We study the stochastic quantisation for the fractional $\varphi^4$ theory. The model has been studied by Brydges, Mitter and Scopola in 2003 as a natural extension of $\phi^4$ theories to fractional sub-critical dimensions. The stochastic quantisation equation is given by the (formal) SPDE
\[
(\partial_t + (-\Delta)^{s}) \varphi = - \lambda \varphi^3 + \xi
\]
where $\xi$ is a space-time white noise over the three dimensional torus. The equation is sub-critical for $s > \frac{3}{4}$.
We derive a priori estimates in the full sub-critical regime $s>\frac{3}{4}$. These estimates rule out explosion in finite time and they imply the existence of an invariant measure with a standard Krylov-Bogoliubov argument.
Our proof is based on the strategy developed for the parabolic case $s=1$ in [Chandra, Moinat, Weber, ARMA 2023]. In order to implement this strategy here, a new Schauder estimate for the fractional heat operator is developed. Additionally, several algebraic arguments from [Chandra, Moinat, Weber, ARMA 2023] are streamlined significantly.
This is joint work with Salvador Esquivel (Münster).