Traces and determinants of strongly stochastic operators.

Periodic orbit theory allows calculations of long-time properties of chaotic systems from traces, dynamical zeta functions, and spectral determinants of deterministic evolution operators, which are in turn evaluated in terms of periodic orbits. For the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral. Techniques for evaluating such path integrals are discussed. Using as an example the logistic map f(x)=lambdax(1-x) with moderate to strong additive Gaussian noise, rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda=5 for all noise levels.