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Multiplicative Schr\"odinger problem and the Dirichlet transport. (arXiv:1807.05649v1 [math.PR])
来源于:arXiv
We consider an optimal transport problem on the unit simplex whose solutions
are given by gradients of exponentially concave functions and prove two main
results. One, we show that the optimal transport is the large deviation limit
of a particle system of Dirichlet processes transporting one probability
measure on the unit simplex to another by coordinatewise multiplication and
normalizing. The structure of our Lagrangian and the appearance of the
Dirichlet process relate our problem closely to the entropic measure on the
Wasserstein space as defined by von-Renesse and Sturm in the context of
Wasserstein diffusion. The limiting procedure is a triangular limit where we
allow simultaneously the number of particles to grow to infinity while the
`noise' goes to zero. The method, which generalizes easily to other cost
functions, including the Wasserstein cost, provides a novel combination of the
Schr\"odinger problem approach due to C. L\'eonard and the related Brownian
particle systems by 查看全文>>