论文标题
平稳性附近线性和非线性分支过程的脊柱过程的时间逆转
Time reversal of spinal processes for linear and non-linear branching processes near stationarity
论文作者
论文摘要
我们考虑一个基于竞争的随机人群模型，影响繁殖和生存以及不断变化的环境。特征的变化是通过跳跃过程来描述的，并且可以通过非线性PDE在大种群中近似具有非局部突变算子的动力学。利用该PDE接收非平凡的固定溶液的事实，只要种群保持接近该平衡，我们就可以通过线性的死亡死亡过程来近似非线性随机种群过程。这使我们能够在人口大时得出，这是在固定时间$ t $中均匀采样的个体的祖先血统所满足的方程式，这是过去$ t \ leq t $的祖先特征所构成的路径。这个过程是一个不均匀的马尔可夫过程，但是我们表明，此过程的时间逆转具有非常简单的结构（例如时间均匀的，独立于$ t $）。这扩展了最近的结果，在其中作者研究了使用拉普拉斯运算符的类似模型，但是这些方法基本上依赖于突变的高斯性质。
We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time $T$, which is the path constituted of the traits of the ancestors of this individual in past times $t\leq T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of $T$). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.
