The project will primarily address invariant measures of random dynamical systems arising as random perturbations of deterministic infinite-dimensional non-linear systems. Those systems include non-linear parabolic stochastic partial differential equations in bounded, unbounded and periodic domains, systems with memory and related. Our basic tool will be probabilistic representations of solutions for those systems also known as Feynman-Kac type formulae.
Our primary goals in studying those systems are:
1) to prove existence and uniqueness results for stationary solutions.
2) to prove existence of determining functionals.
3) to study the corresponding random attractors.
4) to develop probabilistic representation of solutions for a wider class of systems
Probabilistic representations of solutions of partial differential equations have proved to be an important and powerful tool in analysis of partial differential equations both in the coordinate space and Fourier space, and a recent observation is that Feynman-Kac type formulae are particularly suited for analysis of random attractors. For instance, analysis of the original Feynman-Kac formula was successfully applied to the problem of existence/uniqueness of stationary solutions for the stochastic Burgers equations by Sinai (1992) and E. Khanin, Mazel & Sinai (2002). To give another example, a beautiful stochastic cascade representation for solutions of the 3D Navier-Stokes system in the Fourier space which involves branching random walks due to Le Jan and Sznitman (1997) was used by Bakhtin (2004) to construct a stationary solution for the Navier-Stokes system in 3D with random force and prove its uniqueness. This approach promises to be fruitful in pursuing the goals listed above for other systems admitting probabilistic representations such as semilinear parabolic equation studied by Dynkin with collaborators (1990s - 2000s) using superdiffusions.
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