A stochastic quantization equation \[ (\partial_t-\Delta)u=-(u^3-3\infty u)+u+\xi,\quad u|_{t=0}=f\tag{1} \] is considered on \((0,\infty)\times\mathbb T^d\) for \(d\in\{2,3\}\), where \(\xi\) is a space-time white noise and \(\mathbb T^d\) is a \(d\)-dimensional torus of an arbitrary size. A solution \(u\) of this equation is understood in the sense of renormalizations, i.e., \(u\) a nontrivial limit of \(u_\varepsilon\) that solve \[ (\partial_t-\Delta)u_\varepsilon=-(u_\varepsilon^3-3C_\varepsilon u_\varepsilon)+u_\varepsilon+\xi_\varepsilon,\quad u_\varepsilon|_{t=0}=f, \] where \(\xi_\varepsilon=\xi*\rho_\varepsilon\) for a suitable mollifier \(\rho_\varepsilon\) and for suitable renormalization constants \(C_\varepsilon\nearrow\infty\) as \(\varepsilon\searrow 0\) (the limit \(u\) is independent of \(\rho_\varepsilon\)). Let \(\psi_s\) denote the periodic heat kernel on \(\mathbb T^d\) and let \(\alpha>0\). Define \[ \|f\|_{-\alpha}=\sup_{s\in(0,1]}s^\frac{\alpha}{2}\|\psi_s*f\|_{L^\infty(\mathbb T^d)} \] and let \(\mathcal C^{-\alpha}\) be defined as the closure of \(C^\infty\)-period functions with respect to \(\|\cdot\|_{-\alpha}\). Under this set-up, the authors assert the following. Let \(\alpha_0=\mathbf 1_{\{d=3\}}\frac{1}{2}+\theta_0\) for a sufficiently small positive \(\theta_0\). Then there exists \(\lambda_*>0\) such that, for every \(\alpha\in(\alpha_0-\delta,\alpha_0]\), \(0<\delta<\theta_0\) and \(p>\frac{d}{\alpha-\alpha_0+\delta}\) integer \[ \left(\mathbf E\left(\sup_{f_1,f_2\in\mathcal C^{-\alpha_0}}\|u(t;f_2)-u(t;f_1)\|_{-\alpha}\right)^p\right)^\frac{1}{p}\le Ce^{-\frac{\lambda_*}{p}t} \] holds where \(C\) depends on \(\alpha\), \(d\) and \(p\). Moreover, there exists a stationary family of random variables \(\{\eta(t)\}_{t\ge 0}\) in \(L^p(\Omega;\mathcal C^{-\alpha})\) with the law of \(\eta(0)\) being the invariant measure for (1) such that \[ \left(\mathbf E\left(\sup_{f\in\mathcal C^{-\alpha_0}}\|u(t;f)-\eta(t)\|_{-\alpha}\right)^p\right)^\frac{1}{p}\le Ce^{-\frac{\lambda_*}{p}t}. \]