Schrödinger potentials

The Schrödinger potentials are the logarithmic transforms $\phi :=\log f$ and $\psi :=\log g$ of the functions appearing in the Schrödinger system, where $f_0,g_1$ define the (f,g)-transform solution to the Schrödinger bridge problem. They are named by analogy with the Kantorovich potentials of optimal transport, to which they converge in the zero-noise limit.

In the case of reversible Brownian motion ($L=\Delta /2$), the Schrödinger potentials solve second-order Hamilton-Jacobi-Bellman equations:

$\{\begin{array}{ll}({\partial }_t+\Delta /2){\psi }_t+|\nabla {\psi }_t{|}^2/2=0,& (t,x)\in [0,1)\times \mathcal{X}\\ {\psi }_1=\log g_1& \end{array}$

$\{\begin{array}{ll}(-{\partial }_t+\Delta /2){\phi }_t+|\nabla {\phi }_t{|}^2/2=0,& (t,x)\in (0,1]\times \mathcal{X}\\ {\phi }_0=\log f_0& \end{array}$

These are the viscous (second-order) counterparts of the first-order Hamilton-Jacobi equation satisfied by Kantorovich potentials. The gradients $\nabla {\psi }_t$ and $\nabla {\phi }_t$ are respectively the forward and backward drift vector fields of the Schrödinger bridge, and they satisfy the time-reversal relation $\nabla {\psi }_t+\nabla {\phi }_t=\nabla \log {\mu }_t$, connecting them to the score function.

Key Details

• Dual problem: The Schrödinger potentials solve the dual $(D)$: $\int \phi d{\mu }_0+\int \psi d{\mu }_1-\log \int e^{\phi \oplus \psi }d{R}_{01}\to \max$
• Convergence to OT potentials: As diffusivity $1/k\to 0$, the rescaled potentials ${\alpha }_k{\phi }^k,{\alpha }_k{\psi }^k$ converge to the Kantorovich potentials of the limiting Monge-Kantorovich problem
• Analogy: The Schrödinger potentials are “regularized” Kantorovich potentials — smooth everywhere due to the heat semigroup, unlike the merely Lipschitz Kantorovich potentials

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