A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides
A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides
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DOI：
10.1137/1111038
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发表时间：
1966
影响因子：
0.6
通讯作者：
R. Khas'minskii
中科院分区：
文献类型：
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作者：
R. Khas'minskii
如果随机过程F(x，t，co，0)对固定的x满足大数定律，则由这一结果可以得出：对于(X)0，问题(0.1)在O(1/e)阶时的解X()(t，co)不会明显地偏离初始点，如果e很小，则概率接近。更准确地说，在这种情况下，过程X()(et，co)以e-0的概率收敛到零。这一事实表明，一个非平凡的极限分布可能导致过程X)在一个更慢的时间2，RL Stratonovich在[2]中第一次提醒人们在考虑存在噪声的非线性振动问题时需要这样的定理。文献[2]在严格的物理水平上证明了当过程F(x，t，co，)是平稳的时，函数X(E)(82t)在一定意义上收敛于马尔可夫扩散过程e0，并确定了过程的局部性质。在文[3]中，人们试图得到更严格的结果，但在证明极限过程的马尔可夫性时，所引用的函数(1.36)是连续的这一假设是不充分的。[3]中提出的其他一些假设也值得商榷；例如，[3]的级数(1.29)的收敛在实际案例中很难得到证实。
To T if the stochastic process F (x, t, co, 0) satisfies the law of large numbers for fixed x. From this result it followsparticularly that for (x) 0 the solution X ()(t, co) to problem (0.1) at a time of order O (1/e) will not have been displaced noticeably from the initial point with probability close to if e is small. More precisely, in thiscase the process X ()(et, co) converges to zero in probability as e-0. This fact suggests that a non-trivial limit distribution may result for the process X) at a still" slower" time 2t.RL Stratonovich in [2] first called attention to the need for such a kind of theorem when considering nordinear oscillation problems when noise is present. In [2] it is established on a physically rigorous level that when the process F (x, t, co,) is stationary, the function X (e)(82t) converges ina certain sense to a Markov diffusion process as e 0 and the local properties of the process are determined. In [3] an attempt at greater rigor is made but in proving the Markov character of the limit process the assumption that the function (1.36) cited there is continuous is not adequately justified. Some of the other assumptions made in [3 are also open to question; for example, the convergence of the series (1.29) of [3 is difficult to verify in actual cases.