Abstract
It is shown that for an SDE in a Hilbert space, eventual compactness of the driving semigroup together with compact perturbations can be used to establish the existence of an invariant measure. The result is applied to stochastic functional differential equations and the heat equation perturbed by delay and noise, which are both shown to be driven by an eventually compact semigroup.
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O. van Gaans acknowledges the financial support by a “Vidi subsidie” (639.032.510) of the Netherlands Organisation for Scientific Research (NWO).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bierkens, J., van Gaans, O. & Lunel, S.V. Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup. J. Evol. Equ. 9, 771 (2009). https://doi.org/10.1007/s00028-009-0033-7
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DOI: https://doi.org/10.1007/s00028-009-0033-7