Averaging of a Parabolic Partial Differential Equation with Random Evolution
Mamadou Abdoul Diop, Étienne Pardoux · Birkhäuser Basel eBooks · 2004
In this paper, we study the homogenization problem of a semilinear parabolic second order partial differential equation of the type $$\frac{{\partial {u^\varepsilon }}}{{\partial t}}(t,x) = \frac{\partial }{{\partial {x_i}}}{a_{ij}}\left( {\frac{x}{\varepsilon },{\xi _{t/{\varepsilon ^\alpha }}}} \right)\frac{{\partial {u^\varepsilon }}}{{\partial {x_j}}}(t,x) + \frac{1}{{{\varepsilon ^{1 \wedge \tfrac{\alpha }{2}}}}}g\left( {\frac{x}{\varepsilon },{\xi _{t/{\varepsilon ^\alpha }}},{u^\varepsilon }(t,x)} \right)$$ with periodic coefficients rapidly oscillating both in space and time variables. We extend to α 2, the results by Pardoux and Piatnitski [] (the case α = 2), and show that the structure of the limit problem depends crucially on α.