Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type (Q1127690)

A stochastic process of Ornstein-Uhlenbeck type \((X_t)^\infty_0\) on a \(d\)-dimensional space is considered. It can be represented as \[ X_t=e^{-tQ}x+\int^t_0e^{-(t-u)Q}dZ_u, \] where \(Q\) is a real \(d\times d\) matrix all eigenvalues of which have positive real parts, \((Z_t)^\infty_0\) is a Lévy process on \(R^d\), i.e. a stochastically continuous process with stationary independent increments, and \(Z_0=0\). It is proved that \((X_t)\) is recurrent if and only if \((\forall_c>0)\) \[ \int^1_0{dv\over v}\exp\left[\int^1_v{du\over u}\int_{| x|\geq c}\biggl(\exp\bigl(-| e^{(\log u)Q}x|\bigr)-1\biggr)\rho(dx)\right]=\infty, \] where \(\rho\) is a measure of Lévy of the process \((Z_t)\). Some sufficient conditions for this equality to be fulfilled are derived.