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SUMMARY:Mini-course on Geometric Functional Inequalities for Markov Chains
  - Sam Power (University of Bristol)
DTSTART:20241203T140000Z
DTEND:20241203T160000Z
UID:TALK225205@talks.cam.ac.uk
DESCRIPTION:When analysing the long-time behaviour of Markov processes\, m
 any familiar techniques are quite probabilistic in nature\, drawing on 'pa
 thwise' perspectives like couplings\, regenerations\, and similar. A compl
 ementary approach is more functional-analytic in character\, rather focusi
 ng directly on the { generator\, semigroup\, energy functionals\, ... } &n
 bsp\;associated to the process. In some settings\, these techniques can be
  quite appealing\, being robustly applicable to a wide range of processes 
 while often yielding quantitatively sharp bounds.\nThe overall goal of thi
 s mini-course will be to provide a friendly introduction to the functional
 -analytic approach\, to an intended audience who are perhaps more familiar
  with probabilistic techniques. Some more specific goals will be:\n\nto in
 troduce the 'standard' functional inequalities\, to provide intuition for 
 when they 'ought to' hold\, and for their consequences\,\nto clarify the p
 ractical settings to which the functional-analytic approach is well-suited
 \,\nto highlight the simplicity and robustness of the functional-analytic 
 approach in these cases\, and\nto elucidate the compatibility of probabili
 stic techniques with the functional-analytic approach\n\nThe first part of
  the mini-course will review functional inequalities in the classical case
  of reversible diffusion processes\, wherein many of the mathematical obje
 cts of interest are particularly interpretable. The second part of the min
 i-course will then cover how these inequalities apply in the setting of di
 screte-time Markov chains\, which is expected to be of particular interest
  to attendees working on MCMC algorithms.\nDue to time constraints\, I may
  not be able to provide detailed derivations for all results of interest. 
 A secondary goal of the mini-course will then be to provide a suitably ric
 h range of references which will enable the audience to fearlessly enter t
 he literature on functional inequalities in their own time.
LOCATION:Seminar Room 2\, Newton Institute
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