In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Zhao, Junyilang, "Dynamics for a Random Differential Equation: Invariant Manifolds, Foliations, and Smooth Conjugacy Between Center Manifolds" (2018). Theses and Dissertations. 7362.
Wiener process, Wong-Zakai approximations, Multiplicative noise, Random dynamical systems, Invariant manifolds, Foliations, Conjugacy