Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917188363878400 |
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| author | Denisov, Denis Elizarov, Nikita Wachtel, Vitali |
| author_facet | Denisov, Denis Elizarov, Nikita Wachtel, Vitali |
| contents | In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $α\in(0,π]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $α=π/m$ with some integer $m$. Our proof is constructive and allows one to give exact expressions for harmonic polynomials for every integer $m$. Furthermore, we give exact expressions for all finite moments of the exit time, this result is valid for all angles $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03866 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones Denisov, Denis Elizarov, Nikita Wachtel, Vitali Probability 60G50 In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $α\in(0,π]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $α=π/m$ with some integer $m$. Our proof is constructive and allows one to give exact expressions for harmonic polynomials for every integer $m$. Furthermore, we give exact expressions for all finite moments of the exit time, this result is valid for all angles $α$. |
| title | Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones |
| topic | Probability 60G50 |
| url | https://arxiv.org/abs/2601.03866 |