  Ensuring safety through set invariance has proven to be a valuable method in
various robotics and control applications. This paper introduces a
comprehensive framework for the safe probabilistic invariance verification of
both discrete- and continuous-time stochastic dynamical systems over an
infinite time horizon. The objective is to ascertain the lower and upper bounds
of liveness probabilities for a given safe set and set of initial states. The
liveness probability signifies the likelihood of the system remaining within
the safe set indefinitely, starting from a state in the initial set. To address
this problem, we propose optimizations for verifying safe probabilistic
invariance in discrete-time and continuous-time stochastic dynamical systems.
These optimizations are constructed via either using the Doob's nonnegative
supermartingale inequality-based method or relaxing the equations described in
[30,32], which can precisely characterize the probability of reaching a target
set while avoiding unsafe states. Finally, we demonstrate the effectiveness of
these optimizations through several examples using semi-definite programming
tools.
