The Doob h-transform is a technique for conditioning Markov processes on terminal values or events. Given a Markov process with transition density and a positive harmonic function , the h-transform defines new transition probabilities , where . This reweights the transition kernel to bias paths toward the target at time .
In the context of Lévy random bridges, the Doob h-transform naturally arises in the bridge transition probabilities (Corollary A.2 in Goria et al. 2025). The harmonic function is well-defined and positive for , ensuring the transform is valid. This provides a rigorous probabilistic foundation for conditioning processes on endpoints without constructing the bridge dynamics explicitly.
The h-transform is fundamental to the theory of diffusion bridges and appears in both the random bridge framework and in Schrödinger bridge algorithms. It connects the abstract notion of conditioning on a zero-probability event to concrete, computable changes of measure on path space.
The (f,g)-transform (Léonard 2014) generalizes the h-transform to a time-symmetric, two-boundary setting: conditions on both endpoints simultaneously. While the classical h-transform uses a single harmonic function to bias toward the future, the -transform uses a pair of functions solving the Schrödinger system, making it the natural mathematical object for the Schrödinger bridge problem.
Key Details
- Transition probability reweighting via harmonic functions
- Used in bridge sampling algorithms
- Connects to Girsanov’s theorem for measure changes
- The bridge is the h-transformed process with
- Generalized to the (f,g)-transform for two-boundary problems (Schrödinger bridges)