Spectral gap and rate of convergence to equilibrium for a class of conditioned Brownian motions

If a Brownian motion is physically constrained to the interval [ 0 , γ ] by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is π 2 / 2 γ 2 . On the other hand, if Brownian motion is conditioned to remain in ( 0 , γ ) up to time t, then in the limit as t → ∞ one obtains an ergodic process whose exponential rate of convergence to equilibrium is 3 π 2 / 2 γ 2 . A recent paper [Grigorescu and Kang, J. Theoret. Probab. 15 (2002) 817–844] considered a different kind of physical constraint—when the Brownian motion reaches an endpoint, it is catapulted to the point p γ , where p ∈ ( 0 , 1 2 ] , and then continues until it again hits an endpoint at which time it is catapulted again to p γ , etc. The resulting process—Brownian motion physically returned to the point p γ —is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals 2 π 2 / γ 2 . In this paper we define a conditioning analog of the process physically returned to the point p γ and study its rate of convergence to equilibrium.