Burgers-KPZ Turbulence: Göttingen Lectures by Wojbor A. Woyczynski

By Wojbor A. Woyczynski

These lecture notes are woven round the topic of Burgers' turbulence/KPZ version of interface development, a examine of the nonlinear parabolic equation with random preliminary facts. The research is performed commonly within the space-time area, with much less realization paid to the frequency-domain photograph. despite the fact that, the bibliography encompasses a extra whole information regarding different instructions within the box which over the past decade loved a lively growth. The notes are addressed to a various viewers, together with mathematicians, statisticians, physicists, fluid dynamicists and engineers, and comprise either rigorous and heuristic arguments. as a result of multidisciplinary viewers, the notes additionally contain a concise exposition of a few classical issues in chance thought, reminiscent of Brownian movement, Wiener polynomial chaos, etc.

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Define if i ~ { i l , . . , i d } , and k e ( t l , . . idfli ( t l , . . , t d , t) E Ai, x . . x Ai~ x Ai, ( 0 otherwise. Then, by a straightforward computation, Id(f)I(g) 52 LECTURE 4. w(A{d). i~/3{~w(Ah) " . . {d~{jw(Ah)..... , w(A{J d : Id+l(ke) J- ~ d Id-l(f*j g) q- ~ Rj. j=l j=l In view of Property 3 ~, 2 --< (d+ 1)! - #( Aij_~ ) (2#( Aij )) 2#( Ai~+, )-.... id

Recall that E X stands for the mathematical expectation of the random quantity X. 2. WIENER-HERMITE EXPANSIONS 45 of the usual calculus power series expansions. It will be summarized in the next section using an approach borrowed from Kwapien and Woyczynski (1992). ). Denote by /CO the family of all real polynomial chaoses of degree d based on the sequence 71,72,.... , ~n) has a representation d Q(~I,... '~ik. 1 of Kwapien and Woyczynski (1992), the topology of convergence in the second m o m e n t (or any other moment) coincides on/C O with the topology of convergence in probability.

2. WIENER-HERMITE EXPANSIONS 47 | Definition 1. For each d > 1, K:~ k>l, degQ--d}. Each element of ~o is traditionally called a Wiener polynomial chaos of order d. The closure in L2(~, 6, P) of the set/s will be denoted by K:d. The next theorem describes the Wiener chaos decomposition of L2 (f~, g, P). T h e o r e m 1. Let ~o be the linear span of the set {hd(Xh) : h e H, [Ih]l = 1}, and let Tld be the closure o f ~ ~ in L2(f~,g,P). Then oo L2(f~, G, P) = ~ 7-td, d=0 and, for each n = 1, 2 , . .

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