First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

We study the statistics of random functionals Z = integral(T)(0) [x(t)](gamma-2)dt, where x(t) is the trajectory of a one-dimensional Brownian motion with diffusion constant D under the effect of a logarithmic potential V (x) = V-0 ln(x). The trajectory starts from a point x(0) inside an interval entirely contained in the positive real axis, and the motion is evolved up to the first-exit time T from the interval. We compute explicitly the PDF of Z for gamma = 0, and its Laplace transform for gamma not equal 0, which can be inverted for particular combinations of gamma and V-0. Then we consider the dynamics in (0, infinity) up to the first-passage time to the origin and obtain the exact distribution for gamma > 0 and V-0 > -D. By using a mapping between Brownian motion in logarithmic potentials and heterogeneous diffusion, we extend this result to functionals measured over trajectories generated by (x)over dot(t) = root 2D[x(t)](theta)eta(t), where theta < 1 and eta(t) is a Gaussian white noise. We also emphasize how the different interpretations that can be given to the Langevin equation affect the results. Our findings are illustrated by numerical simulations, with good agreement between data and theory.