Moderate Deviations for the Capacity of the Random Walk range in dimension four
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912027463647232 |
|---|---|
| author | Adhikari, Arka Okada, Izumi |
| author_facet | Adhikari, Arka Okada, Izumi |
| contents | In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moderate Deviations for the Capacity of the Random Walk range in dimension four Adhikari, Arka Okada, Izumi Probability 60F15, 60G50 In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation*} |
| title | Moderate Deviations for the Capacity of the Random Walk range in dimension four |
| topic | Probability 60F15, 60G50 |
| url | https://arxiv.org/abs/2310.07685 |