Asymptotic behaviour on the linear self-interacting diffusion driven by α-stable motion. Issue 8 (17th November 2021)

• Record Type: Journal Article
• Authors: Sun, Xichao; Yan, Litan
• Is Part Of: Stochastics. Volume 93:Issue 8(2021)
• Page Start: 1186
• Page End: 1208
• Publication Date: 2021-11-17
• Subjects: α-Stable motion -- self-interacting diffusion -- stable integral -- convergence with probability one; 60G52 -- 60H05 -- 60H10; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2020.1869239
• Languages: English
• ISSNs: 1744-2508

Abstract

Abstract : 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 1 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 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 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 2 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 .