By Gilles Royer

This e-book presents an creation to logarithmic Sobolev inequalities with a few very important functions to mathematical statistical physics. Royer starts by means of accumulating and reviewing the mandatory heritage fabric on selfadjoint operators, semigroups, Kolmogorov diffusion strategies, strategies of stochastic differential equations, and sure different similar issues. There then is a bankruptcy on log Sobolev inequalities with an program to a powerful ergodicity theorem for Kolmogorov diffusion approaches. the rest chapters give some thought to the final environment for Gibbs measures together with lifestyles and specialty matters, the Ising version with actual spins and the applying 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 mostly textual content to similar parts corresponding to Markov chains. Titles during this sequence are co-published with Soci?©t?© Math?©matique de France. SMF contributors are entitled to AMS member discount rates.

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Additional info for An Initiation to Logarithmic Sobolev Inequalities (SMF AMS Texts & Monographs)

Example text

This gives us: E(IXtATn 12) = x2 + E(- rtATn J0 X, V U(X4) ds) + t n T rtAT, x2 + a+ E (J I Xe I2 ds) + (1 + b)+(t n T,,) 0 t x2 + (1 + b)+t + a+ JE(IXSATflI2)ds. Then apply Gronwall's lemma to f (t) = E(IXtAT, 12), to obtain: E(I XtATn I2) < (x2 + (1 + b)+t) e0+t. 2. SEMI-GROUPS 28 When n -+ oo, IXtAT 12 converges to +oo on {T = oo}, thus Fatou's lemma implies that {T < oo} has probability 0. The Cameron-Martin formula. We now wish to study the RadonNikodym derivative of the law of a transformed Brownian motion under a translation, with respect to the law of the Brownian motion itself.

LOGARITHMIC SOBOLEV INEQUALITIES 42 shows that u'(gt) also belongs to H1(µ) with Du'(gt) = u"(gt)Vgt. 15. Let Nt be a Kolmogorov semi-group satisfying the Gross inequality with constant c. 2) g E L"(µ). 2) implies the Gross inequality. PROOF. By a simple density argument it suffices to consider functions g such that Im(g) C [a, b] with a > 0. In this case, a 5 gt < b for gt = Ntg. From gt E D(A) C H' (µ), we conclude that gt /2 E H1(µ). 3) f9log(9i)dii - q log (J 9i dµ) fg' dµ 4 Jg_2JvgtI2 dµ. By applying the previous lemma for q > 0, we see that the function ,D (q,t) = 197 dA is differentiable with respect to the second variable t on R+, and for q > 1 the following inequality holds: -2t(q, t) = -q J99-lA9t dµ = A q -1) fg_2Ivgtl2 dµ 2 J 2(q - 1) dµ) du].

Gross in [Gr75]. In addition, there are more general logarithmic Sobolev inequalities that add a supplementary term ellf II2> with e > 0, to the right-hand side. See, for example, the course of D. Bakry [Bk93]. We refer to the inequality (LS) as the strict logarithmic Sobolev inequality. In [Gr92], L. Gross gives a panoramic view of diverse applications of these inequalities. In this book, unless we state otherwise, a logarithmic Sobolev inequality, or simply Gross inequality, will always refer to an inequality of the form (LS).

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