A specific $N$-Particle Process of Fleming-Viot-Type Alexander Klump Institute of Mathematics May, 2019
Seed for Poisson Process: Seed for Brownian Motion:
  • We consider cells $(X^1 , \ldots , X^N )$ with constant population size $N$
  • Mutation: Cells are damaged and repaired over time
  • Selection:
    • At random times a random cell reproduces itself
      (including their damage)
    • Reproducing causes the death of the most damaged cell
  • Modelling:
    • Selection Times: Poisson Process
    • Mutation: Brownian Motion
  • Question:
    • Can the populations' total damage grow "larger and larger" over time?
    • $S^N_t := X^1_t + \ldots + X^N_t \overset{t\to\infty}{\longrightarrow} \infty$?
  • Without selection the answer would be:
    • No, if $N = 2$
    • Yes, if $N> 2$
  • Negative answer for the selection-process:

    Theorem:
    For all $N$ exists $R > 0$, such that $\max \{ t \geq 0 : S^N_t \leq R\} = \infty$