Conditioning a Diffusion on Its Endpoint

Time runs upward. Each grey path is a Brownian motion in the plane, starting at the black point on the floor. Left alone, the paths spread out more and more as they rise.

Now keep only the paths that end at the red point on the top plane. Doob's h transform says they form a diffusion again, with the same noise and one extra drift: sigma squared times the gradient of the log of h, the probability of still reaching the target.

The rest of the narration is in the book's Companion.