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2020 Gradient estimates and maximal dissipativity for the Kolmogorov operator in $\Phi ^{4}_{2}$
Giuseppe Da Prato, Arnaud Debussche
Electron. Commun. Probab. 25: 1-16 (2020). DOI: 10.1214/20-ECP294

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.

Citation

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Giuseppe Da Prato. Arnaud Debussche. "Gradient estimates and maximal dissipativity for the Kolmogorov operator in $\Phi ^{4}_{2}$." Electron. Commun. Probab. 25 1 - 16, 2020. https://doi.org/10.1214/20-ECP294

Information

Received: 4 October 2019; Accepted: 22 January 2020; Published: 2020
First available in Project Euclid: 30 January 2020

zbMATH: 1434.60151
MathSciNet: MR4066302
Digital Object Identifier: 10.1214/20-ECP294

Subjects:
Primary: 35R15, 60H15, 81S20, NA28C20

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