Abstract
We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence ε_{n} such that nε_{n}→ ∞. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.
Original language  English 

Pages (fromto)  327367 
Number of pages  41 
Journal  Probability Theory and Related Fields 
Early online date  18 Jun 2020 
DOIs  
Publication status  Published  18 Jun 2020 
Keywords
 Dynamical sensitivity
 Exceptional times
 Hausdorff dimension
 Noise sensitivity
 Random walk
ASJC Scopus subject areas
 Analysis
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Matthew Roberts
 Department of Mathematical Sciences  Royal Society University Research Fellow
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching