A discrete approach to Rough Parabolic Equations

Aurélien Deya (1)

1 IECN - Institut Élie Cartan de Nancy

By combining the formalism of \cite{RHE} with a discrete approach close to the considerations of \cite{Davie}, we interpret and solve the rough partial differential equation $dy_t=A y_t \, dt+\sum_{i=1}^m f_i(y_t) \, dx^i_t$ ($t\in [0,T]$) on a compact domain $\mathcal{O}$ of $\R^n$, where $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$ ($p>1$), $f_i(\vp)(\xi):=f_i(\vp(\xi))$ and $x$ is the generator of a $2$-rough path. The (global) existence, uniqueness and continuity of a solution is established under classical regularity assumptions for $f_i$. Some identification procedures are also provided in order to justify our interpretation of the problem.

Mots clés: Rough paths theory; Stochastic PDEs; Fractional Brownian motion

Domaines: Probabilités [math.PR]

Aurélien Deya. A discrete approach to Rough Parabolic Equations. Electronic Journal of Probability, 2011, 16. ⟨hal-00530850v3⟩