Deriving the SPDE from its solution I am interested in finding out the SPDE that satisfies a solution of the form:
$$u(x,t) = u_0(x - ct + \sigma W_t)$$
where $(W_t)_{t\geq 0}$ is the standard Brownian motion and $c,\sigma > 0$. The function $u_0$ can be, for simplicity, a Gaussian kernel (hence this solution is a solution to an advection type equation with a "moving" stochastic wave).
Now I'd like to derive the SPDE that admits this function as a solution. I was advised to use the Ito's formula, but my attempts at deriving it hasn't been successful. Any help will be appreciated!
 A: If $u_0 \in C^2\left(\mathbb{R}\right)$, then $u(x,t) = u_0\left(x - ct + \sigma W_t\right)$ satisfies the following SPDE
\begin{equation}
    \partial_t u = -c\,\partial_x u + \frac{1}{2}\sigma^2 \partial_{xx}u + \sigma
    \partial_x u \dot{W}_t,
\end{equation}
where $\dot{W}_t$ is a Gaussian white noise process in time, i.e. the "time derivative" of Brownian motion. To avoid having the white noise term, we can integrate with respect to time to get
\begin{equation}
    u(x,t) = u_0(x) + \int_0^t \left(-c\,\partial_x u(x,s) + \frac{1}{2}\sigma^2 \partial_{xx}u(x,s)\right)dt + \int_0^t\sigma \partial_x u(x,s)dW_s.
\end{equation}
This formulation already hints at how to show $u$ satisfies the above SPDE.
Applying Itô's formula to $u_0\left(x - ct + \sigma W_t\right)$, we get
\begin{equation}
    u_0\left(x - ct + \sigma W_t\right) = u_0(x) + \int_0^t\left(-c u'(x -cs + \sigma W_s) + \frac{1}{2}\sigma^2 u''(x - cs + \sigma W_s)\right)dt
    + \int_0^t \sigma u'(x-cs +\sigma W_s)dW_s.
\end{equation}
As $\left(x, t\right) \mapsto u_0 \left(x - ct + \sigma W_t\right)$ is smooth in the $x$ argument, we have
$$\partial_x u(x,t) = u'\left(x -ct + \sigma W_t\right) \quad \text{ and } \quad \partial_{xx} u(x,t) = u''\left(x - ct + \sigma W_t\right),$$
which give us the second formulation of the desired SPDE,
\begin{equation}
    u(x,t) = u_0(x) + \int_0^t \left(-c\,\partial_x u(x,s) + \frac{1}{2}\sigma^2 \partial_{xx}u(x,s)\right)dt + \int_0^t\sigma \partial_x u(x,s)dW_s.
\end{equation}
One way to think of this stochastic PDE is as the transport equation with Brownian characteristic lines. A formal rearrangement of the SPDE gives,
$$\left(\partial_t  + \left(c - \sigma \dot{W}_t\right)\partial_x\right) u - \frac{1}{2}\sigma^2 \partial_{xx}u = 0.$$
The first term is analogous to the transport operator, with characteristics lines $\left(x + ct - \sigma W_t;\, t\in \mathbb{R}_+\right)_{x \in \mathbb{R}}$, and the second term is the Itô correction which is a result of $W$.
