Consider a system of n particles moving in three dimensional space under the action of an external force with $C^1$ potential V and coupled to a heat bath causing an external random effect. Then we may consider the following generalization of the Ornstein Uhlenbeck model of physical brownian motion
$dx_t$=$v_tdt$;
$Mdv_t$=-B$v_tdt$-$\nabla$V($x_t)dt$+$\sigma$$dW_t$
where $x_t$, $v_t$ and $W_t$ are 3n dimensional, M, B and $\sigma$ are 3n $\cdot$ 3n dimensional with M symmetric positive definite. The Maxwell-Boltzmann distribution is given by the density function:
$\bar\rho(x,v)=Cexp[-H(x,v)/(kT)]$, $H(x,v)=\frac{1}{2}<v,M_v>+V(x)$,
where C is a normalization constant. Assume that V is such that the stochastic model has a unique solution ($x_t, v_t$) on [$0$; T] for all T > $0$, and $\bar\rho$ is integrable over $R^{6n}$. Prove the following generalization of Einstein fluctuation-dissipation relation:
The Maxwell-Boltzmann density is invariant for our model if and only if
$\sigma\sigma^T=kT(B+B^T)$.