Tome (2003)

Système de processus auto-stabilisants

Samuel Herrmann

GDML_Books, (2003), p.

Résumé

Taking an odd increasing Lipschitz-continuous function with polynomial growth β, an odd Lipschitz-continuous and bounded function ϕ satisfying sgn(x)ϕ(x) ≥ 0 and a parameter a ∈ [1/2,1], we consider the (nonlinear) stochastic differential system ⎧ $X_{t} = X ₀ + B_{t} + a \int_{0}^{t} ϕ * v_{s} (X_{s}) d s - (1 - a) \int_{0}^{t} β * u_{s} (X_{s}) d s$ , (E)⎨ ⎩ $Y_{t} = Y ₀ + B ̃_{t} + (1 - a) \int_{0}^{t} ϕ * u_{s} (Y_{s}) d s - a \int_{0}^{t} β * v_{s} (Y_{s}) d s$ , $ℙ (X_{t} \in d x) = u_{t} (d x)$ and $ℙ (Y_{t} \in d x) = v_{t} (d x)$ , where $β * u_{t} (x) = \int_{ℝ} β (x - y) u_{t} (d y)$ , ${(B_{t})}_{t \geq 0}$ and ${(B ̃_{t})}_{t \geq 0}$ are independent Brownian motions. We show that (E) admits a stationary probability measure, and, under some additional conditions, that $(X_{t}, Y_{t})$ converges in distribution to this invariant measure. Moreover we investigate the link between (E) and the associated system of particles.

Zbl 1027.60058

EUDML-ID : urn:eudml:doc:285971

@book{bwmeta1.element.bwnjournal-article-doi-10_4064-dm414-0-1,
     author = {Samuel Herrmann},
     title = {Syst\`eme de processus auto-stabilisants},
     series = {GDML\_Books},
     year = {2003},
     zbl = {1027.60058},
     language = {fra},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-dm414-0-1}
}

Samuel Herrmann. Système de processus auto-stabilisants. GDML_Books (2003),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-dm414-0-1/