论文标题
通过Harris定理的生长裂缝方程的光谱差距
Spectral gap for the growth-fragmentation equation via Harris's Theorem
论文作者
论文摘要
我们研究了生长裂片方程的长期行为，这是一种非局部线性进化方程，描述了结构化种群动力学中广泛现象的范围。我们在条件下显示了通过使用基于哈里斯定理的方法概括文献中的光谱差距的存在，这是马尔可夫过程平衡的研究。在解决了双重Perron特征值问题之后，通过执行$ H $转换来克服方程式不保守的困难。然后，直接佩隆特征向量的存在是我们方法的结果，这证明了进化方程的指数收缩。此外，从双重特征功能和方程的系数方面可以明确量化收敛速率。
We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.
