A class of diffusions on the line is considered whose drifts are generalized functions (Schwartzian distributions). A probability measure is put on this space of drifts, giving a diffusion in a random environment. An invariance principle is then proven for the rescaled diffusion, generalizing a result of De Masi, Ferrari, Goldstein and the author.
W. David Wick. "Diffusion in a Singular Random Environment." Ann. Probab. 18 (1) 50 - 67, January, 1990. https://doi.org/10.1214/aop/1176990937