Stochastic Mehler kernels via oscillatory path integrals

The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrödinger equation for the harmonic oscillator with a stochastic term of the form (K t)(x) dWt, where K is either the position operator or the momentum operator, and Wt is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel.