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Ian Melbourne

Publications and source records attributed to Ian Melbourne.

At least 37 records · Page 2Linked to original sources
arXiv2021

Sharp Statistical Properties for a Family of Multidimensional NonMarkovian Nonconformal Intermittent Maps

Intermittent maps of Pomeau-Manneville type are well-studied in one-dimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theorem and local limit theorem, Berry-Esseen estimates, large deviation estimates, convergence to stable laws and Lévy processes, and infinite measure mixing.

math.DS↗
arXiv2020

Superdiffusive limits for deterministic fast-slow dynamical systems

We consider deterministic fast-slow dynamical systems on $\mathbb{R}^m\times Y$ of the form \[ \begin{cases} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a(x_k^{(n)}) + n^{-1/α} b(x_k^{(n)}) v(y_k)\;,\quad y_{k+1} = f(y_k)\;, \end{cases} \] where $α\in(1,2)$. Under certain assumptions we prove convergence of the $m$-dimensional process $X_n(t)= x_{\lfloor nt \rfloor}^{(n)}$ to the solution of the stochastic differential equation \[ \mathop{}\!\mathrm{d} X = a(X)\mathop{}\!\mathrm{d} t + b(X) \diamond \mathop{}\!\mathrm{d} L_α \; , \] where $L_α$ is an $α$-stable Lévy process and $\diamond$ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps $f$ of Pomeau-Manneville type.

math.DS↗
arXiv2020

Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1

We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form \[ x_{k+1} = x_k + n^{-1} a_n(x_k,y_k) + n^{-1/2} b_n(x_k,y_k), \quad y_{k+1} = T_n y_k, \] where the fast dynamics is given by a family $T_n$ of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T.) The homogenization results are a consequence of this together with parallel developments on rough path theory in Part 2 by Chevyrev, Friz, Korepanov, Melbourne & Zhang.

math.DS↗
arXiv2020

St. Petersburg paradox for quasiperiodically hypermeandering spiral waves

It is known that quasiperiodic hypermeander of spiral waves almost certainly produces a bounded trajectory for the spiral tip. We analyse the size of this trajectory. We show that this deterministic question does not have a physically sensible deterministic answer and requires probabilistic treatment. In probabilistic terms, the size of the hypermeander trajectory proves to have an infinite expectation, despite being finite with probability one. This can be viewed as a physical manifestation of the classical "St. Petersburg paradox" from probability theory and economics.

nlin.PS↗
arXiv2020

Simulation of non-Lipschitz stochastic differential equations driven by $α$-stable noise: a method based on deterministic homogenisation

We devise an explicit method to integrate $α$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Lévy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct $α$-stable laws.

math.DS↗
arXiv2020

Variance continuity for Lorenz flows

The classical Lorenz flow, and any flow which is close to it in the $C^2$-topology, satisfies a Central Limit Theorem (CLT). We prove that the variance in the CLT varies continuously.

math.DS↗
arXiv2020

Convergence to a Lévy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps

We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.

math.DS↗
arXiv2019

Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional.

math.DS↗
arXiv2019

Multiscale systems, homogenization, and rough paths

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and càdlàg rough path theory.

math.DS↗
arXiv2019

Renewal theorems and mixing for non Markov flows with infinite measure

We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson's methods to the deterministic (i.e. non-i.i.d.) continuous time setting and obtain results on mixing as a consequence. Our results apply to intermittent semiflows and flows of Pomeau-Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet-Eckmann maps with nonintegrable roof function.

math.DS↗
arXiv2018

Rates of mixing for nonMarkov infinite measure semiflows

We develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows. Previously, such results were restricted to the situation where there is a first return Poincaré map that is uniformly expanding and Markov. As illustrations of the method, we consider semiflows over nonMarkov Pomeau-Manneville intermittent maps with infinite measure, and we also obtain mixing rates for semiflows over Collet-Eckmann maps with nonintegrable roof function.

math.DS↗
arXiv2018

Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure

We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgopyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly nonMarkovian) nonuniformly expanding maps hold also for their toral extensions.

math.DS↗