Nonlinear relaxation in the presence of an absorbing barrier

We study the nonlinear relaxation in the presence of multiplicative noise by means of a simple approximation scheme valid outside the critical region and exact asymptotic expansion at the critical point. The theory is developed in the Malthus-Verhulst stochastic model case. We find nonmonotonic growth of fluctuations during the transient. At the critical point we study the statistical properties of the finite time average of the original process. We obtain an exact result for the generating function exhibiting scaling asymptotic behavior at the critical point. We deduce also an asymptotic sum rule for the n-times correlation function of the original process and the asymptotic expression of the two-times correlation function. Our theoretical results are compared with numerical simulations and steady-state known properties.