Conditioned Diffusions which are Brownian Bridges

Let X_t be a one-dimensional diffusion of the form dX_t=dB_t+(X_t)dt. Let Tbe a fixed positive number and let be the diffusion process which is X_t conditioned so that X₀=X_T=x. If the drift is constant, i.e., , then the conditioned diffusion process is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.