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arXiv 2303.09110math.PR
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具有长跳跃的多种群零程过程的平稳涨落

Stationary fluctuations for a multi-species zero range process with long jumps

Linjie Zhao

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摘要：

考虑一维多种群零程过程的平稳涨落，当$x > 0$为$p(x) = c_+ |x|^{-1-α}$，当$x < 0$为$= c_-|x|^{-1-α}$时，转移概率核为$p(x) = c_+ |x|^{-1-α}$.上面的$c_{\pm} \geq 0, α> 0$是参数。证明了对于$0 < α< 3/2$，密度涨落场收敛于一个耦合分数阶Ornstein-Uhlenbeck过程的定态解;对于$α=3/2$，极限点集中于一个耦合分数阶Burgers方程的定态能量解.

英文摘要：

We consider stationary fluctuations for the multi-species zero range process with long jumps in one dimension, where the underlying transition probability kernel is $p(x) = c_+ |x|^{-1-α}$ if $x > 0$ and $= c_-|x|^{-1-α}$ if $x < 0$. Above, $c_{\pm} \geq 0, α> 0$ are parameters. We prove that for $0 < α< 3/2$, the density fluctuation fields converge to the stationary solution of a coupled fractional Ornstein-Uhlenbeck process, and for $α=3/2$, the limit points are concentrated on stationary energy solutions to a coupled fractional Burgers equation.

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