ASYMPTOTIC LAWS FOR ONE-DIMENSIONAL DIFFUSIONS CONDITIONED TO NONABSORPTION

If (X(t)) is a one-dimensional diffusion corresponding to the operator L = 1/2 partial derivative(xx) - alpha partial derivative(x) starting from x > 0 and T-a is the hitting time of a, we prove that under suit able conditions on the drift coefficient the following Limit exists: F or All s > 0, For All Alpha is an element of f(s), lim(t-->infinity)P( x)(X is an element of Alpha\T-0 > t). We characterize this limit as th e distribution of an h-like process, h satisfying Lh = - eta h, h(0) = 0, h'(0) = 1, where eta = -lim(t-->infinity)(1/t)logP(x)(T-0 > t). Mo reover, we show that this parameter eta can only take two values: eta = 0 Or eta = lambda, where lambda is the smallest point of increase of the spectral distribution of the operator l = 1/2 partial derivative (xx) + partial derivative(x)(alpha .).