Stochastic Safety Methods for General Markov Chains

This paper presents a comprehensive approach for assessing the stochastic safety of Markov chains, utilizing properties of the occupation and resolvent kernels as foundational concepts. We examine Markov chains with discrete and continuous state spaces, encompassing both transient and recurrent behaviors. We determine the characterization of the stochastic safety function through the solution of either Dirichlet or Poisson equations. Subsequently, we extract the dynamic programming equations tailored to the discrete or continuous case.