Gradient estimates and maximal dissipativity for the Kolmogorov operator in $\Phi ^{4}_{2}$

Abstract

We consider the transition semigroup $P_{t}$ of the $\Phi ^{4}_{2}$ stochastic quantisation on the torus $\mathbb {T}^{2}$ and prove the following new estimate (Theorem 3.10) \[ |DP_{t} \varphi (x)\cdot h|\le c\,t^{-\beta }\,|h|_{C^{-s}}\|\varphi \|_{0}\,(1+|x|_{C^{- \alpha }})^{\gamma }, \] for some $ \alpha ,\beta ,\gamma ,s$ positive. Thanks to this estimate, we show that cylindrical functions are a core for the corresponding Kolmogorov equation. Some consequences of this fact are discussed in a final remark.