Quadratic Wiener Functionals, Kalman-Bucy Filters, and the KdV Equation | Zendy

Nobuyuki Ikeda | Zendy; Setsuo Taniguchi | Zendy

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Quadratic Wiener Functionals, Kalman-Bucy Filters, and the KdV Equation

Author(s) -

Nobuyuki Ikeda,

Setsuo Taniguchi

Publication year - 2019

Publication title -

advanced studies in pure mathematics

Language(s) - English

Resource type - Conference proceedings

eISSN - 2433-8915

pISSN - 0920-1971

DOI - 10.2969/aspm/04110167

Subject(s) - korteweg–de vries equation , kalman filter , transformation (genetics) , quadratic equation , mathematics , quadratic function , riccati equation , key (lock) , function (biology) , soliton , mathematical analysis , computer science , partial differential equation , physics , statistics , chemistry , nonlinear system , geometry , biochemistry , computer security , evolutionary biology , biology , gene , quantum mechanics

Soliton solutions and the tau function of the KdV equation are studied within the stochastic analytic framework. A key role is played by the It^ o formula and the Cameron-Martin transformation. x Introduction In this paper, we investigate the Korteweg-de Vries (KdV) equation within the framework of stochastic analysis. We shall study soliton solutions with the help of the It^ o formula, whose original form was achieved in 1942 ([9]). The Cameron-Martin transformation, which was established in the early 1940’s ([2, 3]), also plays a key role. Let x > 0 andW n be the space of R n -valued continuous functions on [0;x] starting at the origin, and let P be the Wiener measure onW n . Following the idea of Cameron-Martin [3], we can show that

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