arXiv subjects

Margherita Zanella

Publications and source records attributed to Margherita Zanella.

22 records · Page 2Linked to original sources

arXiv2018

Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in $\mathbb R^n$

On the infinite dimensional space $E$ of continuous paths from $[0,1]$ to $\mathbb R^n$, $n \ge 3$, endowed with the Wiener measure $μ$, we construct a surface measure defined on level sets of the $L^2$-norm of $n$-dimensional processes that are solutions to a class of stochastic gradient system-type equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus by Airault and Malliavin in 1988.

math.PR↗

arXiv2018

Stochastic vorticity equation in $\mathbb R^2$ with not regular noise

We consider the Navier-Stokes equations in vorticity form in $\mathbb{R}^2$ with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the Itô calculus in $L^q$ spaces, $1<q<\infty$. We prove the existence of a unique strong (in the probability sense) solution.

math.PR↗

arXiv2017

Absolute continuity of the law for the two dimensional stochastic Navier-Stokes equations

We consider the two dimensional Navier-Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and coloured in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $ξ$ and we show that, if the initial vorticity $ξ_0$ is continuous in space, then there exists a space-time continuous version of the solution. In addition we show that the solution $ξ(t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.

math.PR↗

arXiv2016

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in $\mathbb R$.

math.PR↗