The asymptotic behavior of fraudulent algorithms

Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m\ge 2,$ satisfying $\min U=0$ and let $\mathcal{U}=\{x\in M:U(x)=0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(\beta )}:=(X^{(\beta )}(t){)}_{t\ge 0}$ defined on $N=M\setminus \mathcal{U},$ whose generator is$U\mathrm{\Delta }\cdot -\beta ⟨\mathrm{\nabla }U,\mathrm{\nabla }\cdot ⟩$, where $\beta \in \text{RR}$ is a real parameter.We show that for $\beta >\frac{m}{2}-1,$ $X^{(\beta )}(t)$ converges a.s.\ as $t\to \mathrm{\infty }$, 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.