随机修正的Camassa-Holm方程的大偏差原理

Large deviation principle of stochastically modified Camassa-Holm equation

为了得到随机修正的Camassa-Holm方程中罕见事件发生概率的指数型估计,研究该方程的大偏差原理。利用弱收敛的方法证明正则化随机修正的Camassa-Holm方程的解满足大偏差原理;然后通过建立正则化方程的解的分布与随机修正的Camassa-Holm方程的解的分布之间的指数等价性,得到随机修正的Camassa-Holm方程的解的大偏差原理。结果表明:当随机修正的Camassa-Holm方程中随机干扰的强度充分小时,罕见事件发生大的偏差的概率是指数量级的小。

In order to obtain the exponential estimation of the probability of rare events in the stochastically modified Camassa-Holm equation, the large deviation principle of the equation is studied. The weak convergence method is used to prove that the solution to the regularized and stochastically modified Camassa-Holm equation satisfies the large deviation principle;and then the large deviation principle of the solution to the original equation is obtained by establishing the exponential equivalence between the distribution of the solution to the regularization equation and the solution to the original equation. The results show that when the intensity of the random interference in the stochastically modified Camassa-Holm equation is sufficiently small, the probability of large deviations in rare events is exponentially small.