Small Noise Perturbations in Multidimensional Case
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912192424574976 |
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| author | Pilipenko, Andrey Proske, Frank Norbert |
| author_facet | Pilipenko, Andrey Proske, Frank Norbert |
| contents | In this paper we study zero-noise limits of $α-$stable noise perturbed ODE's which are driven by an irregular vector field $A$ with asymptotics $% A(x)\sim \overline{a}(\frac{x}{\left\vert x\right\vert })\left\vert x\right\vert ^{β-1}x$ at zero, where $\overline{a}>0$ is a continuous function and $β\in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico \cite{Ba} and Bafico, Baldi \cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in \cite% {PP_self_similar} and based on the analysis of an SDE for $t\longrightarrow \infty $, which is obtained through a transformation of the perturbed ODE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_09935 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Small Noise Perturbations in Multidimensional Case Pilipenko, Andrey Proske, Frank Norbert Probability 60H10, 49N60 In this paper we study zero-noise limits of $α-$stable noise perturbed ODE's which are driven by an irregular vector field $A$ with asymptotics $% A(x)\sim \overline{a}(\frac{x}{\left\vert x\right\vert })\left\vert x\right\vert ^{β-1}x$ at zero, where $\overline{a}>0$ is a continuous function and $β\in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico \cite{Ba} and Bafico, Baldi \cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in \cite% {PP_self_similar} and based on the analysis of an SDE for $t\longrightarrow \infty $, which is obtained through a transformation of the perturbed ODE. |
| title | Small Noise Perturbations in Multidimensional Case |
| topic | Probability 60H10, 49N60 |
| url | https://arxiv.org/abs/2106.09935 |