I’ve obtained some analytical results that I’d like to verify numerically by doing a double path integral. I haven’t done this before with Mathematica and am unsure if it’s possible, I’d like to do the following path integral;

$ \int^{x_{a}}_{{x_{b}}} \int^{y_{a}}_{{y_{b}}}\exp \left[ \frac{i}{\hbar} \left( S[x(t)] – S[y(t)] \right) \right] \exp \left[ \frac{i}{\hbar} \int^{t}_{0}\left( F[x(\tau),y(\tau)] \right) d\tau \right] [dx(t)][dy(t)] $

where

$ S[x(t)] = \int^{t}_{0} \left( \dot{x}(\tau) – v[x(\tau)] \right) d\tau$ ,

$ v[x(\tau)] = x^{2}(\tau)$ ,

and

$ F[x(\tau),y(\tau)] = \left(x(\tau) – y(\tau)\right)\left(\dot{x}(\tau) – \dot{y}(\tau)\right) – \left(x(\tau) – y(\tau)\right)^{2}$

(For context, it’s quantum Brownian motion in which a particle is described within a bath of quantum harmonic oscillators)

Is it possible for Mathematica to do such calculations? If so, could someone point me in a direction to begin?