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%%%%        SUMMARY WORKSHOP MAINARDI 20TH DIC 2002                   %%%%%
%%%% Interdisciplinary Workshop "FROM WAVES TO DIFFUSION AND BEYOND"  %%%%%
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%                         20 December 2002				
%                Department of Physics, University of Bologna,
%                      Via Irnerio 46, 40126 BOLOGNA
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\begin{document}

\centerline{INTERDISCIPLINARY WORKSHOP}
%\vsn
\centerline{{\bf \lq\lq FROM  WAVES TO DIFFUSION AND BEYOND\rq\rq} }
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\centerline{{\bf BOLOGNA, Italy : December 20, 2002}}
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%% \title{Recent Developments on the\\
%%  Stochastic Burgers Equation}
%% \author{Henry McKean, {\it New York University, Courant Institute}
%% \and Michele Baldini, {\it New York University, Physics Dept.}}
%% \date{Dec. 20th, 2002}
%% \maketitle

\centerline{{\bf RECENT DEVELOPMENTS ON THE}}
\vsh
\centerline{{\bf STOCHASTIC BURGERS EQUATION}}
\vsh 
\centerline {\bf Michele BALDINI$^1$ and Henry McKEAN$^2$}
\vsh
\centerline{$^1$ Physics Department, New York University}
\centerline{4 Washington Place, Mailbox 83,
New York, NY 10003 - USA}
%% \centerline{{\tt  mlb257@nyu.edu}}
\vsh
\centerline{
$^2$ Courant Institute of Mathematical Sciences,
New York University}
\centerline{Warren Weaver Hall, Mercer Street, New York, NY 10003 - USA}


\vvs

It is well known among physicists and mathematicians interested in 
fluid dynamics that the Burgers equation 
$v_t+vv_x-\mu v_{xx}=0$ (\cite{burgers}, \cite{leveque}) 
is NOT a good model for the phenomenon of turbulence \cite{Frisch}, 
even though it displays a simultaneous presence of  nonlinear as well 
as dissipative terms, essential features for developing a chaotic behaviour.
The main reason for this inadequacy is that the equation
doesn't contain the pressure, which, thanks to its non-local action, 
is deemed to be the responsible term that introduces the statistic.\\
\indent A different and more propitious situation 
occurs when we add ``by hands'' a stochastic force 
on the right hand side of the equation, 
the simplest being a superposition of independent white noises 
multiplied by a periodic function, namely 

\[
\frac{\partial v}
{\partial t}+v\frac{\partial v}{\partial x}-\mu
\frac{\partial^2 v}{\partial x^2}=e\cdot\frac{db}{dt}
\]

where

\[
e\cdot\frac{db}{dt}=
\sum_{n>0}\{\alpha_n \cos{2\pi nx}+\beta_n\sin{2\pi nx}\}\frac{db_n}{dt}
\]

This, of course, has to be interpreted as a stochastic differential equation
and therefore Ito's stochastic integration theory is needed \cite{mckean}.
In particular we are interested in the statistical properties of the solution 
(\cite{weinan}, \cite{sinai}, \cite{polyakov})
and in the study of the invariant measure \cite{daprato}, 
one of the main 
objects in ergodic theory since it gives a 
representation of a statistical steady state 
in which the equation settles down. 
Here we present briefly two of the ideas involved in this investigation:
%%%%%%%%%%%%%%% 
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\indent 1) The main idea in the proof of an existence 
of an invariant measure goes back to W. Doeblin, 
one of the pioneers of probability theory and great 
investigator of Markov chains. 
The briefness of his approach makes it very elegant and manageable 
\cite{mckean2}.\\
\indent 2) In order to study asymptotic behaviour 
of the process we employ a ``substitution of the randomness''. 
Roughly speaking, a stochastic process 
$t\to X(t)\in\mathcal{R}^n$ 
is a probability measure defined on the infinite dimensional space of paths. 
An explicit representation of this measure is a 
difficult task, even in the ``simplest'' 
case of the Wiener measure. 
We explore the shape of this invariant measure 
replacing the original Brownian motion 
with a different Brownian motion which runs backward in time. 
This operation 
does not spoil the statistic of the process 
but luckily brings the innovation back to the origin and helps 
the path to stay confined.
 
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%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{thebibliography}{9}

\bibitem[B]{burgers} J.M. BURGERS: 
 The nonlinear diffusion equation. 
 Reidel Publishing Company, Dordrecht-Boston (1974).

\bibitem[LV]{leveque} R.J. 
 LEVEQUE: Numerical methods for conservation laws. 
 {\it Lectures in Mathematics-ETH Z\"{u}rich}, 
 Birkh\"{a}user Verlag, Basel-Boston-Berlin (1992).

\bibitem[F]{Frisch} 
 U. FRISCH: Turbulence. Cambridge University Press (1995).

\bibitem[S]{sinai} 
 Y.G. SINAI: 
  Burgers system driven by a periodic stochastic flow. 
 {\it Ito's stochastic calculus and probability theory}, 
  Springer, New York, 347-353 (1996).

\bibitem[EKMS]{weinan} 
 WEINAN E, K. KHANIN, A. MAZEL, Y. SINAI: 
 Invariant measures for Burgers equation with stochastic forcing. 
 {\it Ann. Math.} {\bf 151}, 877-960 (2000).

\bibitem[DPZ]{daprato} 
 G. DAPRATO, J. ZABCZYK: 
 Ergodicity for infinite dimensional systems. 
 {\it London Math. Soc. Lecture Note} {\bf 229}, 
 Cambridge University Press (1996).

\bibitem[MK]{mckean} 
 H. MCKEAN: 
 Stochastic integrals. Academic Press, New York (1969).

\bibitem[P]{polyakov} 
A. POLYAKOV: 
Turbulence without pressure. 
{\it Physical Review E} {\bf 52}(6), 6183-6188 (1995).

\bibitem[MK2]{mckean2} 
H. MCKEAN: 
Turbulence without pressure: existence of the invariant measure. 
{\it To appear}.

\end{thebibliography} 

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