Lectures on Probability Theory by D. Bakry, R. D. Gill, S. A. Molchanov

By D. Bakry, R. D. Gill, S. A. Molchanov

This publication comprises work-outs of the notes of 3 15-hour classes of lectures which represent surveys at the involved subject matters given on the St. Flour chance summer time university in July 1992. the 1st direction, via D. Bakry, is worried with hypercontractivity homes and their use in semi-group thought, particularly Sobolev and Log Sobolev inequa- lities, with estimations at the density of the semi-groups. the second, by way of R.D. Gill, is ready facts on survi- val research; it contains product-integral idea, Kaplan- Meier estimators, and a glance at cryptography and iteration of randomness. The 3rd one, by way of S.A. Molchanov, covers 3 facets of random media: homogenization idea, loca- lization homes and intermittency. every one of those chap- ters presents an creation to and survey of its topic.

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B) Let g be a positive simple function such that f ≥ g. 10 yields i = 1, . . , m. 11 lim μfn = lim n n i=1 m lim μ(fn 1Bi ) ≥ μ(fn 1Bi ) = i=1 n bi μ(Bi ) = μg. 3b. For each k, the function dk ◦ f is simple and f ≥ dk ◦ f . 11, we have lim μfn ≥ μ(dk ◦ f ) n for all k. Letting k lim μfn ≥ μf . → ∞ we obtain the desired inequality that 24 Measure and Integration Chap. 12 Proposition. For f and g in E+ and a and b in R+ , μ(af + bg) = a μf + b μg. The same is true for integrable f and g in E and arbitrary a and b in R.

F ⇒ L(fn ) L(f ). (fn ) ⊂ E+ and fn Proof. Necessity of the conditions is immediate from the properties of the integral: (a) follows from the definition of μf , (b) from linearity, and (c) from the monotone convergence theorem. To show the sufficiency, suppose that L has the properties (a)-(c). 23 A ∈ E. μ(A) = L(1A ), We show that μ is a measure. First, μ(∅) = L(1∅ ) = L(0) = 0. Second, if n A1 , A2 , . . are disjoint sets in E with union A, then the indicator of 1 Ai is n 1 1Ai , the latter is increasing to 1A , and hence, n μ(A) = L(1A ) = lim L( n n 1Ai ) = lim n 1 ∞ n L(1Ai ) = lim n 1 μ(Ai ) = 1 μ(Ai ), 1 where we used the conditions (c) and (b) to justify the second and third equality signs.

30 Transition densities. Let ν be a σ-finite measure on (F, F), and let k be a positive function in E ⊗ F. 2, that is, in differential notation, K(x, dy) = ν(dy) k(x, y). Show that K is a transition kernel. Then, k is called the transition density function of K with respect to ν. 31 Finite spaces. Let E = {1, . . , m}, F = {1, . . , n}, G = {1, . . , p} with their discrete σ-algebras.

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