%% fm60_baldini_slides.tex

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% 1st slide
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\begin{slide}

\begin{center}\Large\bf
Recent Developments on the\\
Stochastic Burgers Equation
\end{center}

\vspace{2cm}

{\bf Michele Baldini}, {\it New York University, Physics Department}\\
{\bf Henry McKean}, {\it New York University, 
Courant Institute of Mathematical Sciences}

\vspace{2cm}

IDEAS:
\begin{itemize} 
\item Testing ground for the Navier-Stokes turbulence
\item Investigation of the invariant measure
\end{itemize}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Navier-Stokes %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\begin{center}
{\bf\Large Turbulence}
\end{center}


The phenomenon of turbulence has been known 
for centuries and the first mathematical formulation goes 
back to the works of Navier-Stokes (1820-1840).\\
It is general folklore that the turbulence is \\
entirely contained in the Navier-Stokes equation

\[
\frac{\partial{\bf v}}{\partial t}+({\bf v\cdot\nabla}){\bf v}-\mu
{\bf \nabla^2 v}=\frac 1{\rho}({\bf f-\nabla}p)
\]

PROBLEM: \underline{extremely complicated}!!! We need simpler models...
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Burgers equation MAIN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
{\bf The Burgers equation}
\end{center}

\[
\frac{\partial v}
{\partial t}+v\frac{\partial v}{\partial x}-\mu
\frac{\partial^2 v}{\partial x^2}=0
\]

where

$v(x,t)$ is the velocity field\\
$\mu$ is the viscosity\\

\vspace{0.5cm}

Features:\begin{itemize}
\item Same type of non-linearity
\item Presence of a dissipative term from which a 
      Reynolds number can be defined
\item Many invariances and conservation laws in common
\end{itemize}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% flaws for Burgers; CHAOS doesn't arise
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
{\bf\large MAIN FLAW}
\end{center}
     
The Burgers equation can be explicitly integrated. Setting 
\[
v=-2\mu\:\left(\frac{W_x}{W}\right)
\]

it becomes the {\it Heat equation}

\[
\frac{\partial W}{\partial t}=\mu \frac{\partial^2 W}{\partial x^2}
\]

Therefore, there is NOT spontaneous arise of randomness by chaotic dynamics.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Burgers equation STOCHASTIC 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
{\bf The STOCHASTIC Burgers equation}
\end{center}

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

where

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

WARNING: this is NOT a partial differential equation in the usual sense 
(because the Brownian motion is not differentiable). 
The correct formulation is in terms of a diffusion:

\[
dv=e\cdot db - \left(v\frac{\partial v}{\partial x}-\mu
\frac{\partial^2 v}{\partial x^2}\right)\, dt
\]


\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%
%% invariant measure 
%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
{\bf INVARIANT MEASURE}
\end{center}

We are interested in the {\it\bf evolution of the \\
statistical properties} for random solutions. 
In particular we want to study the asymptotic \\
behavior of the statistic when time goes to infinity.\\
The limit distribution is called {\it invariant measure} or 
{\it stationary distribution}. If such an object \\
exists, then:

\begin{itemize}
\item The distribution of the system converges 
  to the invariant measure no matter what the initial state is.
\item If the system is initially distributed according to 
the invariant measure, then it remains in that state forever.
\end{itemize}

\end{slide}

%%%%%%%%%%%%%%%%%%
%% Question
%%%%%%%%%%%%%%%%%

\begin{slide}

\large

QUESTIONS:
\begin{itemize}
\item Does the stochastic Burgers \\
equation admit an invariant measure?
\item If so, how does it look like?
\end{itemize}

\end{slide}

%%%%%%%%%%%%%%%%%%%
% general diffusion
%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
\Large DIFFUSION IN $\cal{R}$
\end{center}

A {\it diffusion} is a Markov process with a \\
continuous path. The general form of a diffusion is:


\LARGE
\[
\mbox{"}dX=\sigma (X)db+m(X)dt \mbox{ "}
\]

\large
$\sigma(X)$= dispersion\\
$m(X)$= drift

\vspace{1cm}


\[
X_t-X_0=\int_0^t \sigma (X_s)db_s + \int_0^t m(X_s)ds
\]
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% pathwise behaviour
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

Pathwise, necessary conditions for the existence 
of a ``nice'' (i.e. integrable) invariant measure are:

\begin{itemize}
\item The drift must be strong enough to balance the randomness.
\item The innovation from the random term $\sigma (X)db$ 
ought to be killed for large $t$ and the process must stabilize.
\end{itemize}

This might be not clear at a first sight...

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Ornstein-Uhlenbeck
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
{\bf The ORNSTEIN-UHLENBECK process}
\end{center}

It was introduced to describe the {\it velocity} field of a Brownian 
pollen grain on the surface of a liquid. In fact, 
if $X(t)$ was the position, the velocity would be meaningless.

\[
dX=db-Xdt
\]

The solution is

\[
X_t=e^{-t}X_0 + \int_0^t e^{s-t}db_s
\]

Note:
\begin{itemize}
\item The process rapidly forgets where it came from
\item In the integral term, 
 the innovation at large time enters with amplitude 1
\item The limit for $t\to\infty$ is not defined
\end{itemize}

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Ornstein-Uhlenbeck with substitution
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
{\bf\large Show that the process stabilize}

No HARM in changing the Brownian motion

Fix a time $T$ and substitute: 
\[ b(t)\longrightarrow b^{\downarrow}(t)\equiv b(T)-b(T-t) \]

\[
dX^{\downarrow}=db^{\downarrow}-X^{\downarrow}dt
\]

The solution is

\begin{eqnarray*}
X^{\downarrow}_t  & = & e^{-t}X_0 + \int_0^t e^{s-t}db^{\downarrow}_s\\
& = & e^{-t}X_0 + \int_0^t e^{-s}db_s
\end{eqnarray*}

Note: \underline{the innovation is damped exponentially fast!!}

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}

\begin{center}
{\bf\large CONCLUSIONS}
\end{center}

{\bf Basic hope}: 
every diffusion with a ``nice'' invariant measure features this mechanism.\\
Unfortunately, 
computation becomes horrible but at least it will keep me 
busy for the next year...

\end{slide}



\end{document}




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