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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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