BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//wp-events-plugin.com//7.2.3.1//EN
TZID:Europe/Paris
X-WR-TIMEZONE:Europe/Paris
BEGIN:VEVENT
UID:8879@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20251020T153000
DTEND;TZID=Europe/Paris:20251020T183000
DTSTAMP:20251019T161132Z
URL:https://www.i2m.univ-amu.fr/evenements/marches-aleatoires-sur-des-suit
 es-de-graphes-monotones/
SUMMARY:Nordine Anis Moumeni (I2M): Marches aléatoires sur des suites de g
 raphes monotones
DESCRIPTION:Nordine Anis Moumeni: \n\nThe jury consists of\n\n 	Perla Sousi
  — Professor\, University of Cambridge — Reviewer\n 	Laurent Saloff-Co
 ste — Professor\, Cornell University — Reviewer\n 	Nina Gantert — Pr
 ofessor\, Technical University of Munich — Chair of the Jury\n 	Laurent 
 Miclo — Senior Researcher\, Toulouse School of Economics and Institute o
 f Mathematics — Examiner\n 	Fabienne Castell — Professor\, Aix-Marseil
 le University — Examiner\n 	Bruno Schapira — Professor\, University of
  Lyon 1 — Examiner\n 	Ruojun Huang — Postdoctoral Researcher\, Scuola 
 Normale Superiore di Pisa —  Examiner\n 	Pierre Mathieu — Professor\, 
 Aix-Marseille University — Thesis Supervisor\n\n\nAbstract (pdf)\nIn thi
 s thesis\, we study the merging property of a discrete-time Markov chain o
 n a countable space. Merging refers to the chain's ability to forget its i
 nitial distribution. After an introductory chapter where we present the ba
 sic concepts and necessary vocabulary to understand our results on merging
 \, we dedicate the second chapter to a sufficient condition for establishi
 ng this merging. We assume conditions ensuring irreducibly\, aperiodicity\
 , and the tightness of a sequence of laws with a proper initial probabilit
 y measure. We then prove that the chain merges. Let ((K_t)_{t geq 1}) be a
  sequence of Markov transition operators on the same discrete set (V). Let
  (mu_t^x) denote the law at time (t) starting from (x)\, driven by the seq
 uence ((K_t)_{t geq 1})\, and let (d_{TV}) be the classical total variatio
 n distance. Let (rho) be a proper probability measure on (V). Denote by ((
 rho_t)_{t geq 0}) the sequence of laws driven by the sequence ((K_t)_{t ge
 q 1}) with the initial law (rho). If the sequence ((rho_t)_{t geq 0}) is t
 ight\, then:\n[\nforall (x\,y) in V\, quad lim_{t to +infty} d_{TV}(mu_t^x
 \, mu_t^y) = 0\,.\n]\nThis result is proved using an original tool: the ac
 companying sets process. This random object is derived from the evolving s
 ets introduced by B. Morris and Y. Peres. It also allows us to provide a r
 esult within the framework of ratio theorems\, but under the hypothesis th
 at there exists a finite subset (A) of (V) such that the limit inferior of
  the sequence ((rho_t(A))_{t geq 0}) is strictly positive\, instead of the
  usual tightness assumption. Additionally\, in this same chapter\, we stud
 y the convergence of the laws when the operators or invariant measures con
 verge pointwise.\nIn the second chapter\, we provide quantitative estimate
 s on the merging of time-inhomogeneous Markov chains\, assuming that the i
 nvariant measures can be ordered in an increasing manner. Markov chains ar
 ising from electrical networks provide a class of examples where the invar
 iant measures are explicit. Our approach involves adapting classical funct
 ional inequalities: Poincaré\, Nash\, and hypercontractivity. In the prov
 en bounds\, alongside the classical terms\, we find an additional term tha
 t takes the form of a function of the total masses. Let us illustrate our 
 bounds using the spectral gap and the Poincaré inequality. Let ((K_t)_{t 
 geq 1}) be a sequence of Markov transition operators on the same discrete 
 set (V). Let ((pi_t)_{t geq 1}) be the sequence of their invariant measure
 s\, assumed to be finite. We denote by (Tilde{pi}_t) the probability measu
 re derived from the finite measure (pi_t). We assume that for all (x in V)
 \, the sequence ((pi_t(x))_{t geq 1}) is non-decreasing. For each (t geq 1
 )\, let (gamma_t) be the Poincaré constant of the operator (K_t^*K_t)\, w
 here (K_t^*) is the adjoint of (K_t) in the Hilbert space (ell^2(pi_t)). T
 hen\, we obtain:\n[\nforall (x\,y) in V\, quad d_{TV}(mu_t^x\, mu_t^y) le 
 frac{1}{2} sqrt{frac{pi_t(V)}{pi_1(V)}} left(frac{1}{sqrt{Tilde{pi}_1(x)}}
  + frac{1}{sqrt{Tilde{pi}_1(y)}}right) prod_{s=1}^t sqrt{1 - gamma_s}.\n]
CATEGORIES:Soutenance de thèse,ALEA
LOCATION:Saint-Charles - FRUMAM  (2ème étage)\, 3 Place Victor Hugo\, Mar
 seille\, 13003\, France
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=3 Place Victor Hugo\, Marse
 ille\, 13003\, France;X-APPLE-RADIUS=100;X-TITLE=Saint-Charles - FRUMAM  (
 2ème étage):geo:0,0
END:VEVENT
BEGIN:VTIMEZONE
TZID:Europe/Paris
X-LIC-LOCATION:Europe/Paris
BEGIN:DAYLIGHT
DTSTART:20250330T030000
TZOFFSETFROM:+0100
TZOFFSETTO:+0200
TZNAME:CEST
END:DAYLIGHT
END:VTIMEZONE
END:VCALENDAR