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Multi- to one-dimensional transportation
Chiappori, Pierre-An...
Multi- to one-dimensional transportation by Chiappori, Pierre-André ( Author )
N.A
02-10-2015
Fix probability densities f and g on open sets X⊂Rm and Y⊂Rn with m≥n≥1. Consider transporting f onto g so as to minimize the cost −s(x,y). We give a non-degeneracy condition (a) on s∈C1,1 which ensures the set of x paired with [g-a.e.] y∈Y lie in a codimension n submanifold of X. Specializing to the case m>n=1, we discover a nestedness criteria relating s to (f,g) which allows us to construct a unique optimal solution in the form of a map F:X⟶Y¯¯¯¯. When s∈C2∩W3,1 and logf and logg are bounded, the Kantorovich dual potentials (u,v) satisfy v∈C1,1loc(Y), and the normal velocity V of F−1(y) with respect to changes in y is given by V(x)=v"(f(x))−syy(x,f(x)). Positivity (b) of V locally implies a Lipschitz bound on f; moreover, v∈C2 if F−1(y) intersects ∂X∈C1 transversally (c). On subsets where (a)-(c) can be be quantified, for each integer r≥1 the norms of u,v∈Cr+1,1 and F∈Cr,1 are controlled by these bounds, ||logf,logg,∂X||Cr−1,1,||∂X||C1,1, ||s||Cr+1,1, and the smallness of F−1(y). We give examples showing regularity extends from X to part of X¯, but not from Y to Y¯. We also show that when s remains nested for all (f,g), the problem in Rm×R reduces to a supermodular problem in R×R.
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English
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MYR 0.00
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doi:10.1002/cpa.21707
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