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,[gamma]] by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is [pi]2/2[gamma]2. On the other hand, if Brownian motion is conditioned to remain in (0,[gamma]) up to time t, then in the limit as t-->[infinity] one obtains an ergodic process whose exponential rate of convergence to equilibrium is 3[pi]2/2[gamma]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[gamma], where , and then continues until it again hits an endpoint at which time it is catapulted again to p[gamma], etc. The resulting process--Brownian motion physically returned to the point p[gamma]--is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals 2[pi]2/[gamma]2. In this paper we define a conditioning analog of the process physically returned to the point p[gamma] and study its rate of convergence to equilibrium.