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The Risk-Sensitive Schrödinger Bridge: Is Not a KL Projection

arXiv · AI, language, vision and robotics · article · Sep 23, 2026 · UTC

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 m

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