# Stochastic particle method for the Langevin equation. (SDE, numerical method)

Consider the Langevin equation ( with some initial distribution $$(X_0,V_0)\sim \rho_0$$)

\begin{align} dX(t)&=V(t)dt, \\ dV(t)&=-\partial g(X(t)) dt - \partial f(V(t))dt+\sqrt{2} dW(t),~~~~~~~~~~~~~~~~~~~~(1) \end{align}

where $$W$$ is a 1-dimensional Brownian motion, $$X,Y$$-1 dimensional stochastic processes, here $$g,f$$ usually represent a confining potential and a frictional force respectively. For simplicity assume $$f(s)=g(s)=s^2/2$$ so that $$(1)$$ becomes

\begin{align} dX(t)&=V(t)dt, \\ dV(t)&=-X(t) dt - V(t)dt+\sqrt{2} dW(t).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2) \end{align}

I would like to estimate the distribution of $$(X(t),V(t))$$ at some time $$t$$, using a stochastic particle method. Is there any existing theory that says this is possible?