22-JUL-2023
The Doob $h$-transform is a fundamental technique for conditioning Markov processes on rare events. Given a strictly positive harmonic function $h$ for the generator $\mathcal{A}$ of a Markov process $X$, the $h$-transform reweights the original measure via the local martingale $M_t = h(X_t)/h(X_0)$, producing a new Markov process whose generator is $\mathcal{A}^h f = h^{-1}\mathcal{A}(hf)$. We develop the theory systematically: harmonic functions and Dynkin’s formula, the measure-change construction, conditioning standard Brownian motion to stay positive (yielding the three-dimensional Bessel process $\mathrm{BES}(3)$), conditioning to hit a fixed point (yielding the Brownian bridge), and Doob’s general theorem connecting $h$-transforms to conditional distributions. We conclude with Martin boundary theory, which classifies all positive harmonic functions via minimal harmonic functions and provides the canonical integral representation against the Martin kernel $K(x,\xi)$.
