Investigation of a class of stochastic processes using path-integral techniques

A study is made of the Langevin equation x=-V'( chi )+g( chi ) zeta (t)+ eta (t), where the noises eta (t) and zeta (t) are Gaussian with zero mean and with ( eta (t) eta (t'))=2R delta (t-t'), ( zeta (t) zeta (t'))=(D/ tau ) exp- mod t-t' mod / tau . Path integral representations for conditional probability distributions are given for the two cases tau =0 and tau not=0. For R and D small, but of the same order, the appropriate path integrals are evaluated to leading order using the method of steepest descents, in order to find the stationary probability distribution Pst(x) and the mean relaxation time T for escape from a potential well. Analytical expressions are given for T when tau =0 and when tau is small. For general tau the authors present numerical results for the stationary probability distribution, using the particular forms V(x)=-1/2ax2+1/4Ax4-R 1n x and g(x)=x, which are appropriate to the dye laser.