The asymptotic behavior of fraudulent algorithms

Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(\beta)}:=(X^{(\beta)}(t))_{t\geq 0}$ defined on $N = M \setminus \mathcal{U},$ whose generator is$U \Delta \cdot-\beta\langle \nabla U,\nabla \cdot\rangle$, where $\beta\in\RR$ is a real parameter.We show that for $\beta>\frac{m}{2}-1,$ $X^{(\beta)}(t)$ converges a.s.\ as $t \rightarrow \infty$, toward a point $p \in \mathcal{U}$ and that each $p \in \mathcal{U}$ has a positive probability to be selected. On the other hand, for $\beta < \frac{m}{2}-1,$ the law of $(X^{(\beta)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $\pi_{\beta}$ having density proportional to $U(x)^{-1-\beta}$ with respect to the Riemannian measure.