A probabilistic study of the set of stationary solutions to spatial kinetic-type equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909452520652800 |
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| author | Mentemeier, Sebastian Verovkin, Glib |
| author_facet | Mentemeier, Sebastian Verovkin, Glib |
| contents | In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Using a representation of the collision operator derived in Bassetti, Ladelli, Matthes (2015) and Dolera, Regazzini (2014), we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under assumptions as weak as possible. Our main objective is a characterisation of the set of stationary solutions, e.g. equilibrium solutions for inelastic kinetic-type equations, which we describe as mixtures of multidimensional stable laws. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_05133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A probabilistic study of the set of stationary solutions to spatial kinetic-type equations Mentemeier, Sebastian Verovkin, Glib Probability Primary: 60J85, Secondary: 60F05, 82C40 In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Using a representation of the collision operator derived in Bassetti, Ladelli, Matthes (2015) and Dolera, Regazzini (2014), we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under assumptions as weak as possible. Our main objective is a characterisation of the set of stationary solutions, e.g. equilibrium solutions for inelastic kinetic-type equations, which we describe as mixtures of multidimensional stable laws. |
| title | A probabilistic study of the set of stationary solutions to spatial kinetic-type equations |
| topic | Probability Primary: 60J85, Secondary: 60F05, 82C40 |
| url | https://arxiv.org/abs/2501.05133 |