This article is devoted to define and solve an evolution equation of the form dy t = y t dt + d X t (y t ), where stands for the Laplace operator on a space of the form L p (R n ), and X is a finite dimensional noisy nonlinearity whose typical form is given by) is a γ -Hölder function generating a rough path and each f i is a smooth enough function defined on L p (R n ). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.
In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Lévy area terms.
We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman-Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well-posedness for the corresponding kinetic formulation.
We study a d-dimensional wave equation model (2 ≤ d ≤ 4) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter H = (H 0 , . . . ,, the model must be interpreted in the Wick sense, through a renormalization procedure.Our arguments essentially rely on a fractional extension of the considerations of [12] for the twodimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.Since the pioneering works of Mandelbrot and Van Ness, fractional noises have been considered as very natural stochastic perturbation models, that offer more flexibility than classical white-noise-driven equations. The involvement of fractional inputs first occured in the setting of standard differential equations and, even in this simple context, is known to raise numerous difficulties due to the nonmartingale nature of the process. Sophisticated alternatives to Ito theory must then come into the picture, whether fractional calculus, Malliavin calculus or rough paths theory, to mention just the most standard methods.More recently, fractional (multiparameter) noises have also appeared within SPDE models. A first widely-used example is given by white-in-time colored-in-space Gaussian noises, that can be treated in the classical framework of Walsh's martingale-measure theory [25], or with Da Prato-Zabczyk's infinitedimensional approach to stochastic calculus [6]. Such noise models have thus been applied to a large class
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.