A-priori bounds for singular SPDEs
A talk in the Keine Reihe series by
Hendrik Weber from University of Bath
| Abstract: | The theory of regularity structures is a powerful tool to develop a stable solution theory for a whole class of stochastic PDEs arising in statistical mechanics and quantum field theory. Initiated in Hairer's groundbreaking work in 2013, in only a few years an astonishingly general solution theory covering essentially all equations which satisfy a certain scaling condition (subcriticality or super-renormalizability), has been developed. However, up to now, most results only gave control over solutions for small times and on bounded spatial domains.
The aim if this talk is to present a method to prove a priori estimates in the framework regularity structures. These bounds complement the short time existence and uniqueness theory to obtain control of solutions globally in time and on unbounded domains. Our bounds are implemented in the example of the dynamic $\phi^4$ equation, which is formally given by
\begin{equation*}
(\partial_t - \Delta) u = -u^3 + \infty u +\xi.
\end{equation*}
This equation is subcritical if the distribution $\xi$ is of class $C^{-3+\frac{\delta}{2}}$ for $\delta>0$, and we obtain bounds for all such $\xi$. An analogy to the regularity of white noise suggests to interpret this as a solution theory for $\phi^4$ in all fractional dimensions $<4$.
This is joint work with Ajay Chandra and Augustin Moinat. |