Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations
Authors
A. Coulibaly
- Amadou Mahtar Mbow University of Dakar, Dakar, Senegal.
Abstract
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 (\(1<\alpha<2\)). 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<t,\\
u^{\varepsilon,\delta}(0,x)=u_{0}(x), &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.
Share and Cite
ISRP Style
A. Coulibaly, Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations, Journal of Nonlinear Sciences and Applications, 16 (2023), no. 3, 168--179
AMA Style
Coulibaly A., Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations. J. Nonlinear Sci. Appl. (2023); 16(3):168--179
Chicago/Turabian Style
Coulibaly, A.. "Coupling homogenization and large deviations, with applications to nonlocal parabolic partial differential equations." Journal of Nonlinear Sciences and Applications, 16, no. 3 (2023): 168--179
Keywords
- Homogenization
- large deviations
- nonlocal parabolic PDE
- SDE with jumps
- Feynman-Kac formula
MSC
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