Schrödinger system

The Schrödinger system is the pair of coupled nonlinear integral equations that characterize the solution to the Schrödinger bridge problem. Given a reference path measure $R$ with reversing measure $m$ and endpoint marginal measure $R_{01}$, the system seeks nonnegative measurable functions $f_0,g_1:\mathcal{X}\to [0,\infty )$ such that

$\{\begin{array}{l}{f}_0(x)E_R[g_1(X_1)\mid X_0=x]=d{\mu }_0/dm(x),m\text{-a.e.}\\ g_1(y)E_R[f_0(X_0)\mid X_1=y]=d{\mu }_1/dm(y),m\text{-a.e.}\end{array}$

where ${\mu }_0,{\mu }_1$ are the prescribed initial and terminal marginal distributions. This system was left as an open problem by Schrödinger in 1931; partial solutions were given by Fortet (1940) and Beurling (1960), with the complete solution due to Jamison (1975).

The Schrödinger system is the bridge-theoretic analogue of the Iterative Proportional Fitting (Sinkhorn) equations: solving it is equivalent to finding the correct marginal scalings for the static coupling $\hat{\pi }(dxdy)=f_0(x)g_1(y)R_{01}(dxdy)$. When the reference is Brownian motion, $f_0$ and $g_1$ are solutions of the heat equation run forward and backward respectively, and the system becomes a pair of nonlinear parabolic PDEs.

Key Details

• Solution defines the (f,g)-transform $\hat{P}=f_0(X_0)g_1(X_1)R$, which is the unique solution of the dynamic Schrödinger problem
• The functions $f_t(z)=E_R[f_0(X_0)|X_t=z]$ and $g_t(z)=E_R[g_1(X_1)|X_t=z]$ solve forward/backward parabolic PDEs: $(-{\partial }_t+L)f=0$ and $({\partial }_t+L)g=0$
• Logarithms $\phi =\log f$, $\psi =\log g$ give the Schrödinger potentials
• In the Brownian case: $f_t=e^{tL}{f}_0$ and $g_t=e^{(1-t)L}{g}_1$ with $L=\Delta /2$

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