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

Abstract: If a Brownian motion is physically constrained to the interval by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is . On the other hand, if Brownian motion is conditioned to remain in up to time t, then in the limit as one obtains an ergodic process whose exponential rate of convergence to equilibrium is . 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 , where , and then continues until it again hits an endpoint at which time it is catapulted again to , etc. The resulting process—Brownian motion physically returned to the point —is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals . In this paper we define a conditioning analog of the process physically returned to the point and study its rate of convergence to equilibrium. [Copyright &y& Elsevier]