Convergence to equilibrium for classical and quantum spin systems (Q1902870)

Two general theorems on stochastic equations \(dX = (AX + F(X)) dt + BdW\), where \(A,F\) are operators in a Hilbert space \(H\), having appropriate dissipativity properties, \(W\) is cylindrical Wiener process in a Hilbert space \(U\) and \(B\) a linear operator from \(U\) to \(H\), are applied to the description of evolutions of classical and quantum ``spin systems'' on infinite lattice and/or Euclidean space. A direct construction of the corresponding Markov semigroup is given. Existence of the unique stationary measure and exponential convergence to the equilibrium are established.