| Abstract: |
| The stochastic quantisation equation (also known as stochastic Allen-Cahn equation) in dimensions two and three is given by
\begin{equation}
\begin{cases}
& (\partial_t - \Delta) u = - \left(u^3 - 3\infty u\right) + u + \xi \
& u|_{t=0} = f,
\end{cases}
\tag{SQE} \label{eq:sqe}
\end{equation}
where $\xi$ is space-time white noise and $f$ is some initial condition of suitable regularity. Here,
the term $-3\infty u$ is reminiscent of renormalisation, otherwise \eqref{eq:sqe} is not well-posed
in dimensions two and three due to the low regularity of space-time white noise. It is known that
the deterministic analogon
\begin{equation*}
\begin{cases}
& (\partial_t - \Delta) u = - u^3 + u \
& u|_{t=0} = f
\end{cases}
\end{equation*}
of \eqref{eq:sqe} has finitely many unstable solutions. In this talk I will discuss how the presence of noise
implies uniform synchronisation, that is, any two trajectories approach each other
with speed which is uniform in the initial condition. More precisely, I will explain how a combination
of ``coming down from infinity`` estimates and order-preservation can be used to obtain uniform synchronisation
with rates. This will be a special case of a more general framework which implies quantified synchronisation
by noise for white noise stochastic semi-flows taking values in H\older spaces of negative exponent.
The talk is based on a joint work with Benjamin Gess. |
|