Loading [MathJax]/extensions/TeX/boldsymbol.js

 

 

 

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.