A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process

Allouba, Hassan

Comptes Rendus. Mathématique, Tome 336 (2003) no. 4, pp. 309-314.

Nous introduisons un nouveau processus de temps brownien imaginaire et d'angle brownien, qu'on appelle aussi le processus de Kuramoto–Sivashinsky linéaire (PKSL). En étendant nos techniques développées dans deux articles récents sur la relation entre des processus de temps brownien et des EDPs du quatrième ordre, nous donnons une solution explicite à une version linéaire d'EDP d-dimensionnel de Kuramoto–Sivashinsky : $u_t=-\frac{1}{8}{\Delta }^2\mathrm{u}-\frac{1}{2}\Delta \mathrm{u}-\frac{1}{2}u$. La solution est donnée par une fonctionnelle associée à notre PKSL.

We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto–Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order PDEs, we give an explicit solution to a linearized Kuramoto–Sivashinsky PDE in d-dimensional space: $u_t=-\frac{1}{8}{\Delta }^2\mathrm{u}-\frac{1}{2}\Delta \mathrm{u}-\frac{1}{2}u$. The solution is given in terms of a functional of our LKSP.

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