非自伴 Fokker-Planck 算子的振荡生存概率与特征值
偏微分方程分析
2014-06-02 v1
摘要
我们演示了随机动力学 d \x \eps = \mb a ( \x \eps ) d t + 2 \eps \mb b ( \x \eps ) d \w d\x_\eps=\mb{a}(\x_\eps)\, dt +\sqrt{2\eps}\,\mb{b}(\x_\eps)\,d\w d \x \eps = \mb a ( \x \eps ) d t + 2 \eps \mb b ( \x \eps ) d \w \mb a ( \x ) \mb{a}(\x) \mb a ( \x ) D D D \p D \p D \p D \mb a ( \x ) \mb{a}(\x) \mb a ( \x ) D D D L \eps u ( \x ) = \eps ∑ i , j = 1 2 \p 2 [ σ i , j ( \x ) u ( \x ) ] \p x i \p x j − ∑ i = 1 2 \p [ a i ( \x ) u ( \x ) ] \p x i = − λ \eps u ( \x ) , L_\eps u(\x)=\,\eps\sum_{i,j=1}^2 \frac{\p ^2\left[ \sigma ^{i,j}\left(\x\right) u(\x) \right]}{\p x^i\p x^j}-\sum_{i=1}^2\frac {\p \left[ a^i\left(\x\right) u(\x)\right]} {\p x^i} =-\lambda_\eps u(\x), L \eps u ( \x ) = \eps i , j = 1 ∑ 2 \p x i \p x j \p 2 [ σ i , j ( \x ) u ( \x ) ] − i = 1 ∑ 2 \p x i \p [ a i ( \x ) u ( \x ) ] = − λ \eps u ( \x ) , \mb σ ( \x ) = \mb b ( \x ) \mb b T ( \x ) \mb{\sigma}(\x)=\mb{b}(\x)\mb{b}^T(\x) \mb σ ( \x ) = \mb b ( \x ) \mb b T ( \x ) \eps \eps \eps λ \eps \lambda_\eps λ \eps \eps → 0 \eps\to0 \eps → 0 \eps \eps \eps λ m , n = n ω 1 + m i ω 2 + O ( \eps ) \lambda_{m,n}=n\omega_1+mi\omega_2+O(\eps) λ m , n = n ω 1 + mi ω 2 + O ( \eps ) n = 1 , 2 , … , m = ± 1 , … n=1,2,\ldots,\,m=\pm1,\ldots n = 1 , 2 , … , m = ± 1 , … ω 1 \omega_1 ω 1 ω 2 \omega_2 ω 2 L \eps L_\eps L \eps L \eps ∗ L^*_\eps L \eps ∗
引用
@article{arxiv.1405.7821,
title = {Oscillatory survival probability and eigenvalues of the non-self adjoint Fokker-Planck operator},
author = {David Holcman and Zeev Schuss},
journal= {arXiv preprint arXiv:1405.7821},
year = {2014}
}
备注
16 pages. Asymptotic for the spectrum of non self-adjoint operator. To appear in MMS, SIAM Multiscale Modeling and Simulations 2014