Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

Lower bounds for covering times for reversible Markov chains and random walks on graphs

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Reversible Markov Chains and Random Walks on Graphs

We construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of ℝ2, and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, p
t(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on p
t(x,y).

Brownian motion on the Sierpinski gasket

Brownian motion on nested fractals

We consider the multidimensional Burgers equation with a viscosity term and a random force modelled by a functional of time-space white noise, Wick products. Then we show that the nonlinear equation (B) can be transformed into a linear, stochastic heat equation with a noisy potential. This heat equation is solved explicitly in the following two cases.

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The burgers equation with a noisy force and the stochastic heat equation

/pdf/fractional-brownian-fields-as-integrals-of-white-noise-1eesheb1kc.pdf

Fractional brownian fields as integrals of white noise

We show that there is a close connection between deterministic differential equations of the form dd~t = b(~t) + a(~t) · L (k(t)zk t k (where Zk = Xk + iyk are complex parameters) and Ito-Skorohod stochastic differential equations of the form where b0 , a<> denote the Wick versions of the functions b, a. The connection is provided by the Hermite transform 'H, which maps £ 2 stochastic processes Xt into (deterministic) analytic functions 'H( Xt )( z1 , z2 , • • ·) on c~ = {(z!, Z2, •• ·); Zk E c and 3M with Zj = 0 for j > M}, and by its inverse 1-{-1 , which can be given an explicit form.

Wick multiplication and ito-skorohod stochastic differential equations

We study the multidimensional stochastic (Wick-type) Burgers equation { OUk + A .;:.... u . <> auk = ZJ .6. Uk + Wk (t x) . t > 0 X E R n ot 0 J ox· ' ' ' j=1 J uk(O, x) = 9k(x) ; 1::; k::; n where <> denotes the Wick product, A and v are constants (v > 0, A # 0), .6. denotes the Laplacian and {wk(t,x)}k=1 are (n + 1)-parameter stochastic processes (noise). We prove an existence and uniqueness result for the solution u( t, x) = { uk ( t, x) }k=1 , regarded as an (n + 1)-parameter stochastic process with values in the Kondratiev space (S)-1 of stochastic distributions. CONTENTS §0. INTRODUCTION §1. WHITE NOISE, WICK PRODUCTS AND STOCHASTIC DISTRIBUTIONS §2. FROM THE STOCHASTIC BURGERS EQUATION TO THE STOCHASTIC HEAT EQUATION §3. (S)-1-SOLUTION OF THE STOCHASTIC HEAT EQUATION §4. FROM THE STOCHASTIC HEAT EQUATION TO THE STOCHASTIC BURGERS EQUATION §5. AN EXISTENCE AND UNIQUENESS RESULT REFERENCES 1.) Dept. of Mathematical Sciences, Norwegian Institute of Technology, University of Thondheim, N-7034 Thondheim, Norway 2l Dept. of Mathematics, University of Oslo, Box 1053, Blindern, N-0316 Oslo, Norway 3) Dept. of Mathematics/VISTA, University of Oslo, Box 1053, Blindern, N-0316 Oslo, Norway 4l Dept. of Mathematics, National College of Safety Engineering, Sk denotes the Wick product, ). and v are constants (v > 0, ). =I 0), ..6. denotes the Laplacian and Wk ( t, x) = Wk ( t, x) are ( n + 1 )-parameter generalized stochastic processes; 1 :::; k :::; n. (See details below). The use of the Wick product corresponds to an Ito/Skorohod interpretation of the equation (see e.g. [B1], [L0U 2] and also [HL0UZ 3]). We may regard u = ( u1 , · · · , un) as the velocity field of a vorticity free fluid with viscosity v, being exposed to the stochastic force w = (wr, · · ·, wn)· In [HL0UZ 2] the following was proved: (For definition of functional processes etc. see §1). (Hereand in the followingall gradients are taken w.r.t. x). A. Let N = N(¢, t,x,w) be a functional process and define w =-\IN. Assume that (0.1) has a solution of the form u =-\IX for some functional process X processes X ( ¢, t, x, w). Moreover, assume that the functional (0.2) ). Y :=Exp(-X) 2v as well as Y oX and Y <> X 02 exist in V (J-L) for some p 2: 1. Then Y solves the stochastic heat equation (0.3) { ~~ = v..6.Y + 2: Y <> (N +C); t > 0, X ERn Y(O, x) = Exp( 2:x(o, x)) for some C(t, w) not depending on x. (For simplicity we have written Y(t, x) for Y(¢,x,x,w) etc.) Furthermore, we proved the following: B. Let either N = W(¢, t, x, w) ((n + 1)-parameter white noise) or N = ExpW(¢, t, x, w) ((n + 1)-parameter positive noise), and assume that both f(x) := Y(O,x) and C(t,w) = C ( t) are bounded and deterministic (do not depend on w). Then (0.3) has a unique L 2 (J-L) (respectively L 1 (J-L)) functional process solution Y(t, x) given by t (0.4) Y(t, x) = Ex[f(bat)Exp(J H(s, bas)ds)], 0

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Tom Lindstrøm

Papers

Brownian motion on nested fractals

The burgers equation with a noisy force and the stochastic heat equation

Fractional brownian fields as integrals of white noise

Wick multiplication and ito-skorohod stochastic differential equations

The stochastic wick-type burgers equation