Gaussian deconvolution on $\mathbb R^d$ with application to self-repellent Brownian motion
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918526994874368 |
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| author | Liu, Yucheng |
| author_facet | Liu, Yucheng |
| contents | We consider the convolution equation $(δ- J) * G = g$ on $\mathbb R^d$, $d>2$, where $δ$ is the Dirac delta function and $J,g$ are given functions. We provide conditions on $J, g$ that ensure the deconvolution $G(x)$ to decay as $( x \cdot Σ^{-1} x)^{-(d-2)/2}$ for large $|x|$, where $Σ$ is a positive-definite diagonal matrix. This extends a recent deconvolution theorem on $\mathbb Z^d$ proved by the author and Slade to the possibly anisotropic, continuum setting while maintaining its simplicity. Our motivation comes from studies of statistical mechanical models on $\mathbb R^d$ based on the lace expansion. As an example, we apply our theorem to a self-repellent Brownian motion in dimensions $d>4$, proving its critical two-point function to decay as $|x|^{-(d-2)}$, like the Green function of the Laplace operator $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian deconvolution on $\mathbb R^d$ with application to self-repellent Brownian motion Liu, Yucheng Probability Mathematical Physics 42B10, 60K35, 82B21, 82B27 We consider the convolution equation $(δ- J) * G = g$ on $\mathbb R^d$, $d>2$, where $δ$ is the Dirac delta function and $J,g$ are given functions. We provide conditions on $J, g$ that ensure the deconvolution $G(x)$ to decay as $( x \cdot Σ^{-1} x)^{-(d-2)/2}$ for large $|x|$, where $Σ$ is a positive-definite diagonal matrix. This extends a recent deconvolution theorem on $\mathbb Z^d$ proved by the author and Slade to the possibly anisotropic, continuum setting while maintaining its simplicity. Our motivation comes from studies of statistical mechanical models on $\mathbb R^d$ based on the lace expansion. As an example, we apply our theorem to a self-repellent Brownian motion in dimensions $d>4$, proving its critical two-point function to decay as $|x|^{-(d-2)}$, like the Green function of the Laplace operator $Δ$. |
| title | Gaussian deconvolution on $\mathbb R^d$ with application to self-repellent Brownian motion |
| topic | Probability Mathematical Physics 42B10, 60K35, 82B21, 82B27 |
| url | https://arxiv.org/abs/2411.16058 |