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A Discrete Approach to Rough Parabolic Equations

   
@article{EJP919,
	author = {Aurélien Deya},
	title = {A Discrete Approach to Rough Parabolic Equations},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {16},
	year = {2011},
	keywords = {Rough paths theory; Stochastic PDEs; Fractional Brownian motion},
	abstract = {
By combining the formalism of [8] with a discrete approach close to the considerations of [6], we interpret and we solve the rough partial differential equation $$dy_t=Ay_tdt+\sum_{i=1}^mf_i(y_t)dx_t^i, t\in[0,T]$$ on a compact domain $\mathcal{O}$ of $\mathbb{R}^n$, where  $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$, $p>1$, and $f_i(\varphi)(\xi)=f_i(\varphi(\xi))$, 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.
},
	pages = {no. 55, 1489-1518},
	issn = {1083-6489},
	doi = {10.1214/EJP.v16-919},    
        url = {http://ejp.ejpecp.org/article/view/919}}