Importance sampling, Fokker-Planck and Langevin equations

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

$$\tilde{w}(\mathbf{k},0)=1/2\pi.$$

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