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SUMMARY:The Parisi Formula via Stochastic Analysis
DTSTART:20231103T130000Z
DTEND:20231103T150000Z
DTSTAMP:20240910T214700Z
UID:indico-event-10767@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Elton Hsu (Northwestern University)\n\nProbability a
nd analysis informal seminarThe Parisi formula is a fundamental result in
spin glass theory. It gives a variational characterization of the asymptot
ic limit of the expected free energy. The upper bound is a consequence o
f an interpolation identity due to F. Guerra and the lower bound is a cele
brated result of M. Talagrand. In this talk I will present a new approach
to (an enhanced version of) Guerra's identity using stochastic analysis\,
more specifically Brownian motion and Ito’s calculus. This approach is s
uggested by the form of the Parisi formula in which the solution of a Hami
lton-Jacobi equation is involved. It helps in many ways to illuminate the
original method of Guerra and suggests some possible approaches to the sig
nificantly deeper lower bound\, which has been intensively studied since T
alagrand’s work. Among the techniques from stochastic analysis we will u
se include path space integration by parts for the Wiener measure\, Girsan
ov’s transform (i.e.\, exponential martingales)\, and probabilistic repr
esentation of solutions to (linear) partial differential equations. The ke
y observation is that the nonlinear Hamilton-Jacobi partial differentiatio
n equation figuring in Parisi’s variation formula becomes linear after d
ifferentiating with respect to Guerra’s interpolation parameter\, thus b
ringing the full strength of stochastic analysis based on Ito’s calculus
into play. It is hoped that this approach will shed some lights on the mu
ch more difficult lower bound in the Parisi formula. ========Pour être i
nformé des prochains séminaires vous pouvez vous abonner à la liste de
diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme suj
et: "subscribe seminaire_mathematique PRENOM NOM"(indiquez vos propres pr
énom et nom) et laissez le corps du message vide.\n\nhttps://indico.math.
cnrs.fr/event/10767/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/10767/
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