---
tag: DK3492C42N
kind: definition
name: Doob's h-transform
msc: [60J45, 60J25]
defines: ZN7HJNR2AC
uses: [2AVNDP8BJP, 46436KE7NT, 22735FGJV6, VPA41PWRR1]
sources:
  - url: https://en.wikipedia.org/wiki/Doob%27s_h-transform
---

## Definition

Let $(E, \mathcal{E})$ be a measurable space and let $(P_t)_{t \ge 0}$ be a sub-Markov transition semigroup on $E$.
So the $P_t$ are kernels as for [[Markov kernels|2AVNDP8BJP]] but with $P_t(x, E) \le 1$, and $P_s P_t = P_{s+t}$.
Let $h \colon E \to (0, \infty)$ be a measurable function that is [[excessive|46436KE7NT]] for $(P_t)$.
The *h-transform* of $(P_t)$ is the family of kernels
$$P^h_t(x, dy) = \frac{h(y)}{h(x)} \, P_t(x, dy), \qquad t \ge 0, \ x \in E.$$
A Markov process with transition semigroup $(P^h_t)$ is called an *h-process*.

## Remarks

The kernels $P^h_t$ form a sub-Markov semigroup.
Indeed, $P^h_t(x, E) = P_t h(x)/h(x) \le 1$ since $h$ is excessive, and
$$\int P^h_s(x, dy) P^h_t(y, dz) = \frac{h(z)}{h(x)} \int P_s(x, dy) P_t(y, dz) = \frac{h(z)}{h(x)} P_{s+t}(x, dz).$$
If $P_t h = h$ for all $t$, then each $P^h_t$ is a Markov kernel and the h-process has infinite lifetime.
Example: let $(P_t)$ be [[Brownian motion|22735FGJV6]] on $(0, \infty)$ killed at $0$, and $h(x) = x$.
Then $P_t h(x) = E^x[B_t ; t < \tau_0] = x$ by optional stopping, so $h$ is invariant, and the h-process is the three-dimensional Bessel process, Brownian motion conditioned never to hit $0$.
Non-example: for Brownian motion on $\R$ and $h(x) = 1 + x^2$, one has $P_t h(x) = 1 + x^2 + t > h(x)$, so $h$ is not excessive and $P^h_t(x, \R) > 1$.
If $L$ is the [[infinitesimal generator|VPA41PWRR1]] of $(P_t)$, the generator of the h-process acts by $L^h f = h^{-1} L(h f)$ on suitable $f$.
For $L = \frac12 \Delta$ and harmonic $h$, this is $L^h f = \frac12 \Delta f + \nabla(\log h) \cdot \nabla f$, a Brownian motion with drift $\nabla \log h$.