Conditioned Diffusions and Relative Entropy

What does it cost to make a random path do something unlikely? This book answers with one formula. On Wiener space every equivalent change of measure is a drift, and its price is relative entropy: one half of the expected integral of the squared drift.

Condition a Brownian path on its endpoint, on staying positive or on a rare event, and the conditioned path is Brownian motion with a drift, the drift of Doob’s h-transform; its price is minus the logarithm of the event’s probability. The Boué–Dupuis formula makes that drift optimal and reads importance sampling as its equality case: under the cheapest drift, the estimator has zero variance. Let the noise vanish and the formula becomes the geometry of most likely paths and Kramers’ law. Prescribe the law at both ends and it becomes Schrödinger’s bridge, an h-transform with two ends.

Every theorem is proved in full, and every formula is computed twice, once from each of its sides.

Noa Ben-Dor is a professor of probability who works on change of measure, conditioned diffusions and large deviations.