Abstract
We study asymptotic properties of processes X, living in a Riemannian compact manifold M, solution of the stochastic differential equation (SDE)
with W a Brownian vector field, β(t) = alog(t + 1), \(\mu_t = \frac{1}{t} \int_0^t \delta_{X_s}ds\) and \(V\mu_t(x) = \frac{1}{t}\int_0^t V(x, X_s)ds\), V being a smooth function. We show that the asymptotic behavior of μ t can be described by a non-autonomous differential equation. This class of processes extends simulated annealing processes for which V(x, y) is only a function of x. In particular we study the case \(M = {\mathbb{S}}^n\) , the n-dimensional sphere, and V(x, y) = −cos(d(x, y)), where d(x, y) is the distance on \({\mathbb{S}}^n\) , which corresponds to a process attracted by its past trajectory. In this case, it is proved that μ t converges almost surely towards a Dirac measure.
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Raimond, O. Self-interacting diffusions: a simulated annealing version. Probab. Theory Relat. Fields 144, 247–279 (2009). https://doi.org/10.1007/s00440-008-0147-9
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DOI: https://doi.org/10.1007/s00440-008-0147-9