Equilibrium statistics of a slave estimator in Langevin processes

We analyze the statistics of an estimator, denoted by xi(t) and referred to as the slave, for the equilibrium susceptibility of a one dimensional Langevin process x(t) in a potential phi (x). The susceptibility can be measured by evolving the slave equation in conjunction with the original Langevin process. This procedure yields a direct estimate of the susceptibility and avoids the need, when performing numerical simulations, to include applied external fields explicitly. The success of the method, however, depends on the statistical properties of the slave estimator. The joint probability density function for x(t) and xi(t) is analyzed. In the case where the potential of the system has a concave component the probability density function of the slave acquires a power law tail characterized by a temperature dependent exponent. Thus we show that while the average value of the slave, in the equilibrium state, is always finite and given by the fluctuation-dissipation relation, higher moments and indeed the variance may show divergences. The behavior of the power law exponent is analyzed in a general context and it is calculated explicitly in some specific examples. Our results are confirmed by numerical simulations and we discuss possible measurement discrepancies in the fluctuation dissipation relation which could arise due to this behavior.