Invariance principle for a diffusion in a Markov field (Q2725206)

The authors consider the Itô stochastic differential equation NEWLINE\[NEWLINE dx(t;\omega,\sigma)=V(t, x(t;\omega,\sigma);\omega) dt +\sqrt{2} d w(t;\sigma),\quad x(0;\omega,\sigma)=0, NEWLINE\]NEWLINE where a random velocity field \(V(t,x;\omega):=V(\tau_x(\omega(t)))\) is incompressible in \(x\), i.e., \(\nabla_x V\equiv 0\); \(\tau_x\), \(x\in R^d\), is a stochastically continuous, measurable group of measure preserving transformations acting ergodically on \({\mathcal X}\); \(\omega(t)\) is a Markov process with values in a Polish space \({\mathcal X}\). Under some decorrelation condition on the random field and locally Lipschitz condition in \(x\) for \(V\), the authors prove the invariance principle for the process \(\varepsilon X(t/\varepsilon^2)\), \(t\geq 0,\) as \(\varepsilon\to 0\), i.e. it converges weakly to Brownian motion with a nontrivial covariance matrix.