Synchronisation by noise for the stochastic quantisation equation in dimensions 2 and 3
A talk in the Bielefeld Stochastic Afternoon series by
Pavlos Tsatsoulis from MPI Leipzig
| 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}
\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 (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 (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 Hölder spaces of negative exponent.
The talk is based on a joint work with Benjamin Gess. Please contact stochana@math.uni-bielefeld.de for Meeting-ID and Password. |