THE MECHANISM OF COMPLEX LANGEVIN SIMULATIONS

We discuss conditions under which expectation values computed from a c omplex Langevin process Z will converge to integral averages over a gi ven complex-valued weight function. The difficulties in proving a gene ral result are pointed out. For complex-valued polynomial actions, it is shown that for a process converging to a strongly stationary proces s one gets the correct answer for averages of polynomials if c(tau)(k) =E(e(ikZ(tau))) satisfies certain conditions. If these conditions are not satisfied, then the stochastic process is not necessarily describe d by a complex Fokker-Planck equation. The result is illustrated with the exactly solvable complex frequency harmonic oscillator.