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Système de processus auto-stabilisants

Samuel Herrmann — 2003

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 + B t + a 0 t ϕ * v s ( X s ) d s - ( 1 - a ) 0 t β * u s ( X s ) d s , (E)⎨ ⎩ Y t = Y + B ̃ t + ( 1 - a ) 0 t ϕ * u s ( Y s ) d s - a 0 t β * v s ( Y s ) d s , ( X t d x ) = u t ( d x ) and ( Y t d x ) = v t ( d x ) , where β * u t ( x ) = β ( x - y ) u t ( d y ) , ( B t ) t 0 and ( B ̃ t ) t 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...

Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Samuel HerrmannJulian Tugaut — 2012

ESAIM: Probability and Statistics

In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated to the set of invariant...

Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Samuel HerrmannJulian Tugaut — 2012

ESAIM: Probability and Statistics

In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric...

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