For what you want to do, the simple names Langevin equation and Fokker-Plack equation are misleading and no longer justified. For attempts to create a theory of probability non-conserving generalized Langevin equations and Fokker-Plack equations see, e.g.,
Pollak, E. and Berezhkovskii, A.M., 1993. Fokker–Planck equation for nonlinear stochastic dynamics in the presence of space and time dependent friction. The Journal of chemical physics, 99(2), pp.1344-1346.
Berezhkovskii, A. M., Yu A. D’yakov, and V. Yu Zitserman. "Smoluchowski equation with a sink term: Analytical solutions for the rate constant and their numerical test." The Journal of chemical physics 109, no. 11 (1998): 4182-4189.