Critical for the coexistence of humans and robots in dynamic environments is the capability for agents to understand each other's actions, and anticipate their movements. This paper presents Stochastic Process Anticipatory Navigation (SPAN), a framework that enables nonholonomic robots to navigate in environments with crowds, while anticipating and accounting for the motion patterns of pedestrians. To this end, we learn a predictive model to predict continuous-time stochastic processes to model future movement of pedestrians. Anticipated pedestrian positions are used to conduct chance constrained collision-checking, and are incorporated into a time-to-collision control problem. An occupancy map is also integrated to allow for probabilistic collision-checking with static obstacles. We demonstrate the capability of SPAN in crowded simulation environments, as well as with a real-world pedestrian dataset.