Optimal Extension to Sobolev Rough Paths

被引：2

作者：

Liu, Chong ^{[1
]}

Proemel, David J. ^{[2
]}

Teichmann, Josef ^{[3
]}

机构：

[1] Univ Oxford, Oxford, England

[2] Univ Mannheim, Mannheim, Germany

[3] Swiss Fed Inst Technol, Zurich, Switzerland

来源：

POTENTIAL ANALYSIS | 2023年 / 59卷 / 03期

关键词：

Besov space; Brownian motion; Convex optimization; Lyons-Victoir extension theorem; Sobolev space; Stratonovich integration; Rough path; LEVY AREA; EQUATIONS; THEOREM;

D O I：

10.1007/s11118-022-10017-w

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We show that every R-d-valued Sobolev path with regularity alpha and integrability p can be lifted to a weakly geometric rough path in the sense of T. Lyons with exactly the same regularity and integrability, provided alpha > 1/p > 0. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. This paves a way towards classifying rough path lifts as solutions of optimization problems. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.

引用

页码：1399 / 1424

页数：26

共 26 条

[1] Solving Rough Differential Equations with the Theory of Regularity Structures [J].