A stochastic wireless sensor network consisting of low-duty-cycle, unsynchronized nodes offers a simple, robust, and ultra-low-power solution for many sensor network applications. We demonstrate the feasibility of this approach by proving that such a network exhibits quasi-stable behavior as long as the mean number of active nodes in a neighborhood exceeds a bound between one and two, even in the presence of collisions. As this number increases, the mean lifetime of messages in the network grows rapidly and the average response time decreases. Simulation results correspond closely to the analytical bounds for quasistability, lifetime, and response time. A stochastic sensor network appears to he a promising approach for many wireless sensor network applications with modest data throughput requirements.