标题： 选择的分支扩散模型：从中性Wright-Fisher情形到包括突变的情形 显示英文标题

标题： A branching diffusion model of selection: from the neutral Wright-Fisher case to the one including mutations

摘要： 我们考虑单位区间上的扩散过程 x_{t}。 Doob变换技术包括选择 x_{t}路径的过程。 变换后过程的分布是粒子分支扩散系统的分布，每个粒子像一个新的过程 tilde{x}_{t}一样扩散，在 x_{t}的基础上叠加一个额外的漂移。 tilde{x}_{t}粒子在某些空间依赖速率 lambda 下发生灭绝和/或分支。 对于这个变换后的过程，在分支扩散的类别中，问题在于粒子系统是次临界、临界还是超临界。 在前两种情况下，以概率一发生灭绝。 我们将这些想法应用于种群遗传学中出现的扩散过程。 在这个设置中，过程 x_{t}是一个 Wright-Fisher (WF) 扩散过程，可能是中性的或带有突变。 我们研究一种特定的 Doob 变换，该变换基于通常适应度参数 sigma 的指数函数。 我们考虑这可能是引入选择或适应度在两种 WF 类扩散中的另一种方式，从而引出分支扩散模型的概念。 对于这种适应度的 Doob 变换模型，通常的选择漂移 sigma x(1-x) 应该叠加到 x_{t}的漂移上，以形成 tilde{x}_{t}，这是可以二元分支的过程。 在第一个中性情况下，存在分支事件产生新粒子与边界吸收杀死粒子之间的权衡。 在我们的假设下，分支扩散过程最终在有限时间内以指数尾部全局灭绝。 在第二种有突变的情况下，存在移除系统中一些粒子的杀死事件与边界反射中粒子存活之间的权衡。 这种分支扩散过程也最终在非常长的有限时间内全局灭绝，具有幂律尾部。 我们的方法依赖于 x_{t}和 tilde{x}_{t}的转移概率核的谱展开。 显示英文摘要

摘要： We consider diffusion processes x_{t} on the unit interval. Doob-transformation techniques consist of a selection of x_{t}-paths procedure. The law of the transformed process is the one of a branching diffusion system of particles, each diffusing like a new process tilde{x}_{t}, superposing an additional drift to the one of x_{t}. Killing and/or branching of tilde{x}_{t}-particles occur at some space-dependent rate lambda. For this transformed process, so in the class of branching diffusions, the question arises as to whether the particle system is sub-critical, critical or super-critical. In the first two cases, extinction occurs with probability one. We apply this circle of ideas to diffusion processes arising in population genetics. In this setup, the process x_{t} is a Wright-Fisher (WF) diffusion, either neutral or with mutations. We study a particular Doob transform which is based on the exponential function in the usual fitness parameter sigma. We have in mind that this is an alternative way to introduce selection or fitness in both WF-like diffusions, leading to branching diffusion models ideas. For this Doob-transform model of fitness, the usual selection drift sigma x(1-x) should be superposed to the one of x_{t} to form tilde{x}_{t} which is the process that can branch, binarily. In the first neutral case, there is a trade-off between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally extinct in finite time with exponential tails. In the second case with mutations, there is a trade-off between killing events removing some particles from the system and reflection at the boundaries where the particles survive. This branching diffusion process also gets eventually globally extinct but in very long finite time with power-law tails. Our approach relies on the spectral expansion of the transition probability kernels of both x_{t} and tilde{x}_{t}.