Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

Oleksandr Misiats (), Oleksandr Stanzhytskyi () and Nung Kwan Yip ()
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Oleksandr Misiats: Purdue University
Oleksandr Stanzhytskyi: Kiev National University
Nung Kwan Yip: Purdue University

Journal of Theoretical Probability, 2016, vol. 29, issue 3, 996-1026

Abstract: Abstract In this paper, we investigate the long-time behavior of stochastic reaction–diffusion equations of the type $$\text {d}u = (Au + f(u))\text {d}t + \sigma (u) \text {d}W(t)$$ d u = ( A u + f ( u ) ) d t + σ ( u ) d W ( t ) , where $$A$$ A is an elliptic operator, $$f$$ f and $$\sigma $$ σ are nonlinear maps and $$W$$ W is an infinite-dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $$f$$ f possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure, which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper, we expand the existing classes of nonlinear functions $$f$$ f and $$\sigma $$ σ and elliptic operators $$A$$ A for which the invariant measure exists, in particular in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $$A$$ A is the Shrödinger-type operator $$A = \frac{1}{\rho }(\text {div} \rho \nabla u)$$ A = 1 ρ ( div ρ ∇ u ) where $$\rho = \text {e}^{-|x|^2}$$ ρ = e - | x | 2 is the Gaussian weight.

Keywords: Stochastic Reaction-diffusion Equations; Invariant Measure; Unbounded Domains; Ergodic Behavior; Long-time Behavior (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0606-z

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