Throughout physics Brownian dynamics are used to describe the behaviour of molecular systems. When the Brownian particle is confined to a bounded domain, a particularly important question arises around determining how long it takes the particle to encounter certain regions of the boundary from which it can escape. Termed the first passage time, it sets the natural timescale of the chemical, biological, and physical processes that are described by the stochastic differential equation. Probabilistic information about the first passage time can be studied using spectral properties of the deterministic generator of the stochastic process. In this work we introduce a framework for bounding the leading eigenvalue of the generator which determines the exponential rate of escape of the particle from the domain. The method employs sum-of-squares programming to produce nearly sharp numerical upper and lower bounds on the leading eigenvalue, while also giving good numerical approximations of the associated leading eigenfunction, the quasi-stationary distribution of the process. To demonstrate utility, the method is applied to prototypical low-dimensional problems from the literature.