发表机构
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University; Center for Applied Mathematics, Tianjin University; Department of Financial and Actuarial Mathematics, Xi’an Jiaotong-Liverpool University（山东大学数学与交叉科学研究中心; 天津大学应用数学中心; 西交利物浦大学金融精算数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究相关高斯环境下钉扎模型的标度极限，在特定参数假设下证明其配分函数依分布收敛于分数阶随机热方程的$L^1$解，拓展了$L^2$区域外的相关结果。
AI 中文摘要
本文研究相关高斯环境下钉扎模型的标度极限。模型底层更新过程的尾概率服从指数为$α>0$的多项式衰减。高斯环境$\{ω_n\\_\{n\in\mathbb N\}\}$的协方差满足$\text{Cov}_\mathbb P(ω_n,ω_m)\sim |n-m|^{2H-2}$，其中$H\in(0,1)$。在假设$α\in(0,\frac12]$、$H\in(\frac12,1)$且$α+2H>2$的条件下，我们证明无序钉扎模型的配分函数在适当标度下，依分布收敛于由时间相关且局域在原点的高斯噪声驱动的分数阶随机热方程的$L^1$解。特别地，已知当$α<\frac12$时，该解不具备$L^2$可积性。
英文摘要
In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent $α>0$. The covariance of the Gaussian environment $\{ω_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(ω_n,ω_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $α\in(0,\frac12]$, $H\in(\frac12,1)$ and $α+2H>2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not $L^2$-integrable when $α<\frac12$.
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