Publication 18-CNA-033
Fluctuations of random semi-linear advection equations
Yu Gu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
yug2@andrew.cmu.edu
Tomasz Komorowski
Institute of Mathematics, UMCS, pl. Marii Curie-Sklodowskiej 1, 20-031, Lublin and IMPAN, ul. Sniadeckich 8, 00-956 Warsaw, Poland
komorow@hektor.umcs.lublin.pl
Lenya Ryzhik
Department of Mathematics
Stanford University
Stanford, CA 94305
ryzhik@math.stanford.edu
Abstract: We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian random field, with the randomness affecting both the drift and the nonlinearity. In the linear setting, classical results show that the characteristics converge in distribution to a homogenized Brownian motion, hence the point-wise law of the solution is close to a functional of the Brownian motion. Our main result is that the nonlinearity plays the role of a
random diffeomorphism, and the point-wise limiting distribution is obtained by applying the diffeomorphism to the limit in the linear setting.
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