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Then, we carry out the Cole-Hopf transformation, which consists of supposing that = e U/2 ; then veri es the equation of the linear heat, which we explicitly resolve, hence the expression of U By passing to the limit within this expression when 0, we obtain using a standard technique (Laplace method) the following result (see [EVA 98] for the details of the calculations) Let us suppose, to simplify things, that the initial condition u0 is zero on ] , 0[ and that we have u0 (s) + s 0 for s large enough To have an idea, let us have a look
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at the solution at this instant: t = 1 First, we consider for each x the variable s 0:
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0, the function of
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and we note by a(x, 1) the largest point where the minimum is attained: a(x, 1) = max{s 0 : Fx (s) Fx (s ); for all s } (318)
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The limit solution when 0 with the time t = 1 is then given by: u(x, 1) = x a(x, 1) (319)
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Formula (318) shows that the random process a(x, 1) obtained when the initial condition is a Brownian motion on + (and zero on ) is a subordinator We call a subordinator an increasing L vy process ( x , x 0) A traditional example, which plays an important role later, is that of the rst times of passage of a Brownian motion with derivative More speci cally, let us consider a real standard Brownian motion (Bs , s 0), let us note Xs = Bs + s and let us introduce for x 0: x = inf{s 0, Xs > x}
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Because x is a stopping time and since X x = x, the strong Markov property applied to the Brownian motion implies that the process Xs = Xs+ x x is also a Brownian motion with derivative, which is independent of the portion of the trajectory r x ) For any z [0, x], the rst time of passage z clearly before x , (Xr , 0 0) is independent of only depends on the trajectory before x and hence (Xs , s ( z , 0 z x) The identi cation: x+y x = inf{s 0, Xs+ x > x + y} = inf{s 0, Xs > y}
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then highlights the independence and the homogenity of the incrementation of , which is hence a subordinator Close arguments apply to the increments of function (318) for Burgers equation (317) non-viscous with Brownian initial condition, ie, when we have u(x, 0) = Bx for x 0 Indeed, we verify that (a(x, 1) a(0, 1), x 0) and ( x , x 0) follow the same rule (see [BER 98]) The isolated example of the Burgers equation with initial Brownian data can make us hope that more general results are true and in particular that large classes of non-linear partial differential equations generically develop multifractal solutions However, at present, there are no proven results of this type (however, the reader can consult [VER 94] concerning Burgers equation in several space dimensions)
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333 Random wavelet series Since we are interested in the local properties of the functions, it is equivalent, and also easier, to work with periodic wavelets that are obtained by periodization of a usual base of wavelets (see [MEY 90]) and are de ned on the toric T = /Z The periodic wavelets: j,k (x) = 2j/2
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(320)
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form an orthonormal base of L2 (T) (by adding the constant function equal to 1, see [MEY 90]; we use the same notation than for the wavelets on , which will not lead to any confusion) We also assume that has enough regularity and zero moments Any periodic function f is hence written as follows: f (x) =
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ef (k, j) 2 j/2 j,k (x)
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(321)
where the wavelet coef cients of f are hence given by:
ef (k, j) =