On the Convergence of Diffusion Processes Conditioned to Remain in a Bounded Region for Large Time to Limiting Positive Recurrent Diffusion Processes

Let $X(t)$ be a diffusion process on Rᵈ with generator L = (1/2)∇ · a∇ + b∇ and let ₓ\, x ∈ Rᵈ, be the corresponding measures on paths. Pick 0 < t < T < ∞ and consider the process on the time interval 0, t conditioned to remain in a certain open, connected bounded region G up to time T. We obtain a new process YT(s), 0 s t. Let τG = inf: X(s) ∈ G\. With certain hypotheses on Pₓ(τG > s) (which are always satisfied if a⁻¹b is a gradient function), we show that YT(s) is an inhomogeneous diffusion process and that as T → ∞, YT(s), 0 ≤ s ≤ t converges to a limiting homogeneous positive recurrent diffusion Y(s), 0 ≤ s ≤ t, with state space G. Since t is arbitrary, we actually obtain a limiting process Y(s), 0 ≤ s < ∞. The generator of the limiting process may be written in the form LG = (1/2)∇ · a∇ + b∇ + a(∇ g₀/g₀)∇ - a∇ hg₀ ∇ where g₀ is the square root of the density of a measure μ₀ which minimizes the I-function for the process, over all γ ∈ P(Ḡ), the set of probability measures on Ḡ. The function hg₀ appears in the explicit calculation of I(μ₀) and solves a certain variational equation. The invariant measure for the process is μ₀.