Random Attractors for Non-Autonomous Benjamin–Bona–Mahony Equations Driven by Laplace Multiplier Noise in High-Dimensional Unbounded Channel

<jats:p> We investigate the long-time dynamics of a non-autonomous stochastic Benjamin–Bona–Mahony equation on a high-dimensional unbounded channel domain <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> </jats:inline-formula> , driven by an operator-type Laplace-multiplier noise. We construct a non-autonomous random dynamical system on <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> </jats:inline-formula> and prove the existence of a compact pullback random attractor, which is cocycle-invariant and pullback attracts every tempered random set in <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> </jats:inline-formula> . The lack of compactness of Sobolev embeddings on unbounded domains is overcome by combining spectral decomposition on bounded truncations with uniform tail estimates outside large bounded sets. </jats:p>