Yes, there is a mapping between Langevin and Fokker-Planck equations. I learned about it in the book of Kamenev:

Article Field Theory of Non-Equilibrium Systems

See also this:

Article Many-body theory of non-equilibrium systems

The main step is to write the stochastic dynamics as a path integral through the Martin-Siggia-Rose formalism. This provides a formal expression for the probability density. Then taking derivatives of this expression provides the Fokker-Planck equation.

As to your example. I can not think of a counter example and guess that you are right.

Thank you very much Steven Mathey

I will study your suggested article

Using the FPE, you can recover the statistics that define the evolution of the Langevin Equation, i.e. the first and second moments.

One straightforward way to see the connection between the Langevin and Fokker-Planck Equations is to recast the Langevin Equation as a continuous Markov process and using the Chapman-Kolmogorov Equation to derive the FPE. I'll give a shameless plug to my discussion on phase-locked loops as Markov process in my article found in the link (where I discuss the derivation of the FPE).

Technical Report Markov Chains for Fun and Profit: From Gambler's Ruin to Pha...

Thanks Dr. George Slade and Dr. Steven Mathey

As you said we can prove the duality between the Fokker-Planck equation and the SDEs. Therefore, the FPE gives the time evolution of the measure of the ensemble of the trajectories of the SDE.

but does that mean the asymptotic behavior of the FPE gives the measure of the ensemble of the trajectories of the SDE asymptotically in time?

I mean mathematically proving the duality between FPE and SDE is a proof to my example? or I should prove my example in a different way.

In principle, yes the asymptotic solution of the Fokker-Planck equation is equivalent to the asymptotic Langevin dynamics.

Keep however in mind that 'asymptotic' does not mean that you can forget about time entirely. The probability distribution in the Fokker-Planck equation only depends on one time. It can not be used directly to compute two-times correlation functions (such as ) as is the case for Langevin dynamics. By 'asymptotic', I mean t->infinity but tau finite.

In the Fokker-Planck formulation an observable such as is computed (in the asymptotic, or stationary regime) from the asymptotic probability distribution P(t->infinity,X), which is used as an initial condition to compute the time evolution from t to t+tau. Then P(X(t)) and P(X(t+tau)) become available:

= int_{dX,dX'} P(X,t) P(X->X',t+tau) X X'

The Fokker-Plank equation can be treated as a so-called forward Kolmogorov equation for a certain diffusion process.

To derive a stochastic equation for this diffusion process it is very useful if you know a generator of this process. Finally, to find out a form of the generator you have to consider a PDE, dual to the Fokker-Plank equation which is called the backward Kolmogorov equation. The elliptic operator in the backward Kolmogorov equation coincides with the generator of the required disffusion process. Let me give you an example.

Assume that you consider the Cauchy problem for the Fokker-Plank type equation

u_t=Lu, u(0,x)=u_0(x),

where Lu(t,x)=[A^2(x)u(t,x)]_{xx}-[a(x)u(t,x)]_x.

The dual equation is h_t+L^*h=0, where L^*h= A^2(x)h_{xx}+a(x)h_x.

As a result the required diffusion process x(t) satisfies the SDE

dx(t)=a(x(t))dt+A(x(t))dw(t), x(0)= \xi,

where w(t) is a Wiener process and \xi is a random variable independent on w(t) with the distribution density u_0(x).

You may see the book Bogachev V.I., Krylov N.V., Röckner M., Shaposhnikov S.V. "Fokker-Planck-Kolmogorov equations"