Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations
Alioune Coulibaly · The Journal of Nonlinear Sciences and Applications · 2023
Consider the following nonlocal integro-differential operator of Lévy-type \(\mathcal{L}^{\alpha}_{\varepsilon,\delta}\) given by \[ \mathcal{L}^{\alpha}_{\varepsilon,\delta}f(x):=\int_{\mathbb{R}^{d}\backslash \left\lbrace 0\right\rbrace}\bigg[ f\left( x+\varepsilon\sigma\left(\frac{\scriptstyle x}{\scriptstyle \delta},y\right)\right) - f(x) -\varepsilon\sigma^{i}\left(\frac{\scriptstyle x}{\scriptstyle \delta},y\right)\partial_{i}f(x)\boldsymbol{1}_{B} (y)\bigg] u_{\varepsilon}^{\alpha}(dy)+\left[ \left(\frac{\scriptstyle \varepsilon}{\scriptstyle \delta}\right)^{\alpha-1}b^{i}_{0}\left(\frac{\scriptstyle x}{\scriptstyle \delta}\right)+b^{i}_{1}\left(\frac{\scriptstyle x}{\scriptstyle \delta}\right) \right] \partial_{i}f(x), \] related to stochastic differential equations driven by multiplicative isotropic \(\alpha\)-stable Lévy noise (\(1<\alpha<2\)). We study by using homogenization theory the behavior of \(u^{\varepsilon,\delta}:\mathbb{R}^{d}\longrightarrow\mathbb{R}\) of double perturbed Kolmogorov, Petrovskii and Piskunov (KPP)-type with periodic coefficients varying over length scale \(\delta\) and nonlinear reaction term of scale \(1/\varepsilon\), \begin{equation}\label{eq1} \left\lbrace \begin{array}{ll} \frac{\partial u^{\varepsilon,\delta}}{\partial t}(t,x)=\mathcal{L}^{\alpha}_{\varepsilon,\delta}u^{\varepsilon,\delta}(t,x)+\frac{\scriptstyle 1}{\scriptstyle \varepsilon}f\left(\frac{\scriptstyle x}{\scriptstyle \delta},u^{\varepsilon,\delta}(t,x) \right) , &x\in\mathbb{R}^{d},\ 0