论文标题
一种准平台的方法，用于长期渐进式方程的长期渐进性
A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation
论文作者
论文摘要
在生长裂片系统中，细胞大小缓慢生长，并随机分裂。通常，系统中的细胞数量呈指数增长，细胞大小的分布沉降到平衡的“渐近谱”中。在这项工作中，我们介绍了一种新方法，以证明这种生长裂片方程的这种渐近行为，并表明与渐近分布的收敛是以指数级的速度发生的。我们通过识别相关的亚摩尔多夫过程并通过Lyapunov功能条件研究其准平台行为来做到这一点。通过这样做，我们能够在许多常见案例中简化和推广结果，并为他们的研究提供统一的框架。在这项工作的过程中，我们还能够证明在各种情况下，解决增长方程的解决方案的存在和独特性。
In a growth-fragmentation system, cells grow in size slowly and split apart at random. Typically, the number of cells in the system grows exponentially and the distribution of the sizes of cells settles into an equilibrium 'asymptotic profile'. In this work we introduce a new method to prove this asymptotic behaviour for the growth-fragmentation equation, and show that the convergence to the asymptotic profile occurs at exponential rate. We do this by identifying an associated sub-Markov process and studying its quasi-stationary behaviour via a Lyapunov function condition. By doing so, we are able to simplify and generalise results in a number of common cases and offer a unified framework for their study. In the course of this work we are also able to prove the existence and uniqueness of solutions to the growth-fragmentation equation in a wide range of situations.
