Consider a random walk starting from position $x_0$ on the $1D$ line that takes steps (of step length $r_x$ which is a random variable that depends on its position) at discrete times steps. For this, we can write down the master equation for the probability density $P(x,t)$ of finding the walker at position $x$ at time $t$ as follows:
\begin{equation} P(x,t)=\int_{-\infty}^{\infty}M(x,x')P(x',t-1)dx' \end{equation} where $M(x,x')$ is the probability density of finding the walker the $x$ given that at the previous time-step, it was at position $x'$.
Say we take $M(x,x')$ of the form $\frac{1}{\sqrt{4\pi\Delta t}}e^{\frac{[x-x'(1-\lambda \Delta t)]^2}{4\Delta t}}$ where $\lambda$ and $\Delta t$ are parameters of the model.
Can the RHS of our master equation be considered as a convolution? I would like to solve for $P(x,t)$ and if yes, then using a Fourier transform/Generating function, I can write the convolution as a product and solve easily.
Any help would be greatly appreciated. Thanks :)