Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations

Coulibaly, A.¹

¹ Amadou Mahtar Mbow University of Dakar, Dakar, Senegal

Journal of nonlinear sciences and its applications, Tome 16 (2023) no. 3, p. 168-179.

Résumé

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] \nu_{\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 ($1\alpha2$). 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,\\ u^{\varepsilon,\delta}(0,x)=u_{0}(x), \in\mathbb{R}^{d}. \end{array} \right. \end{equation} The behavior is required as $\varepsilon,\delta$ both tend to $0$. Our homogenization method is probabilistic. Since $\delta$ and $\varepsilon$ go at the same rate, we may apply the large deviations principle with homogenized coefficients.

Détail

DOI : 10.22436/jnsa.016.03.03

Classification : 60H30, 60H10, 35B27, 35R09
Keywords: Homogenization, large deviations, nonlocal parabolic PDE, SDE with jumps, Feynman-Kac formula

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