LiBERA On-line Seminars

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

Joint work with Gerardo Barrera (IST Lisbon), see https://arxiv.org/abs/2412.01381.

Exclusion processes with non-reversible boundary: hydrodynamics and large deviations.

We present an exclusion process defined on a one-dimensional lattice, which we call bulk, that is attached at each boundary point to a stochastic
boundary reservoir. Each reservoir interacts with the bulk on a window of fixed size l close to the respective boundary point at which it is attached. We obtain the hydrodynamic limit for this process whose hydrodynamic equation is described by the heat equation with non-linear Robin boundary conditions. The novelty here is that we do not put any restrictions on the choice of boundary rates apart from guaranteeing the irreducibility of the process, extending the results of [2]. Due to the freedom of choice of rates at the boundary, an interesting phenomenon is observed: we present an example for which we have more than one stationary profile but the hydrodynamic equation has a unique weak solution.

This is joint work with Claudio Landim and João Pedro Mangi.

We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process X follows a spectrally negative Lévy process with a Lévy measure defined by a completely monotone density. Secondly, to verify the optimality of the (2n+1)-barrier strategy when X is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric α-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider α=2 and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe-Yamada technique.

Joint work with Giacomo Lucertini and Stéphane Menozzi (https://arxiv.org/abs/2410.13432)