Timeline for What does the group action of a rough path in a Lie group look like?
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| May 8, 2019 at 12:43 | comment | added | ABIM | What is a good, algebraic introduction to rough path theory? | |
| Sep 22, 2018 at 15:31 | comment | added | Alexander Schmeding | @quallenjäger That is essentially correct. Basically an analysis of the algebraic condition geometric rough paths shows that the paths can be conveniently encoded by a path taking values in the character group of the shuffle algebra (from the shuffle algebras point of view the characters are in the dual which can however be identified with the tensor algebra) The relation for the truncated groups is then as you described. | |
| Sep 20, 2018 at 8:40 | comment | added | quallenjäger | @AlexanderSchmeding Thank you. Let me see if I understood the key idea correctly: For a geometric rough path, one can find the shuffle Hopf algebra on its dual $T(V*)$, where $T(V*)$ is the tensor algebra spanned by the dual base. By truncation, I would have a finite dimensional co-algebra on $T^{[p]}(V*)$ and by dualizing I would obtain the state space of geometric rough path, namely $G^{[p]}$, which is then the Carathedory group. | |
| Sep 20, 2018 at 7:58 | comment | added | Alexander Schmeding | @quallenjäger On your second question: It depends on the flavour of your rough path. For geometric rough paths its the shuffle Hopf algebra, for Gubinelli branched rough paths it the Butcher-Connes-Kreimer Hopf algebra and recently people have begun defining rough paths for different Hopf algebras, see e.g. arxiv.org/pdf/1804.08515.pdf. | |
| Sep 20, 2018 at 7:52 | comment | added | Alexander Schmeding | @quallenjäger (continuing, ran out of characters) The abstract theorem in the paper states that every character group is a pro-Lie group. Reviewing the proof one finds that the steps of the limit can be constructed from finite dimensional coalgebras sitting inside of $\mathcal{H}$. Take the coalgebras $\oplus_{n\leq N} \mathcal{H}_n$. Dualising these finite dimensional coalgebras yields the Caratheordory groups and thus the general proof applies. This is also (secretly) behind our treatment of the Butcher group as an infinite-dimensional Lie group see arxiv.org/pdf/1410.4761.pdf | |
| Sep 20, 2018 at 7:44 | comment | added | Alexander Schmeding | @quallenjäger I hope I understand you correctly, but to clarify: You want to know why the Hopf algebra characters of a graded and connected Hopf algebra are the limit of the Caratheodory Lie groups as in my answer? | |
| Sep 20, 2018 at 0:28 | comment | added | quallenjäger | @AlexanderSchmeding I have read your quoted paper aif.cedram.org/cedram-bin/article/AIF_2016__66_5_2101_0.pdf. However, I can't really see why it implies that the projective limit of finite dimensional Lie-groups, which is mentioned in your answer, can be identified with the Character-group of a Hopf-algebra. The paper mentioned only the converse that the character group of an arbitrary Hopf-algebra is pro-Lie. Could you please be more specific? Moreover, is it the character-group of an arbitrary abstract Hopf-algebra, or is it for some specific hopf-algebra. | |
| Sep 3, 2018 at 10:07 | comment | added | Alexander Schmeding | @quallenjäger My interest stems from the fact that these groups are connected to the infinite-dimensional geometry of rough paths. Based on the work Lyons et al. have done in the field I believe that rough path theory is intrinsically geometric in so far as geometric language is not only a convenient way to talk about rough paths but understanding the geometric properties of the spaces of rough paths should help in the analysis of rough paths and differential equations driven by them. | |
| Sep 3, 2018 at 9:58 | comment | added | quallenjäger | @AlexanderSchmeding How would this problem be interesting for you to investigating? For us, this would enlarge the space of integrable rough path. But unfortunately I posses very poor knowledge in the algebra. I strongly belief that, to attack on this problem, one need to choose a different route. Currently the argument are more geometric. I think from algebra point of view, one could discover some new result. | |
| Sep 3, 2018 at 9:10 | comment | added | Alexander Schmeding | @quallenjäger At the moment of this writing a characterisation is still an open problem. It has been suggested to me now several times as worth investigating. However, there does not seem to be a good idea so far on how to analyse the problem further (if you have any ideas I would definitely be interested). The consensus seems to be that there should be more to find out. All "the community" (aka the people working in rough paths I have spoken to about this) seem to know is that for different types of regularity, the inclusions of the subgroups are strict and that seems to be about it. | |
| Aug 29, 2018 at 10:14 | comment | added | quallenjäger | @AlexanderSchmeding That’s is a nice paper from you. Thank you. Another question from my interest, do you think one can describe the image of the signature ( as subgroup in the character group in the Hopfalgebra) by some unique structure? In other words, let’s say for an bounded variation path, can one some how characterise it’s image under signature mapping? | |
| Aug 29, 2018 at 6:18 | comment | added | Alexander Schmeding | @quallenjäger Thanks for the question, I added more information and some references to the literature to the post. | |
| Aug 29, 2018 at 6:16 | history | edited | Alexander Schmeding | CC BY-SA 4.0 |
Added details and references concerning the lack of some properties of the reduced path subgroup in response to (added in response to the query by @quallenjäger
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| Aug 23, 2018 at 0:09 | comment | added | quallenjäger | @AlexanderSchmeding Can you specify which property of the infinite-dimensional Lie group is not inherited by the reduced path subgroups? Thank you in advance. | |
| Jun 7, 2018 at 9:24 | history | edited | Alexander Schmeding | CC BY-SA 4.0 |
added remarks on the absence of the Lie group structure and some more details on the lift mapping
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| Jun 7, 2018 at 8:33 | history | edited | Alexander Schmeding | CC BY-SA 4.0 |
added 216 characters in body
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| Jun 6, 2018 at 18:19 | comment | added | Alexander Schmeding | It actually is a good question and one that inspires my research since I discovered it last year. Much of Lyons work on rough paths is actually rather geometric in nature (in spirit and in the language used). There seems to be a rather interesting connection to infinite-dimensional Lie theory, not only in rough path theory but also with respect to Hairer's regularity structures for SPDEs (some details are recorded here arxiv.org/pdf/1704.01099.pdf) | |
| Jun 6, 2018 at 17:30 | comment | added | user69208 | Thank you for this excellent answer. I definitely didn't think it was useless. I've been learning more algebra and wanted to see how far we could push Lie theory. The most Like theory I've seen applied to rough paths is in chapter 7 of Multidimensional Stochastic Processes as Rough Paths and was wondering if we could push it further. | |
| Jun 6, 2018 at 13:39 | vote | accept | CommunityBot | moved from User.Id=69208 by developer User.Id=35285 | |
| Jun 6, 2018 at 12:01 | history | answered | Alexander Schmeding | CC BY-SA 4.0 |