Fourier transform in SPDE

\begin{align*} \frac{\partial \theta}{\partial t} = \frac{1}{2} \Delta \theta(t,x) + \theta(t,x) \dot{W}(t,x). \end{align*} We study the existence and uniqueness of the solution under Fourier mode.

Then we apply the similar approach to the turbulent transport of a passive scalar quantity in a stratified, 2-D random velocity field. It is described by the stochastic partial differential equation \begin{align*} \partial_t \theta(t,x,y) = \nu \Delta \theta(t,x,y) + \dot{V}(t,x) \partial_y \theta(t,x,y), \quad t\ge 0\:\: \text{and}\:\: x,y\in \mathbb{R}, \end{align*} where \(\dot{V}\) is some Gaussian noise. We show via a priori bounds that, typically, the solution decays with time. The detailed analysis is based on a probabilistic representation of the solution, which is likely to have other applications as well. This is based on joint work with Davar Khoshnevisan from University of Utah.