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R. Catellier

Publications and source records attributed to R. Catellier.

6 recordsLinked to original sources
arXiv2020

Propagation of chaos for mean field rough differential equations

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.

math.PR↗
arXiv2019

Solving mean field rough differential equations

We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. The analysis requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way.

math.PR↗
arXiv2016

Rough flows and homogenization in stochastic turbulence

We provide in this work a tool-kit for the study of homogenisation of random ordinary differential equations, under the form of a friendly-user black box based on the tehcnology of rough flows. We illustrate the use of this setting on the example of stochastic turbulence.

math.PR↗
arXiv2016

Averaging along irregular curves and regularisation of ODEs

We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path $w$ on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of $ρ$-\tmtextit{irregularity} and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function $w$ sampled according to the law of the fractional Brownian motion of Hurst index $H \in (0,1)$, we prove that almost surely the ODE admits a solution for all $b$ in the Besov-Hölder space $B^{α+1}_{\infty , \infty}$ with $α>-1/2H$. If $α>1-1/2H$ then the solution is unique among a natural set of continuous solutions. If $H>1/3$ and $α>3/2-1/2H$ or if $α>2-1/2H$ then the equation admits a unique Lipschitz flow. Note that when $α<0$ the vector field $b$ is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

math.PR↗
arXiv2015

Average-case complexity of a branch-and-bound algorithm for maximum independent set, under the $\mathcal{G}(n,p)$ random model

We study average-case complexity of branch-and-bound for maximum independent set in random graphs under the $\mathcal{G}(n,p)$ distribution. In this model every pair $(u,v)$ of vertices belongs to $E$ with probability $p$ independently on the existence of any other edge. We make a precise case analysis, providing phase transitions between subexponential and exponential complexities depending on the probability $p$ of the random model.

cs.CC↗