Electronic Journal of Probability

A Discrete Approach to Rough Parabolic Equations

Aurélien Deya

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<p>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 $L^p(\mathcal{O})$, $p&gt;1$, 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.</p>

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 55, 1489-1518.

Accepted: 19 August 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes

Rough paths theory Stochastic PDEs Fractional Brownian motion

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Deya, Aurélien. A Discrete Approach to Rough Parabolic Equations. Electron. J. Probab. 16 (2011), paper no. 55, 1489--1518. doi:10.1214/EJP.v16-919. https://projecteuclid.org/euclid.ejp/1464820224

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