The keywords are diffusion bridges or conditioned diffusions. I like to think of the distribution of these bridges as the stationary distribution of an SPDE on the path space of the diffusion. For an intro to this viewpoint and practical methods for sampling from this distribution, check out the following references:
- M.G. Reznikoff, E. Vanden-Eijnden, C. R. Acad. Sci. Paris, Ser. I 340 (2005). http://www.math1.rwth-aachen.de/~westd/crass.pdf
- Alexandros Beskos, Gareth O. Roberts, Andrew M. Stuart and Jochen Voss: MCMC Methods for Diffusion Bridges. Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350, 2008. https://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart74.pdf
- Hairer, Martin, Andrew M. Stuart, and Jochen Voss. "Sampling conditioned diffusions." Trends in stochastic analysis 353 (2009): 159-186. http://www.hairer.org/papers/HSV07.pdf