On joint stationary probability density function of nonlinear dynamic systems (Q1265359)

Two-dimensional diffusion processes with the drift vector \((x_2, g(x_1,x_2))\) and the diffusion matrix \(\left(\begin{smallmatrix} 0 & 0 \\ 0 & \sigma^2 \end{smallmatrix} \right)\) are considered. Conditions are investigated when the stationary probability density \(p(x_1,x_2)\) has the form \(p(x_1,x_2)= p_1(x_1) p_2(x_2)\). Some examples are discussed in detail.