An Equation Modelling Transport of a Substance in a Stochastic Medium

Abstract

We find an explicit expression for the (unique) solution u = u(t,x, w) of the stochastic partial differential equation

$$\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \frac{1}{2}{{v}^{2}}\Delta u + {{{\vec{W}}}_{x}}\diamondsuit \nabla u;\;\left( {t,x} \right) \in {{\mathbb{R}}^{ + }} \times {{\mathbb{R}}^{d}},} \\ {u\left( {0,\cdot } \right) = f\left( \cdot \right),} \\ \end{array}$$

where Δ and ▽ are the Laplacian and gradient operators respectively, with respect to x = (x ₁,..., x _d ) and ${{\vec{W}}_{x}}$ is d-dimensional white noise in the d parameters (x ₁, x ₂,..., x _d ). The symbol ⋄ denotes the (vector) Wick product, the use of which corresponds to an Itô/Skorohod interpretation of the equation. This equation occurs in many situations. For example, it models the transport of a substance in a turbulent (stochastic) medium.