While searching through this Wikipedia article, I have stumbled uopn the following 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 view this equation. 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.

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.

While searching through this Wikipedia article, I have stumbled uopn the following 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 view this equation. 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 $\xi(\omega)\in L^2(D)$ 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.

While searching through this Wikipedia article, I have stumbled uopn the following 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 view this equation. 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.