Generators of mehler-type semigroups as pseudo-differential operators

Lescot P, Röckner M. Generators of mehler-type semigroups as pseudo-differential operators. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2002;5(3):297-315.We study semigroups (P-t)(tgreater than or equal to0) on a Hilbert space E, given by a Mehler-type formula: P(t)f(x) = integral(E)f(T(t)x + y)mut(dy). Under reasonable assumptions, the LP(E,mu)-generator A(0) of (P-t)(tgreater than or equal to0) turns out to be expressible as a pseudo-differential operator, provided p is an invariant measure for (P-t)(tgreater than or equal to0). The question of L-p-uniqueness is also answered positively