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!

Solution 1:

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 $W$.