The minimization of a relative entropy (with respect to the Wiener measure) is a very old problem which dates back to Schrödinger. C. Leonard has recently proved the connection between this problem and the Monge-Kantorovich problem with quadratic cost (namely the standard Optimal Transport problem). In this talk we will go a step further by showing how second order MFG systems are related to the minimization of a relative entropy (+ a congestion term). Moreover, this new optimization can be efficiently solved by using a generalization of the Sinkhorn algorithm.
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