Ornstein-Uhlenbeck semi-groups on stratified groups
Two analogues of the Ornstein-Uhlenbeck semigroup in the setting of stratified groups \(G\) are constructed, namely, Markovian diffusion semigroups acting on \(L^q(pd\gamma)\), where the invariant density \(p\) is a heat kernel at time 1 on \(G\). The first semigroup is symmetric on \(L^2(pd\gamma)\) with generator \(\sum^n_{i=1} X_i* X_i,\) where \((X_i)^n_{i=1}\) is a linear basis of the first layer of the Lie algebra of \(G\) and \(X_i*\) the adjoint of \(X_i\) with respect to \(L^2(pd\gamma)\), \(i= 1,\dots,n\). The second semigroup, defined by an analogue of Mehler's formula, is compact on \(L^q(pd\gamma)\), \(1< q<\infty\) and non-symmetric on \(L^2(pd\gamma)\). The formal real part of its generator \(N\) is \(\sum^n_{i=1} X_i* X_i.\) Both generators have the same caré du champ, the difference between them being an anti-symmetric first-order differential operator. If polynomials are dense in \(L^2(pd\gamma)\), i.e., if \(G\) has at most four layers, the spectrum of \(N\) is seen to be the set of nonnegative integers, and the corresponding set of eigenspaces is determined. For the case that \(G\) is step two, another description of the eigenspaces is given, resembling the classical definition of Hermite polynomials via their generating function.
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