In this article, we combine the DLA model of Witten and Sander with ideas from optimal transportation. We propose a modification of the DLA model in which the probability of sticking is inversely proportional to the additional transport cost from the point to the root. We used a family of cost functions parametrized by a parameter α as studied in ramified optimal transportation. α < 0 promotes ...

The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp “weighted isoperimetric inequality” and a nonsharp classical inequality similar to the one obtained in [Mi-Si]. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and...

In this paper, we will show that the cost − cosh ◦dHn is a regular cost, meaning that minimizing this cost on hyperbolic space yields a smooth optimal map between two given distributions of mass which satisfies suitable hypotheses. We show this by proving this cost satisfies Ma-Trudinger-Wang’s conditions and by investigating notions of convexity under this cost.

Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented n-dimensional manifold M . Suppose two families of normalized n-forms ω(τ) ≥ 0 and ω̃(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these n-forms represent two evolving distributions of particles over M , the minimum root-mean-squa...

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

We develop an ε-regularity theory at the boundary for a general class of MongeAmpère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on C uniformly convex domains are C up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x · y.

We consider an optimization problem in a given region Q where an agent has to decide the price p(x) of a product for every x ∈ Q. The customers know the pricing pattern p and may shop at any place y, paying the cost p(y) and additionally a transportation cost c(x, y) for a given transportation cost function c. We will study two models: the first one where the agent operates everywhere on Q and ...

Diffusion Magnetic Resonance Imaging (MRI) is used to (noninvasively) study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. Fiber-tracking models rely on how water molecules diffusion is represented in each MRI voxel. The Diffusion Spectrum Imaging ...

• Mathematics Subject Classification: 60E15, 90C40, 49N15 •