色散方程里面，有一个方向是用概率论的办法来证明方程的适定性（well-posedness），不过是在概率意义下的适定。很惭愧，曾经听了好几次也没听明白具体的做法，只记得用到不变测度（invariant Gibbs measure），最后得到的是“几乎（almost sure）”意义下的适定性。对于一些超临界的方程，通常的证明适定性的方法失效了。另外，陶哲轩在介绍色散方程中的孤立子猜想（Soliton Resulotion Conjecture）时提到过
One of the many difficulties with establishing this conjecture is that we except there to be a small class of exceptional solutions which exhibit more exotic (and unstable) behaviour, such as periodic "breather" solutions, or clusters of solitons which diverge from each other only logarithmically. Almost all of the known tools in the subject are deterministic in the sense that if they work at all, they must work for all data in a given class, while to settle this conjecture it may be necessary to develop more "stochastic" techniques that can exclude small classes of exceptional solutions.
大体是说因为孤立子猜想并不是对所有初值都成立，而目前的“确定性”的方法无法精准地排除掉那些例外，所以可能需要“随机性”的办法来排除这些例外。