arXiv subjects

Tommaso Cornelis Rosati

Publications and source records attributed to Tommaso Cornelis Rosati.

5 recordsLinked to original sources

arXiv2021

The spatial $Λ$-Fleming-Viot process in a random environment

We study the large scale behaviour of a population consisting of two types which evolve in dimension d = 1, 2 according to a spatial Lambda- Fleming-Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher-KPP equation in a random potential. The proofs build on two-scale Schauder estimates and semidiscrete approximations of the Anderson Hamiltonian.

math.PR↗

arXiv2021

Synchronization for KPZ

We study the longtime behavior of KPZ-like equations: $$ \partial_{t}h(t,x) = Δ_{x} h (t, x) + | \nabla_{x}h (t,x)|^{2} + η(t, x), \qquad h(0, x) = h_0(x), \qquad (t, x) \in (0, \infty) \times \mathbb{T}^{d} $$ on the $d-$dimensional torus $\mathbb{T}^{d}$ driven by an ergodic noise $η$ (e.g. space-time white in $d= 1$. The analysis builds on infinite-dimensional extensions of similar results for positive random matrices. We establish a one force, one solution principle and derive almost sure synchronization with exponential deterministic speed in appropriate Hölder spaces.

math.PR↗

arXiv2020

A Rough Super-Brownian Motion

We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the unique weak solution to the stochastic PDE: \[\partial_t μ= (Δ{+} ξ) μ{+} \sqrt{2νμ} \tildeξ\] for independent space white noise $ξ$ and space-time white noise $\tildeξ$. In dimension $2$ the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-Brownian motion.

math.PR↗

arXiv2019

Killed Rough Super-Brownian Motion

This note extends the results in [8] to construct the rough super-Brownian motion on finite volume with Dirichlet boundary conditions. The backbone of this study is the convergence of discrete approximations of the parabolic Anderson model (PAM) on a box.

math.PR↗

arXiv2019

The KPZ Equation on the Real Line

We prove existence and uniqueness of distributional solutions to the KPZ equation globally in space and time, with techniques from paracontrolled analysis. Our main tool for extending the analysis on the torus to the full space is a comparison result that gives quantitative upper and lower bounds for the solution. We then extend our analysis to provide a path-by-path construction of the random directed polymer measure on the real line and we derive a variational characterisation of the solution to the KPZ equation.

math.PR↗