To preface, this problem arose from my research of noise (stochastic variable) in physics, but I do not have a mathematical treatment nor approach for it.
Suppose I have a continuous, differentiable variable $\phi_{n}(t)$ and some noise $\xi_{n}(t)$ where $t$ is time and $n$ denotes that these quantities are still in the discrete, spatial limit. I have the following equation involving $\phi_{n}$ and $\xi_{n}$ such that
$$ C\frac{d^{2}\phi_{n}}{dt^{2}} - \frac{1}{L}\left(\phi_{n+1}+\phi_{n-1}-2\phi_{n}\right) = \frac{2}{\sqrt{R}}\left(\xi_{n}-\xi_{n-1}\right) $$
where $C$, $L$, and $R$ are constants. Usually, in the absence of noise $\xi_{n}$, one can take the continuum limit in $\phi_{n}$ such that $\phi_{n}(t)\longrightarrow \phi(x,t)$. Through finite difference:
$$ \phi_{n+1} + \phi_{n-1} - 2\phi_{n} \approx a^{2}\frac{\partial^{2}\phi(x,t)}{\partial x} $$
where $a$ is the discretization length (this amounts to just a constant). So the equation now reads
$$ \frac{\partial^{2}\phi(x,t)}{\partial t^{2}} = \frac{a^{2}}{LC}\frac{\partial\phi(x,t)}{\partial x^{2}}. $$
This is but the wave equation, and it is simply solvable. However, in the presence of noise $\xi_{n}(t)$, how should I take its continuum limit? Generically, for the finite difference formula, I have that $\xi_{n}(t) - \xi_{n-1}(t) \rightarrow \xi(x,t)-\xi(x-a,t)$ such that
$$ \xi(x,t)-\xi(x-a,t) \approx a\frac{\partial\xi}{\partial x} - \frac{1}{2}a^{2}\frac{\partial^{2}\xi}{\partial x^{2}} $$
I understand that in Ito (which is what I'm using), I need to retain up to the second-order derivative. But the noise is stochastic, so I am not even sure if the derivatives are defined.
I could try to define the Wiener process, $W(x,t) = \int_{0}^{x}\xi(x^{\prime},t)dx^{\prime}$. Since the initial equation is in $\phi_{n}(t)$ and $\xi_{n}(t)$, one could take the continuum limit of the equation and integrate both sides with respect to $t$ to introduce $W(x,t)$ and see where that leads to. However, it is unclear on how to even take the continuum limit of $\xi_{n}(t)$. How should I approach the problem here?
Thanks