Bi-Parameter Stability of Invariant Measures of Delay Selkov Systems with Locally Lipschitz Noise and Lattice p-Laplacian

Abstract We consider a wide class of stochastic lattice Selkov systems with three new features: 1) The discrete p -Laplace operator is defined on a high-dimensional unbounded integer set $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , and has a superlinear growth rate $$p&gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ; 2) The coupled drift terms are locally Lipschitz from $$\ell ^2\times \ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> to $$\ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , and have arbitrary polynomial growth rates; 3) The diffusion terms have time-delay effects, and are locally Lipschitz from $$\ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> to $$\ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . The existence of invariant measures of the stochastic systems in the Hilbert space $$(\ell ^2\times \ell ^2)\times L^2((-\rho ,0),\ell ^2\times \ell ^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are established by driving the tightness of a family of probability distributions of the solutions based on the idea of uniform tail-end estimates, the method of high-order moment uniform estimates, the technique of diadic division, and the Arzelà-Ascoli theorem. By improving these uniform estimates for large enough times and bounded initial data, we also establish the tightness of the collection of all invariant measures with respect to the noise intensity and the delay parameter. Then we show that the weak limiting point of any sequence of invariant measures must be an invariant measure of the corresponding limiting system as the noise intensity and the delay parameter tend to zero simultaneously. Our results are new even when the discrete p -Laplacian ( $$p&gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ) is replaced by the standard discrete Laplacian ( $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ), and the arbitrary order growth rate of the drift term reduced to the cubic growth. Our methods can be used for discussing the existence and stability of invariant measures of the systems in the Banach space $$C([-\rho ,0],\ell ^2\times \ell ^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup>