Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding
By Chunrong Feng, Huaizhong Zhao. 2012
70 citations recorded by the source corpus.
Abstract In this paper, we study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward–backward infinite horizon stochastic integral equations on L 2 ( D ) in general cases. For this we use Mercerʼs Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener–Sobolev spaces in C 0 ( [ 0 , T ] , L 2 ( Ω × D ) ) and generalized Schauderʼs fixed point theorem to prove the existence of a solution of the integral equations. This is the first paper in literature to study random periodic solutions of SPDEs. Our result is also new in finding semi-stable stationary solution for non-dissipative SPDEs, while in literature the classical method is to use the pull-back technique so researchers were only able to find stable stationary solutions for dissipative systems.
Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9ee98ba9d46ab874a356190ecdf1262706304bd0
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Citations in the corpus
- Random periodic processes, periodic measures and ergodicity (2014, background): “The definition of random periodic paths (solutions) for stochastic semi-flow was given in [22], [23]:”
- Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “The reader is referred to [23, 24, 53] for details regarding random periodic solutions.”
- Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise (2015, methodology): “The existence was studied for cocycles in [33] using a qualitative method, and for stochastic semiflows in [13],[14] using some analysis tools.”