Applications of stochastic calculus to the KdV equation and hierarchy

对随机微积分理论进行了深入的研究，旨在将随机振荡积分应用于KdV方程和KdV族。在随机微积分中得到了几个新的结果：随机表示在无反射势收敛中的应用，随机振荡积分的具体表示，非对易谐振子的Wiener积分表示方法。此外，还研究了布朗单的一种新的维纳泛函。最后，为了从代数分析的角度理解随机振荡积分，研究了随机振荡积分与雅可比方程之间的关系。

A deep study of the theory of Stochastic Calculus was made, aiming to apply stochastic oscillatory integrals to the KdV equation and the KdV hierarchy. Several new results in Stochastic Calculus are obtained : an application of stochastic representation to the convergence of reflectionless potentials, concrete expression of stochastic oscillatory integral, a Wiener integral representation approach to non commutative harmonic oscillator. Moreover, a new Wiener functional of the Brownian sheet was investigated. Finally, the relationship between stochastic oscillatory integrals and Jacobi equations was studied to understand stochastic oscillatory integrals from the point of view of algebraic analysis.