Discretization of Wasserstein Gradient Flow

We consider the local model and the model with non-local interactions with linear diffusion as equivalent problems to the gradient flow along a convex functional. In the first part, the local model is considered. We propose the particle approximation of the solution with operator splitting technique into the proximal (implicit) and adding Gaussian (diffusion) steps. We prove the uniform in time bounds  of the approximation in terms of Wasserstein distance under the assumption that a unique minimiser of the functional exists. In the second part, the propagation of chaos is used to approximate the non-local model with the system of local models and then we have the ability to apply the same technique as before. The talk will be based on the joint work with Iwona Chlebicka, Jørgen Endal and Błażej Miasojedow.