it seems that with solving the stationary form of forward Fokker Planck equation we can find the equilibrium solution of stochastic differential equation.
is the above statement true?is it a conventional way to find the equilibrium solution of a SDE? and do SDEs always have equilibrium solution?
Ali Namadchian ,
As commented above, stationary solutions do not exist in all cases, but I believe your question goes a little bit further, so I'll try to answer. Equilibrium solutions of Forkker-Planck equations exist when a mathematical condition called detailed balance is satisfied. One can grasp the physical intuition of what detailed balance means by looking at systems with discrete states, or diffusive systems with positive-definite and constant diffusion matrix. In both cases, detailed balance means that the local stationary probability current vanishes everywhere. In other words, the flow of probability entering one state exactly cancels the flow of probability leaving the state and this is valid for any of the microstates characterizing the system. In the case of diffusion matrices which are only positive semi-definite, and/or non-constant, the condition is a little bit harder to state with words, however the important point here is that detailed balance is what determines (in some texts is even taken as the very definition of) equilibrium.
Best wishes,
Reinaldo.
Dear Dr. Yann Lanoiselée and Dr. Reinaldo García-García
Thank you for your answers, I always compare and SDE with ODE,
As in ODEs that we have both stable or convergent solution and divergent solution, intuitively I think in SDEs we must have the same definitions.
For example I think the Dr. Yanns example (Brownian motion) is not a stable stochastic process.
as a control engineers We always seek to control a system with ODE governing equation and stabilize it.
Consider x_dot=A*x+Bu where u is a control function. choosing a proper u for example u=-k*x the system turns to x_dot=(A-BK)*x and if we choose K such that A-BK has negative eigenvalues the system will be stable and has a convergent solution. (it was a linear system, in nonlinear systems we can use Lyapunov control functions). do you think these definitions are expanded for SDEs framework, for example we can add some control function, so that we have a stable SDE?
I should say that we do not have any control on the diffusion part of the SDEs (except when we have a non-constant diffusion matrix such as dx=f(x,u)dt+g(x)dwt, although u does not appear in diffusion part directly, with its effect on x it can affect the diffusion part)
The question about ``equilibrium solution" and probably also many other questions concerning Fokker-Planck equation are answered in the book of H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications. For this one search the key-word detailed balance, as suggested above. The question abut stability of SDE is discussed for example here http://www6.cityu.edu.hk/ma/ws2010/doc/mao_notes.pdf .
Thanks Dr. Viktor Holubec
the book of H. Risken that you introduced is very useful.
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