---
tag: GPK4F1K971
kind: definition
name: Conditioned Brownian motion
msc: [60J65, 60J45]
defines: H0NDB85RCD
uses: [22735FGJV6, ZN7HJNR2AC, EYMQ82Z8XJ, 0Q4CBD3HX0, TCYSWCTYYM]
sources:
  - url: https://en.wikipedia.org/wiki/Doob%27s_h-transform
---

## Definition

Let $d \ge 1$ and let $D \subseteq \R^d$ be a [[domain|0Q4CBD3HX0]].
Let $p^D_t(x, y)$ be the transition density of [[Brownian motion|22735FGJV6]] in $\R^d$ killed on leaving $D$.
Let $h \colon D \to (0, \infty)$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $D$.
The *h-conditioned Brownian motion* in $D$ is the [[Doob h-transform|ZN7HJNR2AC]] of killed Brownian motion by $h$.
It is the Markov process on $D$ with transition densities
$$p^h_t(x, y) = \frac{h(y)}{h(x)} \, p^D_t(x, y), \qquad t > 0, \ x, y \in D.$$

## Remarks

A positive harmonic function on $D$ is excessive for killed Brownian motion, so the h-transform is defined; the h-process may have a finite lifetime.
Example: $D = (0, \infty)$ and $h(x) = x$.
Here $h$ is positive and harmonic, and the h-process is the three-dimensional Bessel process: Brownian motion conditioned never to hit $0$.
Example: for the unit disc and $h(z) = (1 - \abs{z}^2)/\abs{\zeta - z}^2$, the [[Poisson kernel|TCYSWCTYYM]] with pole at $\zeta$ on the unit circle, the h-process is Brownian motion conditioned to exit the disc at $\zeta$.
Non-example: $h(x) = x$ on $D = \R$ is harmonic but takes negative values, so the formula does not give a family of positive kernels.
More generally, conditioning to exit at a point of the Martin boundary uses the minimal harmonic function with pole at that point.