Stability of stochastic differential equations with additive persistent noise

We present a stability result for stochastic differential equations subject to additive persistent noise. Specifically, we propose a Lyapunov test for noise-to-state stability in pth moment with respect to an arbitrary subspace. To check the hypotheses of our result, we develop a method that exploits equivalence relations between positive semidefinite functions and a family of seminorms. With this method, we can translate inequalities between two positive semidefinite functions into separate sets of geometric conditions that relate each of them to a seminorm.