Nonlinear evolution equations with gradient coupled noise
The letter deals with nonlinear partial differential equations perturbed by noise of the form \[ {\partial\over\partial t}u(t,x)+F\bigl(t,u(t,x)\bigr)={\partial\over\partial x}u(t,x)\dot X_t,\quad u(0,x)=u_0(x), \] where \(F\) depends on \(t,u\), and the space derivatives of \(u\) up to some order \(N\). The noise \(X_t\) is the solution of an Itô equation of the form \[ dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t,\quad X_0=x, \] where \(B_t\) is a Brownian motion and the coefficients are sufficiently smooth. The term \({\partial\over\partial x}u(t,x)\dot X_t\) is interpreted both in the Itô and Stratonovich sense, comparing the results. Examples treated by the authors include Burgers equation and Korteweg-de Vries equation. Moreover, the general problem is treated in the multidimensional case, and for systems of nonlinear equations. The paper is based on the transformation \(u(t,x)=\overline u(t,x+X_t)\), where \(\overline u(t,x)\) is the solution of a suitable deterministic nonlinear partial differential equation, of different form depending on the Itô or Stratonovich interpretation. The equations for \(\overline u(t,x)\) and \(u(t,x)\) are related by the Itô-Ventzel formula. By means of this transformation the authors prove existence and uniqueness results, and compare the Itô and the Stratonovich models, pointing out the competition between the noise term and the diffusion term (when it is present).
stochastic partial differential equations
scientific article; zbMATH DE number 1154759