Integration of controlled rough paths via fractional calculus

2017 ◽  
Vol 29 (5) ◽  
pp. 1163-1175 ◽  
Author(s):  
Yu Ito
Keyword(s):  
Fractional Calculus ◽  
Path Analysis ◽  
Path Integration ◽  
Rough Paths ◽  
Rough Path

AbstractWe develop a fractional calculus approach to rough path analysis, introduced by Y. Hu and D. Nualart [6], and show that our integration can be generalized so that it is consistent with the rough path integration introduced by M. Gubinelli [5].

QUASI-SURE EXISTENCE OF BROWNIAN ROUGH PATHS AND A CONSTRUCTION OF BROWNIAN PANTS

2006 ◽  
Vol 09 (04) ◽  
pp. 513-528 ◽  
Author(s):  
YUZURU INAHAMA
Keyword(s):  
Loop Space ◽  
Compact Manifold ◽  
Continuous Process ◽  
Rough Paths ◽  
Finite Dimensional ◽  
Rough Path

In this paper we will prove the quasi-sure existence of the Brownian rough path for finite-dimensional cases. As an application we will give a construction of Brownian pants, that is a certain continuous process on the continuous loop space over a compact manifold.

scholarly journals Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

2021 ◽  
Author(s):  
Kistosil Fahim ◽  
Erika Hausenblas ◽  
Debopriya Mukherjee
Keyword(s):  
Maximal Regularity ◽  
Correction Term ◽  
Regularity Property ◽  
One Dimension ◽  
Rough Paths ◽  
Point Of Interest ◽  
Convergence Results ◽  
Regular Approximation ◽  
Rough Path ◽  
Rough Path Theory

AbstractWe adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

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