The asymptotic behavior of fraudulent algorithms

Let U be a Morse function on a compact connected m-dimensional Riemannian manifold, m≥2, satisfying minU=0 and let U={x∈M:U(x)=0} be the set of global minimizers. Consider the stochastic algorithm X(β):=(X(β)(t))t≥0 defined on N=M∖U, whose generator isUΔ⋅−β⟨∇U,∇⋅⟩, where $\beta\in\RR$ is a real parameter.We show that for β>m2−1, X(β)(t) converges a.s.\ as t→∞, toward a point p∈U and that each p∈U has a positive probability to be selected. On the other hand, for β<m2−1, the law of (X(β)(t)) converges in total variation (at an exponential rate) toward the probability measure πβ having density proportional to U(x)−1−β with respect to the Riemannian measure.