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

Consider the following nonlocal integro-differential operator of L´evy-type L α ε , δ given by L αε , δ f ( x ) := (cid:90) R d \{ 0 } (cid:20) f (cid:16) x + εσ (cid:16) x δ , y (cid:17)(cid:17) − f ( x ) − εσ i (cid:16) x δ , y (cid:17) ∂ i f ( x ) 1 B ( y ) (cid:21) ν αε ( dy ) + (cid:20)(cid:16) ε δ (cid:17) α − 1 b i 0 (cid:16) x δ (cid:17) + b i 1 (cid:16) x δ (cid:17)(cid:21) ∂ i f ( x ) , related to stochastic differential equations driven by multiplicative isotropic α -stable L´evy noise (1 < α < 2). We study by using homogenization theory the behavior of u ε , δ : R d −→ R of double perturbed Kolmogorov, Petrovskii and Piskunov (KPP)-type with periodic coefficients varying over length scale δ and nonlinear reaction term of scale 1 /ε , (cid:14)