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arXiv · 2605.03595

Almost Sure Reachability in Continuous-time Stochastic Systems

Arash Bahari Kordabad·Rupak Majumdar·Sadegh Soudjani
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Abstract

We provide certificates for almost sure reachability of continuous-time stochastic systems governed by stochastic differential equations (SDEs). We first show that a standard Euler-Maruyama discretization may fail to preserve almost sure reachability property of the system using a double-well Langevin system. This observation motivates us to develop certificates for almost sure reachability directly on the continuous-time system. We introduce a pair of certificates, a drift function and a variant function, and prove necessity and sufficiency for almost sure reachability of an open bounded target set. Using these certificates, for linear SDEs, we give a characterization of almost sure reachability in terms of the spectral structure of the system matrices. For polynomial SDEs, we fix a polynomial template for the drift function and choose the variant function template as an exponential function composed with a polynomial. This allows us to translate the conditions in the certificates into sum-of-squares (SOS) constraints. We then propose an alternating scheme to resolve bilinearities. We illustrate the approach on the double-well Langevin example, showing that continuous-time SOS certificates recover almost sure reachability that is lost under time discretization. Moreover, we verify the SOS approach on a polynomial system.

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Arash Bahari Kordabad, Rupak Majumdar, Sadegh Soudjani. 2026-05-05. Almost Sure Reachability in Continuous-time Stochastic Systems. https://arxiv.org/abs/2605.03595

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