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?


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