Sharp convergence rates for singular SPDEs - the case of KPZ
Conference, From discrete systems to SPDEs, Vienna, Austria
For subcritical singular SPDEs, the theory of regularity structures typically yields, as the mollification scale \(\varepsilon\to0\), a convergence rate of order \(\varepsilon^\gamma\) for some small \(\gamma>0\). The exponent \(\gamma\) is limited by the gap between the optimal regularity of the driving noise and the regularity in which the noise is measured; in this sense, one can trade regularity of the solution for convergence rate. For instance, under the subcritical and no variance blow-up conditions, in the KPZ equation driven by space-time white noise one obtains \(\gamma<\frac14\), while for \(\Phi^4_3\) one gets \(\gamma<\frac13\). In this work, we show that the optimal exponent for KPZ is \(\gamma=\frac12\). More precisely, we prove if \(u_\varepsilon\) denotes the KPZ solution driven by mollified white noise at scale \(\varepsilon\), then \(\varepsilon^{-1/2}(u_\varepsilon - u_{\varepsilon/2})\) is bounded in \(L^p(\Omega)\) for all \(p \ge 2\) and converges in law to a non-trivial limit \(w\), which solves another singular SPDE driven by an independent white noise. This CLT-type result identifies the optimal convergence rate for KPZ. We also conjecture that the same mechanism yields the optimal rates \(\gamma=1\) for \(\Phi^4_3\) and \(\gamma=2\) for \(\Phi^4_2\), which will be studied in forthcoming work.