We can solve the equation for w(\mathbf{y},t) by making a Fourier transform to momentum space. The PDF w(\mathbf{x},t) is related to its Fourier transform \tilde{w}(\mathbf{k},t) through
w(\mathbf{x},t) = \int_{-\infty}^{\infty}d\mathbf{k} \exp{(i\mathbf{kx})}\tilde{w}(\mathbf{k},t),and using the definition of the \delta -function
\delta(\mathbf{x}) = \frac{1}{2\pi} \int_{-\infty}^{\infty}d\mathbf{k} \exp{(i\mathbf{kx})},we see that
\tilde{w}(\mathbf{k},0)=1/2\pi.