Linear stability analysis for bifurcations in spatially extended systems with fluctuating control parameter

We study the threshold in systems which exhibit a symmetry breaking instability, described by a PDE that is of first order in time like the Ginzburg-Landau or Swift-Hohenberg equation, with the control parameter fluctuating in space and time. From the zero-dimensional case it is known that due to the long tails of the probability distributions all the moments of the linearized equation have different thresholds and none of them coincides with the threshold of the full nonlinear equation, which in fact has no noise-induced shift. We show that in the spatially extended case and with noise of finite correlation length there is a noise-induced threshold shift. At linear order in the fluctuation strength epsilon the threshold coincides with that of the first (and second) moment of the linearized equation. At order epsilon(2) differences in the thresholds arise. We introduce a method to obtain the threshold of the full system up to order epsilon(2) from the stability exponents of the first and second moment of the linearized equations. A preliminary account of this work has been given elsewhere [A. Becker and L. Kramer, Phys. Rev. Lett. 73 (1994) 955].