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We prove uniform synchronisation by noise with rates for the stochasticquantisation equation in dimensions two and three. The proof relies on acombination of coming down from infinity estimates and the framework oforder-preserving Markov semigroups derived in [Butkovsky, Scheutzow; 2019]. Inparticular, it is shown that this framework can be applied to the case of statespaces given in terms of H\"older spaces of negative exponent.

Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$