Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators
Let \(H\) be a real separable Hilbert space with inner product \(\langle \cdot, \cdot \rangle\) and norm \(|\cdot |\), and \(A\) be a nonnegative definite self-adjoint operator on \(H\) such that \(A^{-1}\) is trace-class. Suppose that \(F\) is bounded and Lipschitz continuous. Sobolev space \(W^{1,2} (H,\mu_0)\) is defined as follows: Let \({\mathcal F}C^\infty_b (H)\) be the set of all functions \(\varphi\) on \(H\) of the form \[ \varphi(x) = f(\langle e_1, x\rangle, \dots, \langle e_n,x\rangle), \qquad x \in H, \] where \(n \in \mathbb{N}\), \(f \in C^\infty_b (\mathbb{R}^n)\), and \(\{e_j\mid j \in \mathbb{N}\}\) is an eigenbasis of \(A\) which we fix from now on. For \(\varphi \in {\mathcal F} C^\infty_b (H)\) let \(\varphi'\) denote its Fréchet derivative, and for \(x \in H\) let \(D_x \varphi(x)\) denote the element corresponding to \(\varphi'(x) \in H^*\). Then \(W^{1,2} (H, \mu_0)\) is defined as the completion of \({\mathcal F} C^\infty_b(H)\) with respect to the norm given by \[ |\varphi|^2_{W^{1,2} (H, \mu_0)} = |\varphi|^2_{L^2(H, \mu_0)} + \int_H |D_x \varphi(x)|^2 \mu_0(dx), \] where \(\mu_0\) is Gaussian measure and \(\mu_0 = {\mathcal N}(0, {1\over 2} A^{-1})\). For \(\varphi \in {\mathcal F} C^\infty_b(H)\) let \(\varphi''\) denote its second Fréchet derivative, and for \(x \in H\) let \(D_{xx} \varphi (x)\) denote the corresponding element in \({\mathcal L}(H)\). Note that \(D_{xx} \varphi(x)\) is trace-class and that \[ D_x \varphi = \sum^n_{j=1} \partial_j f(\langle e_1, \cdot\rangle, \dots, \langle e_n, \cdot \rangle) e_j, \] if \(\varphi = f(\langle e_1, \cdot \rangle, \dots, \langle e_n, \cdot\rangle)\) and \(\partial_jf(y) = {\partial f\over \partial y_j}(y)\), \(y \in \mathbb{R}^n\). Hence \(D_x \varphi(x) \in D(A)\) for all \(x \in H\) and we can define \[ {\mathcal A}_0 \varphi(x) = \text{Tr} [D_{xx} \varphi(x)] + \langle x, AD_x \varphi(x)\rangle, \qquad x \in H, \] and \[ {\mathcal A}_F \varphi = {\mathcal A}_0 \varphi(x) + \langle F, D_x \varphi(x) \rangle, \qquad x \in H. \] Let \(F : H \to H\) be Borel measurable. A probability \(\mu\) on \({\mathcal B} (H)\) (:= Borel \(\sigma\)-algebra on \(H\)) is called a generalized invariant measure if \[ \int_H {\mathcal A}_F \varphi (x) \mu(dx) = 0 \text{ for all }\varphi \in {\mathcal F} C^\infty_b(H), \qquad \int_H \langle e_j, x\rangle^2 \mu(dx) < + \infty \text{ for all } j \in \mathbb{N} \] and \[ \int_H |F(x)|^2 \mu(dx) < + \infty \] hold. In this case we write \({\mathcal A}^*_F \mu = 0\). The authors prove the following main result: Let \(F : H \to H\) be Borel measurable and let \(\mu\) be a probability measure on \({\mathcal B}(H)\) such that \({\mathcal A}^*_F \mu = 0\). Then \(\mu = \rho \mu_0\) with \(\sqrt{\rho} \in W^{1,2} (H,\mu_0)\).
scientific article; zbMATH DE number 885329