The reciprocal class R(Q) of a reference Markov measure Q is the set of all path measures that share the same bridge (conditional law given endpoints) as Q. A measure Π is reciprocal to Q if Π = Π_{0,T} · Q_{|0,T} — it has its own joint endpoint distribution but uses Q’s bridge to fill in the intermediate path. This means that conditioned on the initial and terminal values, the law of the path is entirely determined by the reference process Q, regardless of how the endpoints themselves are distributed.
The reciprocal projection takes a Markov process M, samples joint endpoints (X₀, X_T) from M, and reconstructs the path using Q’s bridge kernel. This operation replaces the internal dynamics of M with those of Q while retaining M’s endpoint coupling.
The Schrödinger bridge is uniquely characterized as the measure that is simultaneously Markov, in R(Q), and satisfies the boundary conditions. IMF alternates between Markovian and reciprocal projections to converge to this unique intersection. Reciprocal processes need not be Markov — the Markov property is an additional structural constraint beyond reciprocality.
Key Details
- R(Q) = {Π : Π_{|0,T} = Q_{|0,T}} (same bridge as Q)
- Reciprocal projection replaces bridge while keeping endpoints
- SB is unique intersection of Markov ∩ R(Q) ∩ boundary constraints
- Reciprocal processes need not be Markov
- Originated in the work of Bernstein and Jamison