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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.04792 |
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| _version_ | 1866917979100282880 |
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| author | Avneri, Eran Mytnik, Leonid |
| author_facet | Avneri, Eran Mytnik, Leonid |
| contents | The symbiotic branching model in $\mathbb{R}$ describes the behavior of two branching populations migrating in space $\mathbb{R}$ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter $ρ$, which takes values in $[-1,1]$ and governs the correlation between the branching mechanisms of the two populations. While existence and uniqueness for this system were established for $ρ\in [-1,1)$, weak uniqueness for the completely positively correlated case of $ρ= 1$ has been an open problem. In this paper, we resolve this problem, establishing weak uniqueness for the corresponding system of stochastic partial differential equations. The proof uses a new duality between the symbiotic branching model and the well-known parabolic Anderson model. Furthermore, we use this duality to investigate the long-term behavior of the completely positively correlated symbiotic branching model. We show that, under suitable initial conditions, after a long time, one of the populations dies out. We treat the case of integrable initial conditions and the case of bounded non-integrable initial conditions with well-defined mean. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04792 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model Avneri, Eran Mytnik, Leonid Probability The symbiotic branching model in $\mathbb{R}$ describes the behavior of two branching populations migrating in space $\mathbb{R}$ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter $ρ$, which takes values in $[-1,1]$ and governs the correlation between the branching mechanisms of the two populations. While existence and uniqueness for this system were established for $ρ\in [-1,1)$, weak uniqueness for the completely positively correlated case of $ρ= 1$ has been an open problem. In this paper, we resolve this problem, establishing weak uniqueness for the corresponding system of stochastic partial differential equations. The proof uses a new duality between the symbiotic branching model and the well-known parabolic Anderson model. Furthermore, we use this duality to investigate the long-term behavior of the completely positively correlated symbiotic branching model. We show that, under suitable initial conditions, after a long time, one of the populations dies out. We treat the case of integrable initial conditions and the case of bounded non-integrable initial conditions with well-defined mean. |
| title | Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model |
| topic | Probability |
| url | https://arxiv.org/abs/2504.04792 |