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This volume focuses on quasi-stationary distributions (QSDs) for killed processes, specifically in the context of diffusions with boundary killing and dynamical systems with traps. The authors present QSDs as tools for describing long-term behavior conditioned on survival. The research traces back to Kolmogorov and Yaglom and has gained significant attention in recent decades. Key findings include the exponential distribution property of killing time for QSDs, general results on their existence, and the analysis of trajectories that survive indefinitely. The work covers birth-and-death chains and diffusions, detailing the existence of single or multiple QSDs, convergence to extremal QSDs, and classification of survival processes. Additionally, the authors explore Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. These findings are pertinent to researchers in Markov chains, diffusions, potential theory, and dynamical systems, particularly where extinction is a key concept. The volume includes numerous examples to illustrate the theory and uniquely addresses the distribution behavior of individuals in decaying populations over extended periods. It also lays the groundwork for applications in mathematical ecology, statistical physics, computer science, and economics.
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Quasi-Stationary Distributions, Pierre Collet
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- 2014
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