Solutions of kinetic-type equations with perturbed collisions
We study a class of kinetic-type differential equations \(\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t\), where \(\widehat{\mathcal{Q}}\) is an inhomogeneous smoothing transform and, for every \(t\geq 0\), \(\phi_t\) is the Fourier--Stieltjes transform of a probability measure. We s...
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| creator | Buraczewski, Dariusz Dyszewski, Piotr Marynych, Alexander |
| description | We study a class of kinetic-type differential equations \(\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t\), where \(\widehat{\mathcal{Q}}\) is an inhomogeneous smoothing transform and, for every \(t\geq 0\), \(\phi_t\) is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on \(\widehat{\mathcal{Q}}\) the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to \(\widehat{\mathcal{Q}}\). Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as \(t\to\infty\). |
| doi_str_mv | 10.48550/arxiv.2208.09498 |
| format | Article |
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We show that under mild assumptions on \(\widehat{\mathcal{Q}}\) the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to \(\widehat{\mathcal{Q}}\). Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as \(t\to\infty\).</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.2208.09498</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Asymptotic properties ; Characteristic functions ; Differential equations ; Mathematical analysis ; Mathematics - Probability ; Random walk ; Stochastic processes</subject><ispartof>arXiv.org, 2023-09</ispartof><rights>2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). 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| subjects | Asymptotic properties Characteristic functions Differential equations Mathematical analysis Mathematics - Probability Random walk Stochastic processes |
| title | Solutions of kinetic-type equations with perturbed collisions |
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