NONLINEAR RELAXATION IN THE PRESENCE OF AN ABSORBING BARRIER

We study the nonlinear relaxation in the presence of multiplicative no ise by means of a simple approximation scheme valid outside the critic al region and exact asymptotic expansion at the critical point. The th eory is developed in the Malthus-Verhulst stochastic model case. We fi nd nonmonotonic growth of fluctuations during the transient. At the cr itical point we study the statistical properties of the finite time av erage of the original process. We obtain an exact result for the gener ating function exhibiting scaling asymptotic behavior at the critical point. We deduce also an asymptotic sum rule for the n-times correlati on function of the original process and the asymptotic expression of t he two-times correlation function. Our theoretical results are compare d with numerical simulations and steady-state known properties.