An exponential decay of a stochastic oscillatory integral with phase function determined as a stochastic line integral of a 1-form is studied. A sufficient condition for such an integral to decay exponentially fast is given in terms of the exterior derivative of the 1-form, i.e., the magnetic field.
Setsuo TANIGUCHI. "Exponential decay of stochastic oscillatory integrals on classical Wiener spaces." J. Math. Soc. Japan 55 (1) 59 - 79, January, 2003. https://doi.org/10.2969/jmsj/1196890842