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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 ⎧$X_t=X₀+B_t+a{\int }_0^t\varphi *v_s\left(X_s\right)ds-(1-a){\int }_0^t\beta *u_s\left(X_s\right)ds$, (E)⎨ ⎩$Y_t=Y₀+B{̃}_t+(1-a){\int }_0^t\varphi *u_s\left(Y_s\right)ds-a{\int }_0^t\beta *v_s\left(Y_s\right)ds$, $\mathbb{P}\left(X_t\in dx\right)=u_t\left(dx\right)$ and $\mathbb{P}\left(Y_t\in dx\right)=v_t\left(dx\right)$, where $\beta *u_t\left(x\right)={\int }_{ℝ}\beta (x-y)u_t\left(dy\right)$, ${\left(B_t\right)}_{t\ge 0}$ and ${\left(B{̃}_t\right)}_{t\ge 0}$ 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...

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...

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...