  Robots deployed in unstructured, real-world environments operate under
considerable uncertainty due to imperfect state estimates, model error, and
disturbances. Given this real-world context, the goal of this paper is to
develop controllers that are provably safe under uncertainties. To this end, we
leverage Control Barrier Functions (CBFs) which guarantee that a robot remains
in a ``safe set'' during its operation -- yet CBFs (and their associated
guarantees) are traditionally studied in the context of continuous-time,
deterministic systems with bounded uncertainties. In this work, we study the
safety properties of discrete-time CBFs (DTCBFs) for systems with discrete-time
dynamics and unbounded stochastic disturbances. Using tools from martingale
theory, we develop probabilistic bounds for the safety (over a finite time
horizon) of systems whose dynamics satisfy the discrete-time barrier function
condition in expectation, and analyze the effect of Jensen's inequality on
DTCBF-based controllers. Finally, we present several examples of our method
synthesizing safe control inputs for systems subject to significant process
noise, including an inverted pendulum, a double integrator, and a quadruped
locomoting on a narrow path.
