A Liouville Property for Isotropic Diffusions in Random Environment

We obtain a Liouville property for stationary diffusions in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Precisely, we prove that, on a subset of full probability, the constant functions are the only strictly sub-linear maps which are invariant with respect to the evolution of the diffusion. And, we prove that the constant functions are the only bounded, ancient maps which are invariant under the evolution. These results depend upon the previous work of Fehrman [3] and Sznitman and Zeitouni [7] and, in the first case, our methods are motivated by the work, in the discrete setting, of Benjamini, Duminil-Copin, Kozma and Yadin [1].Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1404.527

Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu, R\"ockner; JDE, 2011], [Barbu, R\"ockner; JDE, 2018+], and [Gess, AoP, 2014].Comment: 37 page

A central limit theorem for nonlinear conservative SPDEs

We prove a central limit theorem characterizing the small noise fluctuations of stochastic PDEs of fluctuating hydrodynamics type. The results apply to the case of nonlinear and potentially degenerate diffusions and irregular noise coefficients including the square root. In several cases, the fluctuations of the solutions agree to first order with the fluctuations of certain interacting particle systems about their hydrodynamic limits

Green function and invariant measure estimates for nondivergence form elliptic homogenization

We prove quantitative estimates on the the parabolic Green function and the stationary invariant measure in the context of stochasic homogenization of elliptic equations in nondivergence form. We consequently obtain a quenched, local CLT for the corresponding diffusion process and a quantitative ergodicity estimate for the environmental process. Each of these results are characterized by deterministic (in terms of the environment) estimates which are valid above a random, ``minimal'' length scale, the stochastic moments of which we estimate sharply.Comment: 60 page

Biocolloid retention in partially saturated soils

Unsaturated soils are considered excellent filters for preventing the transport of pathogenic biocolloids to groundwater, but little is known about the actual mechanisms of biocolloid retention. To obtain a better understanding of these processes, a number of visualization experiments were performed and analyzed

Well-Posedness of the Dean-Kawasaki and the Nonlinear Dawson-Watanabe Equation with Correlated Noise

Fehrman B, Gess B. Well-Posedness of the Dean-Kawasaki and the Nonlinear Dawson-Watanabe Equation with Correlated Noise. Archive for Rational Mechanics and Analysis. 2024;248(2): 20.In this paper we prove the well-posedness of the generalized Dean–Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally 1/2-Hölder continuous, including the square root. This solves several open problems, including the well-posedness of the Dean–Kawasaki equation and the nonlinear Dawson–Watanabe equation with correlated noise