Asymptotic behaviour on the linear self-interacting diffusion driven by α-stable motion.

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 α = M t α − θ ∫ 0 t ∫ 0 s (X s α − X r α) d r d s + ν t , where θ ≠ 0 , ν ∈ R and M α is an α-stable motion on R ( 0 < α ≤ 2). The process is an analogue of the self-attracting diffusion (see Durrett-Rogers, Prob. Theory Related Fields92 (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 α for 1 2 < α ≤ 2. When θ > 0 we show that ψ α (t) (X t α − X ∞ α) converges to an α-stable random variable in distribution, as t tends to infinity, where ψ α (t) = t 1 / α for 1 ≤ α ≤ 2 and ψ α (t) = t 2 − 1 α for 1 2 < α < 1. When θ < 0 , for all 1 2 < α ≤ 2 we show that, as t → ∞ , J t α (θ , ν , 0) := t e 1 2 θ t 2 X t α converges to ξ ∞ α − ν θ and J t α (θ , ν , n) : = − θ t 2 (J t α (θ , ν , n − 1) − (2 n − 3) ! ! (ξ ∞ α − ν θ)) → (2 n − 1) ! ! (ξ ∞ α − ν θ) a.s. for all n ≥ 1 , where (− 1) ! ! = 1 and ξ ∞ α = ∫ 0 ∞ s e 1 2 θ s 2 d M s α . [ABSTRACT FROM AUTHOR]