We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers law as an approximation of
a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via $L log L$ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.
We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation
law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.
This thesis analyze the Wasserstein gradient flow of a functional defined as a double convolution of a non-smooth repulsive interaction potential. To be more precise, the potential under investigation has a -|x| behavior close to the origin. The alre
ady existent machinery of Wasserstein gradient flow is well posed for lambda-convex potential. In this case this property is lost, but it is proven that in the one dimensional case existence and uniqueness of the solution is still achieved.
