Let $E$ be a $\mathbb R$-Banach space, $\rho$ be a metric on $E$, $\delta_x$ denote the Dirac measure on $(E,\mathcal B(E))$ at $x$ for $x\in E$ and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$ satisfying the following property: If $\delta>0$, there is a $t_0\ge0$ so that for all $t\ge t_0$ there is a $a>0$ with $$\inf_{\substack{(x,\:y)\:\in\:E^2\\\left\|x\right\|_E,\:\left\|y\right\|_E\:\le\:c}}\sup_{\substack{\gamma\\\gamma\text{ is a coupling of }\delta_x\kappa_t\text{ and }\delta_y\kappa_t}}\gamma\left(\left\{\rho<\delta\right\}\right)\ge a\tag1.$$
In which sense is this an "irreducibility" condition on $(\kappa_t)_{t\ge0}$? If $(\Omega,\mathcal A,\operatorname P)$ is a probability space and $X:\Omega\times[0,\infty)\times E$ is a stochastic flow, $X^x_t(\omega):=X(\omega,t,x)$ for $(\omega,t,x)\in\Omega\times[0,\infty)\times E$ and $$\kappa_t(x,B)=\operatorname P\left[X^x_t\in B\right]\;\;\;\text{for all }(x,B)\in E\times\mathcal B(E)\text{ and }t\ge0\tag2,$$ how can we describe $(1)$ in plain English. I've got some problems to understand the intuitive meaning of $(1)$ for the flow.