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Displaying similar documents to “Question 147. Trouver la développée de a 2 x 2 + b 2 y 2 = ( x 2 + y 2 ) 2

Système de processus auto-stabilisants

Samuel Herrmann

Similarity:

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...