A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions

In this paper we introduce a new transportation distance between non-negative measures inside a domain $Omega$. This distance enjoys many nice properties, for instance it makes the space of non-negative measures inside $Omega$ a geodesic space, without any convexity assumption on $Omega$. Moreover, we will show that the gradient flow of the entropy functional w.r.t. this distance coincides with the heat equation, subject to the Dirichlet boundary condition equal to 1