双时间尺度的随机偏微分方程
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摘要
自然科学和工程中的一些复杂系统可以由多时间尺度的随机常微分方程或随机偏微分方程来刻画.这类随机系统的定性分析引起了研究者广泛的兴趣.本文研究双时间尺度的随机偏微分方程,主要包含两个方面的内容：双时间尺度随机偏微分方程的强收敛意义下的平均化原理和双时间尺度随机偏微分方程的随机惯性流形.
本文的安排如下：
在第一章中,介绍一些随机过程理论和随机动力系统理论中的基本的概念,定义,以及著名的Ito公式.
在第二章中,研究由加性噪声驱动的随机FitzHugh-Nagumo系统.在耗散性条件下,可以证明快运动方程对应的“固定”方程存在惟一满足指数混合性的不变测度.由此,通过“平均化“慢运动方程的漂移系数,可以导出一个慢运动方程的有效方程.接下来,通过估计慢运动方程的解过程和有效方程的解过程在适当空间中的偏差,可以证明强收敛意义的随机平均化原理成立.
在第三章中,考虑一个有界开区间上的双时间尺度随机抛物方程.这类模型源于状态空间被噪声扰动的物理系统,且此处所考虑的乘性噪声对快慢两个运动都有扰动.应用类似于第二章中的技巧,证明强收敛意义下的平均化原则对于这类随机抛物方程是成立的,由于考虑更为一般的乘性噪声,因而其证明过程有所不同,且比较复杂.在适当的条件下,当固定慢变量时,可以得到快变量方程的一个“固定”方程,其指数混合性的不变测度是存在且唯一的.由此可以得到刻画慢运动性态的一个平均化方程.最后,应用Burkholder-Davis-Gundy不等式,可以证明当时间尺度参数趋于零时,慢运动方程的解过程强收敛于这个平均化方程的解过程.
在第四章中,研究无穷维空间中双时间尺度的随机动力系统的不变流形,这类随机系统由一类耦合的快-慢随机发展方程所生成的,其具体的形式可以是耦合的随机偏微分方程,也可以是耦合的随机偏微分方程-常微分方程.在适当的条件下,可以证明上述的系统存在一个具有指数吸引性的随机不变流形.进一步还证明当时间尺度参数趋于零时,这个不变流形趋于一个所谓的慢流形.通过几个例子,上面的结论可以适用于耦合的抛物-双曲型偏微分方程,耦合的抛物-常微分方程以及耦合的双曲-双曲型偏微分方程.
在最后一章中,对全文进行总结并指出本文需要改进的工作.
Some complex systems in science and engineering are described by stochastic ordinary differential equations (SODEs) or stochastic partial differential equations (SPDEs) with multiple time scales. The qualitative analysis for such stochastic sys-tems has been drawing more and more attention. This thesis mainly studies the asymp-totic behavior of SPDEs with two characteristic time-scales. It consists of two topics: the strong convergence in stochastic averaging principle for two time-scales SDPEs and the inertial manifolds reduction for two time-scales stochastic evolutionary sys-tems.
This thesis is organized as follows:
In Chapter 1, some primary definition, notion and the well-known It o formula in theory of stochastic processes and random dynamical systems is presented.
In Chapter 2, the stochastic FitzHugh-Nagumo system subjected with additive noise is explored. Under the dissipative conditions, it can been shown that the "frozen" equation with respect to the fast motion has a unique invariant measure with exponen-tial mixing property. As a result, an effective equation for the slow motion can be de-rived by averaging its drift coefficient. And then, the strong convergence in stochastic averaging principle is proved by estimating the difference between the solution pro-cess of slow equation and that of the effective equation in suitable space.
In Chapter 3, a stochastic parabolic equations with two time-scales on a bounded open interval is studied. The present model arise from some physical systems with noise perturbations to the state space. In this case, the noise is included in both fast motion and slow motion and it is of multiplicative type. The strong convergence in stochastic averaging principle for such stochastic parabolic equations is established by techniques similar to Chapter 2. But since more regular multiplicative noise is con-sider, the proof is thus modified and is of complexity. Under suitable conditions, the existence of an exponential mixing invariant measure for the fast equation with frozen slow variable is proved. As a consequence, the averaged equation which captures the dynamic of the slow motion can be obtained. Finally, with aid of the Burkholder- Davis-Gundy inequality, it is proved that the solution process of the slow equation strong convergence to the solution of the effective equation when the time scale pa-rameter tends to zero.
In Chapter 4, invariant manifolds for infinite dimensional random dynamical sys-tems with two time-scales is studied. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations, which could be coupled SPDEs, or coupled SPDEs-SODEs. Under suitable conditions, it is proved that an exponen-tially tracking random invariant manifold exists, eliminating the fast motions for this coupled system. It is further shown that if the time scale parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differen-tial equations and coupled hyperbolic-hyperbolic partial differential equations.
In the final chapter, a summary of this thesis is made and some questions that need to be improved and perfected in the future study are raised.
本文的安排如下：
在第一章中,介绍一些随机过程理论和随机动力系统理论中的基本的概念,定义,以及著名的Ito公式.
在第二章中,研究由加性噪声驱动的随机FitzHugh-Nagumo系统.在耗散性条件下,可以证明快运动方程对应的“固定”方程存在惟一满足指数混合性的不变测度.由此,通过“平均化“慢运动方程的漂移系数,可以导出一个慢运动方程的有效方程.接下来,通过估计慢运动方程的解过程和有效方程的解过程在适当空间中的偏差,可以证明强收敛意义的随机平均化原理成立.
在第三章中,考虑一个有界开区间上的双时间尺度随机抛物方程.这类模型源于状态空间被噪声扰动的物理系统,且此处所考虑的乘性噪声对快慢两个运动都有扰动.应用类似于第二章中的技巧,证明强收敛意义下的平均化原则对于这类随机抛物方程是成立的,由于考虑更为一般的乘性噪声,因而其证明过程有所不同,且比较复杂.在适当的条件下,当固定慢变量时,可以得到快变量方程的一个“固定”方程,其指数混合性的不变测度是存在且唯一的.由此可以得到刻画慢运动性态的一个平均化方程.最后,应用Burkholder-Davis-Gundy不等式,可以证明当时间尺度参数趋于零时,慢运动方程的解过程强收敛于这个平均化方程的解过程.
在第四章中,研究无穷维空间中双时间尺度的随机动力系统的不变流形,这类随机系统由一类耦合的快-慢随机发展方程所生成的,其具体的形式可以是耦合的随机偏微分方程,也可以是耦合的随机偏微分方程-常微分方程.在适当的条件下,可以证明上述的系统存在一个具有指数吸引性的随机不变流形.进一步还证明当时间尺度参数趋于零时,这个不变流形趋于一个所谓的慢流形.通过几个例子,上面的结论可以适用于耦合的抛物-双曲型偏微分方程,耦合的抛物-常微分方程以及耦合的双曲-双曲型偏微分方程.
在最后一章中,对全文进行总结并指出本文需要改进的工作.
Some complex systems in science and engineering are described by stochastic ordinary differential equations (SODEs) or stochastic partial differential equations (SPDEs) with multiple time scales. The qualitative analysis for such stochastic sys-tems has been drawing more and more attention. This thesis mainly studies the asymp-totic behavior of SPDEs with two characteristic time-scales. It consists of two topics: the strong convergence in stochastic averaging principle for two time-scales SDPEs and the inertial manifolds reduction for two time-scales stochastic evolutionary sys-tems.
This thesis is organized as follows:
In Chapter 1, some primary definition, notion and the well-known It o formula in theory of stochastic processes and random dynamical systems is presented.
In Chapter 2, the stochastic FitzHugh-Nagumo system subjected with additive noise is explored. Under the dissipative conditions, it can been shown that the "frozen" equation with respect to the fast motion has a unique invariant measure with exponen-tial mixing property. As a result, an effective equation for the slow motion can be de-rived by averaging its drift coefficient. And then, the strong convergence in stochastic averaging principle is proved by estimating the difference between the solution pro-cess of slow equation and that of the effective equation in suitable space.
In Chapter 3, a stochastic parabolic equations with two time-scales on a bounded open interval is studied. The present model arise from some physical systems with noise perturbations to the state space. In this case, the noise is included in both fast motion and slow motion and it is of multiplicative type. The strong convergence in stochastic averaging principle for such stochastic parabolic equations is established by techniques similar to Chapter 2. But since more regular multiplicative noise is con-sider, the proof is thus modified and is of complexity. Under suitable conditions, the existence of an exponential mixing invariant measure for the fast equation with frozen slow variable is proved. As a consequence, the averaged equation which captures the dynamic of the slow motion can be obtained. Finally, with aid of the Burkholder- Davis-Gundy inequality, it is proved that the solution process of the slow equation strong convergence to the solution of the effective equation when the time scale pa-rameter tends to zero.
In Chapter 4, invariant manifolds for infinite dimensional random dynamical sys-tems with two time-scales is studied. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations, which could be coupled SPDEs, or coupled SPDEs-SODEs. Under suitable conditions, it is proved that an exponen-tially tracking random invariant manifold exists, eliminating the fast motions for this coupled system. It is further shown that if the time scale parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differen-tial equations and coupled hyperbolic-hyperbolic partial differential equations.
In the final chapter, a summary of this thesis is made and some questions that need to be improved and perfected in the future study are raised.
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