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Erratic behaviour for one-dimensional random walks in a generic quasi-periodic environment

Hamiltonian systems, from topology to applications through analysis II November 26, 2018 - November 30, 2018

November 30, 2018 (02:00 PM PST - 03:00 PM PST)
Speaker(s): Maria Saprykina (Royal Institute of Technology (KTH))
Location: MSRI: Simons Auditorium
  • Random walk in random environment

  • limit theorem

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification



Consider a Markov chain on a one-dimensional torus $\mathbb T$, where a moving point jumps from $x$ to $x\pm \alpha$ with probabilities $p(x)$ and $1-p(x)$, respectively, for some fixed function $p\in C^{\infty}(\mathbb T, (0,1))$ and $\alpha\in\mathbb R\setminus \mathbb Q$. Such Markov chains are called random walks in a quasi-periodic environment. It was shown by Ya.~Sinai that for Diophantine $\alpha$ the corresponding random walk has an absolutely continuous invariant measure, and the distribution of any point after $n$ steps converges to this measure. Moreover, the Central Limit Theorem with linear drift and variance holds. In contrast to these results, we show that random walks with a Liouvillian frequency $\alpha$ generically exhibit an erratic statistical behaviour. In particular, for a generic $p$, the corresponding random walk does not have an absolutely continuous invariant measure, both drift and variance exhibit wild oscillations (being logarithmic at some times and almost linear at other times), Central Limit Theorem does not hold. These results are obtained in a joint work with Dmitry Dolgopyat and Bassam Fayad.

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