Complementability and maximality in different contexts: ergodic theory, Brownian and poly-adic filtrations

International audienceThe notions of complementability and maximality were introduced in 1974 by Ornstein and Weiss in the context of the automorphisms of a probability space, in 2008 by Brossard and Leuridan in the context of the Brownian filtrations, and in 2017 by Leuridan in the context of the poly-adic filtrations indexed by the non-positive integers. We present here some striking analogies and also some differences existing between these three contexts

Filtrations poly-adiques, standardité, complémentabilité, maximalité

Version française de l'article ``Poly-adic filtrations, standardness, complementability and maximality'' à paraître aux Annals of ProbabilityInternational audienceÉtant donn{é}e une filtration (\zc_n)_{n \le 0} index{é}e par les entiers n{é}gatifs, nous introduisons la notion de compl{é}mentabilit{é} pour les filtrations incluses dans (\zc_n)_{n \le 0}. Dans le cas de filtrations poly-adiques, nous d{é}finissons aussi la notion de maximalit{é}, dont nous donnons plusieurs caract{é}risations. Lorsque (\zc_n)_{n \le 0} est poly-adique, nous montrons que toute filtration compl{é}mentable par une filtration kolmogorovienne est maximale dans (\zc_n)_{n \le 0}. Nous montrons que la r{é}ciproque est fausse, mais qu'une r{é}ciproque partielle est vraie. Cette r{é}ciproque partielle {é}tend le th{é}or{é}me d'isomorphisme lacunaire de Vershik dans le cas des filtrations poly-adiques

Can mechanisms really replace laws of nature?

Today, mechanisms and mechanistic explanation are very popular in philosophy of science and are deemed a welcome alternative to laws of nature and deductive-nomological explanation. Starting from Mitchell's pragmatic notion of laws, I cast doubt on their status as a genuine alternative. I argue that (1) all complex-systems mechanisms onto-logically must rely on stable regularities, while (2) the reverse need not hold. Analogously, (3) models of mechanisms must incorporate pragmatic laws, while (4) such laws themselves need not always refer to underlying mechanisms. Finally, I show that Mitchell's account is more encompassing than the mechanistic accoun

Causal discovery and the problem of ignorance. An adaptive logic approach

AbstractIn this paper, I want to substantiate three related claims regarding causal discovery from non-experimental data. Firstly, in scientific practice, the problem of ignorance is ubiquitous, persistent, and far-reaching. Intuitively, the problem of ignorance bears upon the following situation. A set of random variables V is studied but only partly tested for (conditional) independencies; i.e. for some variables A and B it is not known whether they are (conditionally) independent. Secondly, Judea Pearl's most meritorious and influential algorithm for causal discovery (the IC algorithm) cannot be applied in cases of ignorance. It presupposes that a full list of (conditional) independence relations is on hand and it would lead to unsatisfactory results when applied to partial lists. Finally, the problem of ignorance is successfully treated by means of ALIC, the adaptive logic for causal discovery presented in this paper

Densité des orbites des trajectoires browniennes sous l'action de la transformation de Lévy

International audienceLet T be a measurable transformation of a probability space $(E,\mathcal {E},\pi)$, preserving the measure π. Let X be a random variable with law π. Call K(⋅, ⋅) a regular version of the conditional law of X given T(X). Fix $B\in \mathcal {E}$. We first prove that if B is reachable from π-almost every point for a Markov chain of kernel K, then the T-orbit of π-almost every point X visits B. We then apply this result to the Lévy transform, which transforms the Brownian motion W into the Brownian motion |W| − L, where L is the local time at 0 of W. This allows us to get a new proof of Malric's theorem which states that the orbit under the Lévy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets

Filtrations at the threshold of standardness

A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time $-\infty$, called non-standardness. For example, two dyadic filtrations with trivial tail $\sigma$-field are not necessarily isomorphic. Yet, any essentially separable filtration indexed by the non-positive integers becomes standard when sufficiently many integers are skipped. In this paper, we focus on the non standard filtrations which become standard if (and only if) infinitely many integers are skipped. We call them filtrations at the threshold of standardness, since they are as close to standardardness as they can be although they are non-standard. Two class of filtrations are studied, first the filtrations of the split-words processes, second some filtrations inspired by an unpublished example of B. Tsirelson. They provide examples which disproves some naive intuitions. For example, it is possible to have a standard filtration extracted from a non-standard one with no intermediate (for extraction) filtration at the threshold of standardness. It is also possible to have a filtration which provides a standard filtration on the even times but a non-standard filtration on the odd times