Temporal breakdown and Borel resummation in the complex Langevin method

Understand

We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4 measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between Euclidean and Lorentzian signature.

• Our main result is that the asymptotics for short stochastic times t encapsulates information also about the equilibrium aspects.
• The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t -> 0 asymptotic expansion which is shown to be Borel summable.
• The Borel transform correctly reproduces the time dependent moments of the complex measure for all t, including their t -> infinity equilibrium values.