Let
- $D:=(0,a)$ for some $a>0$
- $H:=L^2(D)$, $$e_n(x):=\sqrt{\frac 2a}\sin\frac{n\pi x}a\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ and $$\lambda_n:=\left(\frac{n\pi}a\right)^2\;\;\;\text{for }n\in\mathbb N$$
- $D(A):=H^2(D)\cap H_0^1(D)$ and $$A:D(A)\to H\;,\;\;\;u\mapsto-u''$$
- $Q$ be a bounded, linear, nonnegative and symmetric operator on $H$ with finite trace and $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N$$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $\left(W(t)\right)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$ and $$B_n:=\frac{\langle W,e_n\rangle_H}{\sqrt{\lambda_n}}\;\;\;\text{for }n\in\mathbb N$$
I would like to formulate a stochastic version of the wave equation $$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}\tag 1$$ with $c>0$. More concretely, I want to obtain a SDE of type as studied in A Concise Course on Stochastic Partial Differential Equations.
Ignoring rigor, I would like to consider an equation of the form $$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}+\varepsilon\frac{\partial W}{\partial t}\tag 2$$ with $\varepsilon>0$. If $u$ is a $H$-valued solution of this (formal) equation, then $$u(t,x)=\sum_{n\in\mathbb N}u_n(t)e_n(x)\;\;\;\text{for all }t\ge 0\text{ and }x\in D\tag 3$$ and hence, since $$\left\|W_n(t)-W(t)\right\|_{L^2(\operatorname P,\:H)}\stackrel{n\to\infty}\to 0\;\;\;\text{for all }t\ge 0\tag 4$$ with $$W_n:=\sum_{i=1}^n\sqrt{\lambda_i}B_ie_i\;\;\;\text{for }n\in\mathbb N\;,$$ we obtain $$\frac{{\rm d}^2u_n}{{\rm d}t}(t)=-c^2\lambda_nu_n(t)+\varepsilon\sqrt{\lambda_n}\frac{{\rm d}B_n}{{\rm d}t}(t)\tag 5$$ with $$u_n(t):=\langle u(t,\;\cdot\;),e_n\rangle_H\;\;\;\text{for }n\in\mathbb N\text{ and }t\ge 0\;.$$
Question: How do we need to understand $(2)$ rigorously? And is my derivation correct or are there crucial points where we need to be more careful? Can we recast $(3)$ as an equation of type as described in the book?
[Please note that I've asked for the solution of $(6)$ in another question]