In this paper, as an attempt we consider the linear self-interacting diffusion driven by an α-stable motion, which is the solution to the equation $X_t^{\alpha }=M_t^{\alpha }-\theta {\int }_0^t{\int }_0^s(X_s^{\alpha }-X_r^{\alpha })\mathrm{d}r\mathrm{d}s+\nu t,$where $\theta \ne 0$, $\nu \in \mathbb{R}$ and $M^{\alpha }$ is an α-stable motion on $\mathbb{R}$ ($0<\alpha \le 2$). The process is an analogue of the self-attracting diffusion (see Durrett-Rogers, Prob. Theory Related Fields 92 (1992), 337–349, and Cranston-Le Jan, Math. Ann. 303 (1995), 87–93.). The main object of this paper is to prove some limit theorems associated with the solution process $X^{\alpha }$ for $\frac{1}{2}<\alpha \le 2$. When $\theta >0$ we show that ${\psi }_{\alpha }(t)(X_t^{\alpha }-X_{\mathrm{\infty }}^{\alpha })$ converges to an α-stable random variable in distribution, as t tends to infinity, where ${\psi }_{\alpha }(t)=t^{1/\alpha }$ for $1\le \alpha \le 2$ and ${\psi }_{\alpha }(t)=t^{2-\frac{1}{\alpha }}$ for $\frac{1}{2}<\alpha <1$. When $\theta <0$, for all $\frac{1}{2}<\alpha \le 2$ we show that, as $t\to \mathrm{\infty }$, $J_t^{\alpha }(\theta ,\nu ,0):=t{e}^{\frac{1}{2}\theta t^2}{X}_t^{\alpha }$ converges to ${\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta }$ and $\begin{array}{rl}{J}_t^{\alpha }(\theta ,\nu ,n):& =-\theta t^2(J_t^{\alpha }(\theta ,\nu ,n-1)-(2n-3)!!({\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta }))\\ & \to (2n-1)!!({\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta })\end{array}$a.s. for all $n\ge 1$, where $(-1)!!=1$ and ${\xi }_{\mathrm{\infty }}^{\alpha }={\int }_0^{\mathrm{\infty }}s{e}^{\frac{1}{2}\theta s^2}\mathrm{d}{M}_s^{\alpha }$.

在本文中，作为一种尝试，我们考虑由α 稳定运动驱动的线性自相互作用扩散，这是方程的解$X_{吨}^{\alpha }={米}_{吨}^{\alpha }-\theta {\int }_0^{吨}{\int }_0^{秒}(X_{秒}^{\alpha }-X_r^{\alpha })\mathrm{d}r\mathrm{d}秒+\nu 吨,$在哪里 $\theta \ne 0$, $\nu \in \mathbb{电阻}$ 和 ${米}^{\alpha }$是α 稳定运动$\mathbb{电阻}$ ($0<\alpha \le 2$）。该过程类似于自吸扩散（参见 Durrett-Rogers, Prob. Theory Related Fields 92 (1992), 337–349 和 Cranston-Le Jan, Math. Ann. 303 (1995), 87–93。 ）。本文的主要目的是证明一些与求解过程相关的极限定理$X^{\alpha }$ 为了 $\frac{1}{2}<\alpha \le 2$. 什么时候$\theta >0$ 我们表明 ${\psi }_{\alpha }(吨)(X_{吨}^{\alpha }-X_{\mathrm{\infty }}^{\alpha })$收敛于分布中的α 稳定随机变量，因为t趋于无穷大，其中${\psi }_{\alpha }(吨)={吨}^{1/\alpha }$ 为了 $1\le \alpha \le 2$ 和 ${\psi }_{\alpha }(吨)={吨}^{2-\frac{1}{\alpha }}$ 为了 $\frac{1}{2}<\alpha <1$. 什么时候$\theta <0$， 对所有人 $\frac{1}{2}<\alpha \le 2$ 我们证明，作为 $吨\to \mathrm{\infty }$, $J_{吨}^{\alpha }(\theta ,\nu ,0):=吨{\mathrm{电子}}^{\frac{1}{2}\theta {吨}^2}{X}_{吨}^{\alpha }$ 收敛到 ${\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta }$ 和 $\begin{array}{rl}{J}_{吨}^{\alpha }(\theta ,\nu ,n)：& =-\theta {吨}^2(J_{吨}^{\alpha }(\theta ,\nu ,n-1)-(2n-3)！！({\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta }))\\ & \to (2n-1)！！({\xi }_{\mathrm{\infty }}^{\alpha }-\frac{\nu }{\theta })\end{array}$至于所有 $n\ge 1$， 在哪里 $(-1)！！=1$ 和 ${\xi }_{\mathrm{\infty }}^{\alpha }={\int }_0^{\mathrm{\infty }}秒{\mathrm{电子}}^{\frac{1}{2}\theta {秒}^2}\mathrm{d}{米}_{秒}^{\alpha }$.