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Professor Alex Mesiats
Purdue University
Invariant measures for stochastic reaction-diffusion equaitions
January 16, 2015
3-4 PM, 646 PGH
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Abstract
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Abstract: We study the long-time behavior of systems governed by nonlinear reaction-diffusion type equations
du = (Au + f(u))dt + σ(u) dW(t),
where A is an elliptic operator, f and σ are nonlinear maps and W is an infinite
dimensional nuclear Wiener process. This equation is known to have a uniformly bounded (in time) solution
provided f(u) possesses certain dissipative properties. The existence of a bounded solution implies, in turn, the
existence of an invariant measure for this equation, which is an important step in establishing the ergodic behavior of the underlying physical system. In my presentation I will talk about expanding the existing class of nonlinearities f and σ, for which the invariant measure exists. We also
show that the equation has a unique invariant measure if A is a Shrodinger-type operator A = 1/ρ(div ρ ∇ u) where
ρ = e-|x|^2 is the Gaussian weight. In this case the source of dissipation comes from the operator A instead of the nonlinearity f. The main idea is to show that the reaction-diffusion equation has a unique
bounded solution, defined for all t ∈ R, i.e. that can be extended backwards in time. This solution is an analog of the
trivial solution for the linear heat equation.
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