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Continuous flows driving branching processes and their nonlinear evolution equations
Advances in Nonlinear Analysis ( IF 3.2 ) Pub Date : 2022-01-01 , DOI: 10.1515/anona-2021-0229
Lucian Beznea 1 , Cătălin Ioan Vrabie 2
Affiliation  

We consider on M (ℝ d ) (the set of all finite measures on ℝ d ) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFk F \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d -dimensional Brownian motion and the non-local branching process on the finite configurations of M (ℝ d ), induced by the sequence ( b k ) k ⩾1 of positive numbers such that ∑k⩾1bk⩽1 \sum\nolimits_{k \geqslant 1} b_k \leqslant 1 . It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝ d instead of the d -dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.

中文翻译:

连续流动驱动分支过程及其非线性演化方程

我们在 M (ℝ d ) 上(ℝ d 上所有有限度量的集合)考虑与非线性算子 F↦ΔF′+∑k⩾1bkFk F \mapsto \Delta F' + \sum\nolimits_{k 相关的演化方程\geqslant 1} b_k F^k ,其中 F' 是 F 的变分导数,我们证明它有一个由 d 维布朗运动的分布和有限配置上的非局部分支过程表示的解的 M (ℝ d ),由正数的序列 ( bk ) k ⩾1 诱导,使得 ∑k⩾1bk⩽1 \sum\nolimits_{k \geqslant 1} b_k \leqslant 1 。事实证明,对于得到的方程的解,该表示也适用于相同的分支过程,用 ℝ d 上的马尔可夫过程的生成器代替拉普拉斯算子而不是 d 维布朗运动;更一般的,我们可以在 Lusin 拓扑空间上使用右马尔可夫过程的生成器。我们首先研究驱动分支过程的连续流。我们表明,如果超过程的分支机制与空间变量无关,则通过在度量上的右连续流的时间演化中引入分支来获得超过程,该分支由右连续流作为空间运动典型地诱导。相应的结果适用于空间运动状态空间的所有有限配置的集合上的非局部分支过程。然后通过在度量上引入右连续流的时间演化中的分支来获得超过程,该分支由右连续流作为空间运动典型地诱导。相应的结果适用于空间运动状态空间的所有有限配置的集合上的非局部分支过程。然后通过在度量上引入右连续流的时间演化中的分支来获得超过程,该分支由右连续流作为空间运动典型地诱导。相应的结果适用于空间运动状态空间的所有有限配置的集合上的非局部分支过程。
更新日期:2022-01-01
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