(f,g)-transform

The $(f,g)$-transform is a time-symmetric generalization of Doob’s h-transform that characterizes solutions to the Schrödinger bridge problem. Given a Markov reference measure $R\in {\mathrm{M}}_{+}(\Omega )$ and two nonnegative measurable functions $f_0,g_1:\mathcal{X}\to [0,\infty )$ satisfying $E_R(f_0(X_0)g_1(X_1))=1$, the $(f,g)$-transform is defined as the path measure

$P:=f_0(X_0)g_1(X_1)R\in \mathrm{P}(\Omega ).$

This construction was identified by Föllmer (1988), Föllmer-Gantert (1997), and Nagasawa (1989) as a time-symmetric version of the classical h-transform. While Doob’s h-transform $P=h(X_1)R$ conditions on the future only, the $(f,g)$-transform conditions on both endpoints simultaneously, making it the natural mathematical object for boundary-value problems on path space.

The key structural result (Léonard 2014, Theorems 2.9, 2.12) is that the unique solution $\hat{P}$ to the dynamic Schrödinger bridge problem is an $(f,g)$-transform of the reference measure, where $f_0$ and $g_1$ solve the Schrödinger system. The solution inherits the Markov property from $R$ (Proposition 2.10).

Key Details

• Born’s formula: The time-marginal density of an $(f,g)$-transform satisfies $P_t=f_t{g}_{t}m$, where $f_t(z)=E_R[f_0(X_0)|X_t=z]$ and $g_t(z)=E_R[g_1(X_1)|X_t=z]$ — a classical analogue of Born’s rule in quantum mechanics
• Forward/backward generators: The $(f,g)$-transform has forward generator ${\vec{A}}_t=L+\Gamma (g_t,\cdot )/g_t$ and backward generator ${\overleftarrow{A}}_t=L+\Gamma (f_t,\cdot )/f_t$, where $\Gamma$ is the carré du champ operator
• For reversible Brownian motion ($L=\Delta /2$): the forward drift is $\nabla {\psi }_t$ and backward drift is $\nabla {\phi }_t$, where $\phi ,\psi$ are the Schrödinger potentials
• The $(f,g)$-transform simultaneously generalizes conditioning on the past (via $f_0$) and future (via $g_1$) — it is not merely a product of two h-transforms

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