## 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

### Citations:

Zbl 0194.40801; Zbl 0573.93076
Full Text:

### References:

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