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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Stability of optimal transport
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This chapter is devoted to the following theme: Consider a family of geodesic spaces
Xkich
converges to some geodesic space
X
; does this imply that certain basic objects in the theory of optimal transport on
Xk
“pass to the limit”? In this chapter I shall show that the answer is affirmative: One of the main results is that the Wasserstein space
P2(Xk)
converges, in (local) Gromov—Hausdorff sense, to the Wasserstein space
P2(X)
. Then I shall consider the stability of dynamical optimal transference plans, and related objects (displacement interpolation, kinetic energy, etc.). Compact spaces will be considered first, and will be the basis for the subsequent treatment of noncompact spaces.