The gradient flow of the double well potential and its appearance in interacting particle systems

In this work we are interested in the existence of solutions to parabolic partial differential equations associated to gradient flows which involve the so-called double well potential, which is a nonconvex and nonconcave functional. Therefore the formal L2-gradient flow of the double well potential leads to a so-called forward-backward parabolic equation, which is not well-posed: it may fail to admit local in time classical solutions, at least for a large class of initial data. We discretize this forward-backward parabolic equation in space and prove convergence of the scheme for a suitable class of initial data. Moreover we identify the limit equation and characterize the long-time behavior of the limit solutions. Then we view such discrete-in-space schemes as systems of particles driven by the double-well potential and add a perturbation by independent Brownian motions to their dynamics. We describe the behaviour of a particle system with long-range interactions, in which the range of interactions is allowed to depend on the size of the system. We give conditions on the interaction strength under which the scaling limit of the particle system is a well-posed stochastic PDE and characterize the long-time behavior of this stochastic PDE

On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved

Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow

We prove that solutions of a mildly regularized Perona-Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.Comment: 19 page