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\title{Action Minimization and Sharp Interface Limits
for the Allen-Cahn Equation}

\author{Maria G. Reznikoff, University of Bonn}
\date{4 April, 2005}
\maketitle

\noindent  Phase transformation which would never occur in the
deterministic setting becomes possible in the presence of
stochastic perturbations. For instance, the stable state $u\equiv
-1$ of the deterministic Allen-Cahn equation can be stochastically
driven to the opposing state, $u\equiv +1$.  We consider the
Allen-Cahn phase transformation problem and the related large
deviation action functional,
\begin{align*}
\int_0^T\int_{[0,L]^d} |\dot u+(-\Delta u+V'(u))|^2\,dx\,dt.
\end{align*}
When $L$ is fixed and $T\to\infty$, the action minimizer is easy
to identify.  When time is short or joint limits are considered,
however, the story is more complicated.  In the particularly
interesting case of the sharp interface limit, the competing costs
of interface nucleation and interface
propagation lead to a reduced functional with a special form.\\


\noindent Includes joint work with Bob Kohn, Felix Otto, Yoshihiro
Tonegawa, and Eric Vanden-Eijnden.


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