On nonlocal PDEs with small equipollent parameters

Abstract The paper deals with the large deviation principle for a family of nonlocal nonlinear partial differential equations with Neumann boundary condition in the half space. Specially, we study the behavior of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> </m:mrow> </m:msup> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:mrow> </m:math> u^{\varepsilon{,\delta}}\colon\mathbb{R}^{d}\to\mathbb{R} , solution of equations with periodic coefficients varying over length scale 𝛿 and nonlinear reaction term of scale <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>ε</m:mi> </m:mrow> </m:math> 1/\varepsilon . Our method is fully probabilistic. Since the problem depends on two equipollent parameters (𝜀 and 𝛿), both tending to zero, we may apply the large deviations principle with homogenized coefficients. The main finding of the paper is new and is an extension of many known results in the literature.