Książka - szczegóły

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
⎧$X_{t} = X₀ + B_{t} + a ∫_{0}^{t} ϕ*v_{s}(X_{s})ds - (1-a) ∫_{0}^{t} β*u_{s}(X_{s})ds$,
(E)⎨
⎩$Y_{t} = Y₀ + B̃_{t} + (1-a) ∫_{0}^{t} ϕ*u_{s}(Y_{s})ds - a∫_{0}^{t} β*v_{s}(Y_{s})ds$,
$ℙ(X_{t} ∈ dx) = u_{t}(dx)$ and $ℙ(Y_{t} ∈ dx) = v_{t}(dx)$,
where $β*u_{t}(x) = ∫_{ℝ} β(x-y)u_{t}(dy)$, $(B_{t})_{t≥0}$ and $(B̃_{t})_{t≥0}$ 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 the link between (E) and the associated system of particles.