The square of white noise as a Jacobi field

Bayesian inverse problems for Burgers and Hamilton-Jacobi equations with white-noise forcing

The paper formulates Bayesian inverse problems for inference in a topological measure space given noisy observations. Conditions for the validity of the Bayes formula and the well-posedness of the posterior measure are studied. The abstract theory is then applied to Burgers and Hamilton-Jacobi equations on a semi-infinite time interval with forcing functions which are white noise in time. Inference is made on the white noise forcing, assuming the Wiener measure as the prior.

The projection spectral theorem and Jacobi fields

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to Lévy white noise measures.

On stochastic differential equations driven by the renormalized square of the Gaussian white noise

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the no…

白噪声驱动的连续随机流及其对偶

Continuous stochastic flows driven by white noise and their duals

研究一类由时空白噪声驱动的连续随机流，通过显式随机微分方程刻画其对偶流，并证明系数近似下解的收敛性，应用于平方贝塞尔流、雅可比流及多项式自排斥流。

Percolation of even sites for enhanced random sequential adsorption

Consider random sequential adsorption on a chequerboard lattice with arrivals at rate $1$ on light squares and at rate $λ$ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites {\em black}, and the remaining squares {\em white}. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability $p$; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs $(λ, p)$, containing the pair $(1,0.…

Homogenization of a stochastically forced Hamilton-Jacobi equation

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity o…