Renormalization Group Approach to SDEs with Nonlinear Diffusion Terms

Renormalization Group (RG) method has been recognized as an unified and effective approach for singularly perturbed problems in many fields, such as oscillation and boundary layer problems in ordinary differential equations (ODEs), center manifolds in dynamical systems, and long-time asymptotic behavior in partial differential equations (PDEs), etc. In this paper, we are going to investigate the application of RG method to the asymptotic behavior of a class of stochastic differential equation with multiplicative noise. We will first show how to formulate the RG equation and the approximate solution. Then, the rigorous estimation of the error between the approximate solution and the real one will be presented.