On the Ornstein-Uhlenbeck operator in convex subsets of Banach spaces. (English) Zbl 1412.35376
The author investigates the Ornstein-Uhlenbeck operator \(L^{\Omega}\) defined as the self-adjoint operator associated to the quadratic form \[ \int_{\Omega}\langle \nabla_Hu,\nabla_Hv\rangle_H\,\gamma(dx),\quad u,\,v\in W^{1,2}(\Omega,\gamma), \] where \(\Omega\) is an open convex subset of an infinite-dimensional separable Banach space \(X\) endowed with a centered non-degenerate Gaussian measure, \(W^{1,2}(\Omega,\gamma)\) is the Sobolev space, and \(\nabla_H\) is the gradient along the Cameron-Martin space \(H\). By using the cylindrical approximation of \(\Omega\), he approximates \(L^{\Omega}\) by finite-dimensional Ornstein-Uhlenbeck operators, and proves some inequalities for the semigroup \((T^{\Omega}(t))_{t\ge 0}\) generated by \(L^{\Omega}\). Then the author proves that \((T^{\Omega}(t))_{t\ge 0}\) is a submarkovian semigroup, and shows the Poincaré inequality and the logarithmic-Sobolev inequality. Spectral properties of \(L^{\Omega}\) are finally deduced.
Reviewer: Rodica Luca (Iaşi)
MSC:
35R20 | Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) |
35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
39B62 | Functional inequalities, including subadditivity, convexity, etc. |
47D07 | Markov semigroups and applications to diffusion processes |
References:
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[2] | F.-Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory, Science Press, Beijing, 2004. |
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