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Invariant measures for generalized Langevin equations in conuclear spaces. (English) Zbl 0997.60067

A complete description of invariant measures for a linear stochastic differential equation in \(\mathbb R^{d}\) was found by M. Zakai and J. Snyders [J. Differ. Equations 8, 27-33 (1970; Zbl 0194.40801)], an extension of this result to infinite-dimensional Hilbert spaces is due to J. Zabczyk [in: Mathematical control theory. Banach Cent. Publ. 14, 591-609 (1985; Zbl 0573.93076)]. In the paper under review, the same problem is solved for linear stochastic differential equations in conuclear spaces.
Let \(\Phi \) be a Fréchet nuclear space with a dual space \(\Phi ^\prime \), \(A\) a generator of a \(C_0\)-semigroup \((T_{t})\) of linear operators on \(\Phi \), and \(q\) a continuous Hilbertian seminorm on \(\Phi \). Let \(W\) be a generalized time-homogeneous Wiener process in \(\Phi ^\prime \) associated with \(q\), i.e. \(\mathbf E(\langle W_{s}, \varphi \rangle \langle W_{t},\psi \rangle) = (s\land t)q(\varphi , \psi)\) for all \(s,t\in \mathbb R_{+}\), \(\varphi ,\psi \in \Phi \). The authors show that there exists an invariant measure for a Langevin equation \[ dX_{t} = A^\prime X_{t} dt + dW_{t}\tag{1} \] in \(\Phi ^\prime \) if and only if \[ \int ^\infty _0 q^{2}(T_{t}\varphi) dt<\infty \tag{2} \] for each \(\varphi \in \Phi \). If the condition (2) is satisfied, then the centered Gaussian measure \(\nu \) on \(\Phi ^\prime \) with variance functional \(\int ^\infty _0 q^2(T_{t}\varphi) dt\) is an invariant measure for (1), and a probability measure \(\mu \) on \(\Phi ^\prime \) is an invariant measure for (1) if and only if \(\mu = \nu \ast \gamma \) for some \((T^\prime _{t})\)-invariant measure \(\gamma \) on \(\Phi ^\prime \).
Further, a particular case \(\Phi ^\prime = {\mathcal S}^\prime (\mathbb R^{d})\) (the space of tempered distributions) and \(A = -(-\Delta) ^{\alpha /2}\), \(\alpha \in ]0,2[\), (the fractional Laplacian) is investigated. Since \({\mathcal S}(\mathbb R^{d})\) is not invariant under the semigroup generated by \(A\), the general theorem is not applicable, nevertheless, the authors find sufficient conditions for the existence of an invariant measure if the Wiener process \(W\) is either space-homogeneous, or associated with a seminorm \(q\) of the form \[ q(\varphi ,\psi) =\int _{\mathbb R^{2d}} A(x,y)\varphi (x)\psi (y) dx dy, \] the kernel \(A\) being tempered, \[ \int _{\mathbb R^{2d}} |A(x,y)|(1+|x|^2) ^{-k}(1+|y|^2)^{-k} dx dy <\infty \] for a \(k<\frac {d}2-\frac {\alpha }4\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
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