Asymptotic arc-sine laws for finite-dimensional interacting diffussion
Consider finite-dimensional interacting diffusion processes governed by \[ dX_i(t) = \alpha(X_i(t)) dW_i(t) + \sum_{j \in S} A_{ij} (X_j(t)-X_i(t)) dt ,\quad i \in S, t \geq 0, \] where \((W_i)_{i \in S}\) is a system of independent, one-dimensional standard Wiener processes, \((A_{ij})\) a matrix with nonnegative elements, \(S\) a finite set and \(\alpha : \mathbb{R} \to \mathbb{R}_+\) is a Borel-measurbale function of linearly bounded growth. The author proves that the distribution of related occupation time at the first quadrant \(\mathbb{R}_+^S = [0,\infty)^S\) of \(\mathbb{R}^S\) given by \[ \frac{1}{t} \int^t_0 I_{\mathbb{R}_+^S} (X(u)) du \] converges to a generalized arc-sine law as \(t\) tends to \(+\infty\).
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