In many social networks, whether two agents incorporate each other's opinion or not depends on the proximity of their opinions. The bounded confidence model of opinion dynamics captures this by introducing a parameter called the influence range and by stating than an agent incorporates the opinion of another agent if the distance between their opinions is less than the influence range. We generalize the bounded confidence opinion dynamics model by allowing a stochastic influence range that varies across time and agents, and by incorporating the effect an additive noise on the update rule. We establish the conditions under which this stochastic dynamics is stable in an appropriate sense. Our main observation is that the the presence of heavy tailed influence ranges is critical for stability.