The Risk-Sensitive Schrödinger Bridge: Is Not a KL Projection
Abstract
The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value Jθ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly θ/2, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.
Related
- Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching
- Schrödinger-Bass Bridge Barycenter Models for Measure-Valued Data
- Gradient Discrete Schrodinger Bridge Matching for Peptide Generation
- A potential projection method for the nonlinear Schrödinger equation with low regularity potential
- E-ROBOT: a dimension-free method for robust statistics and machine learning via Schrödinger bridge.
BibTeX
@misc{behjoo2026risksensitive,
title = {{The Risk-Sensitive Schrödinger Bridge: Is Not a KL Projection}},
author = {Hamidreza Behjoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27250},
eprint = {2609.27250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27250},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Risk-Sensitive Schrödinger Bridge: Is Not a KL Projection
AU - Hamidreza Behjoo
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27250
UR - https://arxiv.org/abs/2609.27250
AB - The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value J_θ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly θ/2, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.
ER -