Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator
This work deals with properties of the semigroup related to the Ornstein-Uhlenbeck operator \[ L\phi(x)=\textstyle{{1\over 2}}\text{ Tr }QD^2\phi(x)+ \langle Ax,D\phi(x)\rangle \] in a real separable Hilbert space \(H\). We assume that \(A\) is the generator of a \(C_0\)-semigroup \(S(t)\), \(t\geq 0\), of bounded operators on \(H\), \(Q\) is bounded, selfadjoint, and nonnegative. By \(D\phi\) we denote the Fréchet derivative of a function \(\phi:H\to\mathbb{R}\). Notice that \(L\phi(x)\) is well-defined for every \(x\in H\), at least for appropriately chosen cylindrical functions. In this paper, we require that \[ \int^\infty_0\text{tr } S(u)QS^*(u)du<\infty. \] If this is satisfied, we can define on \(H\) the family of Gaussian measures \(\mu_t\), \(t\geq 0\), and \(\mu\) with the mean zero and the covariance operators \[ Q_t= \int^t_0 S(u)QS^*(u)du \] and \[ Q_\infty= \int^\infty_0 S(u)QS^*(u)du, \] respectively. For simplicity of presentation we assume that \(\ker Q_\infty=\{0\}\). Let \[ R_t\phi(x)= \int_H\phi(S(t)x+ y)\mu_t(dy). \] Then the family of operators \(R_t\), \(t\geq 0\), forms a strongly continuous semigroup of contractions on \(L^p(H,\mu)\) for every \(p\geq 1\). Moreover, it can be shown that \(L\) has a unique extension to a generator of a \(C_0\)-semigroup on \(L^p(H,\mu)\) \((p\geq 1)\) which coincides with \(R_t\). The semigroup \(R_t\) can be identified as the transition semigroup corresponding to the Ornstein-Uhlenbeck process on \(H\): \[ Z^x(t)= S(t)x+ \int^t_0 S(t- s)dW(s), \] where \(W\) is a Wiener process on \(H\) with the covariance operator \(Q\) that is \(R_t\phi(x)= E\phi(Z^x(t))\). In this paper, we investigate the hypercontractivity, compactness, and space-time regularity of the semigroup \(R_t\) and give necessary and sufficient conditions for each case. The results obtained are important for the investigation of uniqueness and ergodic properties of the semigroups related to more general operators of the form \[ L_F\phi(x)=\textstyle{{1\over 2}}\text{ tr }QD^2\phi(x)+ \langle Ax+ F(x),D\phi(x)\rangle. \]
scientific article; zbMATH DE number 989760