A time-reversed representation for the tail probabilities of stationary reflected Brownian motion

We consider the exponential decay rate of the stationary tail probabilities of reflected Brownian motion X in the N-dimensional orthant R+N having drift b, covariance matrix A, and constraint matrix D. Suppose that the Skorokhod or reflection mapping associated with the matrix D is well-defined and Lipschitz continuous on the space of continuous functions. Under the stability condition D−1b<0, it is known that the exponential decay rate has a variational representation V(x). This representation is difficult to analyze, in part because there is no analytical theory associated with it. In this paper, we obtain a new representation for V(x) in terms of a time-reversed optimal control problem. Specifically, we show that V(x) is equal to the minimum cost incurred to reach the origin when starting at the point x, where the constrained dynamics are described in terms of another constraint matrix D̄, and the cost is quadratic in the control as well as the “local time” or constraining term. The equivalence of these representations in fact holds under the milder assumption that the matrices D and D̄ satisfy what is known as the completely-S condition. We then use the time-reversed representation to identify the minimizing large deviation trajectories for a class of RBMs having product form distributions. In particular, we show that the large deviation trajectories associated with product form RBMs that approximate open single-class networks or multi-class feedforward networks do not cycle.