We prove an invariance principle for Brownian motion in Gaussian or Poissonian random scenery by the method of characteristic functions. Annealed asymptotic limits are derived in all dimensions, with a focus on the case of dimension $d=2$, which is the main new contribution of the paper.
"An invariance principle for Brownian motion in random scenery." Electron. J. Probab. 19 1 - 19, 2014. https://doi.org/10.1214/EJP.v19-2894