Proposition 5.2.3.label Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^{d})$ with transition functions $P_{s, t}(x, \cdot)$, $\mu$ be its initial distribution, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^{d} \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, then there exists a probaiblity measure $\bp^{h}$ on $D([0, \infty); \real^{d})$ such that:
- (1)
For each $t \ge 0$,
\[\frac{d \bp^{h}}{d \bp}\bigg |_{\cf_{t^+}}= \frac{\one_{\bracs{X_0 \in A_0}}}{\mu(A_{0})}\frac{h(t, X_{t})}{h(0, X_{0})}\]where $A_{0} = \bracs{h(0, \cdot) > 0}$.
- (2)
Under $\bp^{h}$, $X_{t}$ is a Markov process with transition function
\[P_{s, t}^{h}(x, dy) = \begin{cases}\frac{h(t, y)}{h(s, x)}P_{s, t}(x, dy)&h(s, x) > 0 \\ 0&h(s, x) = 0\end{cases}\] - (3)
If $\bp$ is the classical Wiener measure, then the process
\[\tilde X_{t} = X_{t} - \int_{0}^{t} \frac{\partial_{x} h(s, X_{s})}{h(s, X_{s})}ds\]is a $\bp^{h}$-Brownian motion.
The distribution $\bp^{h}$ is the $h$-transform of $\mathbf{P}$.