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We prove gradient bounds for the Markov semigroup of the dynamic$\varphi^4_2$-model on a torus of fixed size $L>0$. For sufficiently large mass$m>0$ these estimates imply exponential contraction of the Markov semigroup.Our method is based on pathwise estimates of the linearized equation. Tocompensate the lack of exponential integrability of the stochastic drivers weuse a stopping time argument and the strong Markov property in the spirit ofCass--Litterer--Lyons. Following the classical approach of Bakry-\'Emery, as acorollary we prove a Poincar\'e/spectral gap inequality for the$\varphi^4_2$-measure of sufficiently large mass $m>0$ with almost optimalcarr\'e du champ.

Gradient-type estimates for the dynamic $\varphi^4_2$-model