A comparison of homogenization and large deviations, with applications to wavefront propagation (Q1613603)

A stochastic differential equation \[ dX^{x,\varepsilon ,\delta }_{t} = B^{\varepsilon ,\delta } \left (\frac {X^{x,\varepsilon ,\delta }_{t}}{\delta }\right) dt + \varepsilon ^{1/2}\sum ^{d}_{j=1} A_{j} \left (\frac {X^{x,\varepsilon ,\delta }_{t}}{\delta }\right) dW^{j}_{t},\quad X^{x,\varepsilon ,\delta }_{0}=x, \tag{1} \] in \(\mathbb R^{d}\) is considered. It is assumed that \(A_{j}, B^{\varepsilon ,\delta }: \mathbb R^{d}\to \mathbb R^{d}\) are continuous functions periodic of period 1 in each variable, the matrix \(A(x) = \sum ^{d}_{j=1} A_{j}(x)A^{\text{T}}_{j}(x)\) is uniformly positive definite, \(B^{\varepsilon ,\delta } = \delta ^{-1} \varepsilon B_0 + B_1 + B^{\varepsilon ,\delta }_2\), and \(\sup _{x}|B^{\varepsilon ,\delta }_2(x)|\) tends to zero as \(\varepsilon ,\delta \to 0\). The behaviour of solutions to (1) as \(\varepsilon \searrow 0\) (\(\delta >0\) fixed) and as \(\delta \searrow 0\) (\(\varepsilon >0\) fixed) has been studied thoroughly in the literature, the limit process being described by the large deviations theory and the homogenization theory, respectively. The combined effect of large deviations and homogenization is investigated, and large deviations type results are proven for the family \(\{X^{x,\varepsilon , \delta (\varepsilon)}_{T}\), \(\varepsilon >0\}\) of random variables in \(\mathbb R^{d}\). The rate function is shown to depend on the value \(c\) of the limit \(\lim _{\varepsilon \to 0} \delta (\varepsilon) / \varepsilon \), the three different regimes corresponding to \(c=0\), \(c\in ]0,\infty [\) and \(c= + \infty \), respectively. These results are extended to large deviations estimates in the path space, and applied to the study of the wavefront propagation in a partial differential equation \[ \frac {\partial u^{\varepsilon ,\delta }}{\partial t} = \frac {\varepsilon }2\text{Tr}\left (A(\delta ^{-1}x) ^{\text{ T}}D^2_{x}u^{\varepsilon ,\delta }(t,x)\right) + \bigl \langle B^{\varepsilon ,\delta }(\delta ^{-1}x), \nabla _{x}u^{\varepsilon ,\delta }(t,x)\bigr \rangle + \frac 1{\varepsilon }f(u^{\varepsilon ,\delta }), \] with a nonlinear term \(f\) of the Kolmogorov-Petrovskij-Piskunov type, and in related reaction-diffusion systems.