An Initiation to Logarithmic Sobolev Inequalities (SMF AMS by Gilles Royer

By Gilles Royer

This publication offers an creation to logarithmic Sobolev inequalities with a few vital functions to mathematical statistical physics. Royer starts by way of accumulating and reviewing the mandatory heritage fabric on selfadjoint operators, semigroups, Kolmogorov diffusion techniques, ideas of stochastic differential equations, and likely different similar issues. There then is a bankruptcy on log Sobolev inequalities with an software to a robust ergodicity theorem for Kolmogorov diffusion techniques. the rest chapters ponder the overall environment for Gibbs measures together with life and distinctiveness matters, the Ising version with actual spins and the appliance of log Sobolev inequalities to teach the stabilization of the Glauber-Langevin dynamic stochastic types for the Ising version with genuine spins. The routines and enhances expand the fabric broadly speaking textual content to similar components akin to Markov chains. Titles during this sequence are co-published with Soci?©t?© Math?©matique de France. SMF individuals are entitled to AMS member mark downs.

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PROOF. Let p be any probability measure on Rd. 4) 2ff2 log (l(I dp<, ) J(f2 log(f2) - f2 Iog(t2) - f2 +t2) dp. 5) f2(x)log(f2(x)) - f2(x)log(t2) - f2(x) + t2 '> 0, for all t and x. 5) implies that the integrand is positive and we are able f2 + If II (,,)) dv. to write: 2 f2log J Ifl IIf110(v) dv < z e-infV (f2 log(f2) - fI logllf I1i2(N) - f2 +IIf Il2L2(p)) dp 2 e- infV z Z f f2 log V1 l I e-infVfIvfl2

2. KOLMOGOROV SEMI-GROUPS 23 We are then able to define the stochastic integral: Aft = fo e(s) dBs, which is characterized by the following properties: (1) The process Aft is almost surely continuous and is a squareall u < t. integrable (Ft)-martingale: lE(Mt I Fu) = N fu (2) It is linear in e and the following isometry holds: 1E(MM) _ 1E(fo e2(s) ds). (3) For any subdivision 0 = to < t1 < < tk_1 < tk = t and any family of Xt-measurable random variables ek, we have: n t Ee 1k- 1 II1tk-1,tkl dBs = 0 k=1 E ek-1(Btk - Btk-1 ) k=1 We next generalize the above to the case of measurable and adapted sto- chastic integrands that satisfy the weaker condition: t (BL2) almost surely.

Subdivide the interval [0, t] into n subintervals of the same length and set: = n E(B)Lt/n - B(k-1)t/n)2 - t An(t) k=1 Prove that An(t) tends to 0 in L2(P) when n formula: 2 oo. Deduce from this the ft Bs dBs = Bt - t. , a constant. 2. 15 (Ito). Let e and f satisfy respectively (BL2) and (BL1) and consider the process: Xt = Xo+J te(s)dB,+J f(s)ds. ) f (s) + 2 axe u(s, X, )e2 (s)] ds. J It is clear that the two preceding integrands satisfy, respectively, the conditions (BL2) and (BL1). The reader will find a proof of Ito's formula, for example, in [KS88].

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