On geometric convergence rate of Markov search towards the fat target

Let $f:A\to \mathbb{R}$ be a continuous function with the minimal value $f^{?}$, where $A$ is the compact metric space. Let ${\left\{X_t\right\}}_{t\in \mathbb{N}}$ be a Markov chain which represents the global optimization process on $A$. We present sufficient conditions for very strong, geometric convergence mode of the form $Ef\left(X_t\right)?f^1\le c^t?(Ef\left(X_0\right)?f^1)$, where $c\in (0,1)$ is some constant. This convergence mode is natural if the set of global minima is fat.