Abstract
We discuss conditions under which expectation values computed from a complex Langevin processZ will converge to integral averages over a given complex-valued weight function. The difficulties in proving a general result are pointed out. For complex-valued polynomial actions, it is shown that for a process converging to a strongly stationary process one gets the correct answer for averages of polynomials ifc τ(k)≡ E(eikZ(τ)) satisfies certain conditions. If these conditions are not satisfied, then the stochastic process is not necessarily described by a complex Fokker-Planck equation. The result is illustrated with the exactly solvable complex frequency harmonic oscillator.
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Gausterer, H., Lee, S. The mechanism of complex Langevin simulations. J Stat Phys 73, 147–157 (1993). https://doi.org/10.1007/BF01052754
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DOI: https://doi.org/10.1007/BF01052754