What is the formal definition of a stochastic PDE and a solution to a stochastic PDE?

Question

While searching through this Wikipedia article, I have stumbled uopn the following 'stochastic' heat equation $$\partial_tu=\Delta u+\xi,$$ where $\xi$ is the space-time white noise. However, I don't get in what sense one should interpret this equation, and how it relates to first order SDEs.

For the sake of simplicity, let us just consider the time independent equation $$\begin{cases} -\Delta u=\xi, &\text{ on }D\subset \mathbb{R}^d, \;\;\text{ with }d\geq2,\\ u=0, &\text{ on }\partial D. \tag{1}\end{cases}\label{1}$$ How should I interpret equation \eqref{1}? Is it just the usual deterministic Poisson equation with an extra $\omega$ paramter? I.e are we just assuming that $\xi:\Omega\to L^2(D)$ is a random function and we are looking for a solution $u:\Omega\to H^1(D)$ corresponding to this random function? This seems too trivial because already know that for each $\omega$ and $\xi(\omega)\in L^2(D)$, there is a unique $u(\omega)\in H^1_0(D)$ that solves \eqref{1}. So I don't believe this is the right interpretation.

Note. Please bear in mind that I only know deterministic PDEs and first order SDEs, so I apologize beforehand if this question is trivial or my reasoning is naive.

Answer 1

Since I don't see any other answers, I move my comments to answer.

Here are some good introductory references for SPDEs, by Hairer: hairer.org/notes/SPDEs.pdf, Bréhier: hal.archives-ouvertes.fr/hal-00973887v1/document , and Da Prato: "Stochastic Equations in Infinite Dimensions"

There they go over the notion of mild solutions which is analogous to weak solution from PDEs eg. chapter 5 in Hairer's notes: for linear stochastic pde

$$dx = Lx dt + Q dW(t) , x(0) = x_0 (5.1)$$

where $x$ take values in a separable Banach space $B$, $L$ is the generator of a $C_0$ semigroup on $B$, $W$ is a cylindrical Wiener process on some Hilbert space $K$, and $Q: K \to B$ is a bounded linear operator

Let $S$ be the $C_0$-semigroup generated by $L$. A B-valued process $x(t)$ is said to be a mild solution to (5.1) if the identity

$$x(t) = S(t)x_0 +\int_{0}^{t} S(t − s)QdW(s) , (5.3)$$

holds almost surely for every $t > 0$.