Stochastic partial differential equation

Definition

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, let $D \subseteq \R^d$ be open and let $T > 0$. Let $L$ be a linear differential operator in the variable $x \in D$, and let $F$ and $G$ be functions. Let $\xi$ be a random noise on $[0, T] \times D$, for example space-time white noise, the formal time derivative of a cylindrical Wiener process $W$. A stochastic partial differential equation is an equation for a random field $u(t, x, \omega)$ of the form $$\partial_t u = L u + F(u) + G(u) \, \xi, \qquad t \in (0, T], \ x \in D,$$ with initial and boundary conditions. In Itô form it reads $du = (L u + F(u)) \, dt + G(u) \, dW$. A solution is a stochastic process, adapted to the filtration of $W$, that satisfies the equation in a weak or mild sense. Classical derivatives are not used, because $\xi$ is too irregular.