers a theoretical foundation for safety verification in continuous-time stochastic control systems.