TY - JOUR

T1 - Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

AU - Fahim, Kistosil

AU - Hausenblas, Erika

AU - Mukherjee, Debopriya

N1 - Publisher Copyright:
© 2021, The Author(s).

PY - 2021/12

Y1 - 2021/12

N2 - We 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.

AB - We 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.

KW - Ferromagnetism

KW - Landau–Lifshitz–Gilbert equations

KW - Partial differential equation

KW - Rough paths theory

KW - Wong–Zakai approximation

UR - http://www.scopus.com/inward/record.url?scp=85112592944&partnerID=8YFLogxK

U2 - 10.1007/s00245-021-09808-1

DO - 10.1007/s00245-021-09808-1

M3 - Article

AN - SCOPUS:85112592944

SN - 0095-4616

VL - 84

SP - 1685

EP - 1730

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

ER -