Timeline for Uniqueness of solutions of Young differential equations
Current License: CC BY-SA 4.0
30 events
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| Jan 7 at 23:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 8, 2023 at 21:45 | answer | added | Thomas Kojar | timeline score: 1 | |
| Mar 2, 2023 at 6:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 30, 2023 at 23:50 | comment | added | user479223 | @ThomasKojar I agree with Oleg. You are not paying attention to the question that is being asked. | |
| Jan 30, 2023 at 23:39 | comment | added | Oleg | @ThomasKojar I really do not understand what you do not understand in my question. We are given two processes $Y$ and $X$, $Y$ is regular $X$ is not. They solve a certain equation. Question: does this mean that $Y=0$ or there is another regular process $\widetilde Y$ such that $d\widetilde Y=\widetilde Y dX$. You are asking a different question, but which part in my question you do not understand? I am talking here about standard ODEs not RDEs. You are trying to answer a different question. | |
| Jan 30, 2023 at 23:34 | comment | added | Oleg | In other words: I am asking for uniqueness only among $C^\alpha$ solutions (not among all solutions) for which we can always define this as a Young integral. Is it more clear now? | |
| Jan 30, 2023 at 22:35 | comment | added | Thomas Kojar | @Oleg I updated the question | |
| Jan 30, 2023 at 21:54 | comment | added | Oleg | @ThomasKojar: no it is possible. If $Y=0$, $X=t^\alpha$, then regularity of $0$ is infinity, and regularity of $X$ is $\alpha$ which can be as small as possible. Again: we are given_ two processes $X$ and $Y$ with this properties. Given. Someone gave them to me. Question: is it true that $Y$ is identically $0$, and why? | |
| Jan 30, 2023 at 21:51 | comment | added | Oleg | @ThomasKojar Let me rephrase my question. We are given two processes. $Y$ which is very regular and $X$ which is less regular. It is known that $dY_t=Y_td X_t$ (in the usual ODE, not RDE sense, all the integrals are well-defined Young integrals). Question: is it true that $Y\equiv0$? | |
| Jan 30, 2023 at 21:48 | comment | added | Oleg | @ThomasKojar What do you mean I cannot impose a different regularity? Of course I can, this is the whole point of the question. If we know in advance that $Y$ is very regular, show that $Y$ is identically zero (or provide a counter-example). | |
| Jan 30, 2023 at 21:38 | comment | added | Thomas Kojar | @Oleg The RDE forces the regularity of Yt to also be less than 1/2. You cannot impose a different regularity. | |
| Jan 30, 2023 at 17:28 | history | edited | Oleg | CC BY-SA 4.0 |
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| Jan 30, 2023 at 17:27 | comment | added | Oleg | @user479223 sorry I forgot to mention that we fix the initial condition Y(0)=0. | |
| Jan 30, 2023 at 12:58 | comment | added | user479223 | As stated this is not true. For example let $X(t)=t$ and $Y(t)=e^t$. Then $X\in C^{.6}$ and $Y\in C^{.1}$. You probably need some kind of NOT Holder condition. | |
| Dec 30, 2022 at 0:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Aug 3, 2020 at 17:02 | history | bounty ended | CommunityBot | ||
| S Aug 3, 2020 at 17:02 | history | notice removed | CommunityBot | ||
| Jul 28, 2020 at 16:15 | comment | added | fedja | Then just ask a separate question with full details and I (or somebody else) will try to answer. Just make sure that you describe the space in which you are looking for $Y$ precisely :-) | |
| Jul 28, 2020 at 16:11 | comment | added | Oleg | I was hoping though that one can solve this problem by iterating the inequalities I wrote. My motivation is that i am stuck with a similar problem, but for PDEs. I want to show that dY/dt=\Delta Y+YdX has a unique zero solution (where now Y=Y(x,t)) under the same regularity assumptions. In this case one cannot write the solutions explicitly, but one can get very similar bounds to the ones presented in my question. | |
| Jul 28, 2020 at 16:00 | comment | added | Oleg | @fedja thanks! that's indeed a great solution. If you would formally post is as a solution, I would be very happy to award you the promised bounty. | |
| Jul 27, 2020 at 12:50 | comment | added | fedja | "Maybe I am missing something here, but the proof definitely does not follow immediately from your comment." It does. Let $t>0$. Choose small $\varepsilon>0$ and consider the last moment $t_1\in[0,t]$ when $|Y_t|\le\varepsilon$. Then $|Y_s|\ge\varepsilon$ on $[t_1,t]$, so $X\in Hol_\alpha([t_1,t])$ and then $Y_s=Y_{t_1}\exp(X_s-X_{t_1})$ for $s\in[t_1,t]$. In particular $|Y_t|\le\varepsilon\exp(2\|X\|_{C([0,t])})$. But $\varepsilon>0$ is arbitrarily small! | |
| Jul 26, 2020 at 21:55 | comment | added | Oleg | @fedja I agree and I also thought about using this fact, but unfortunately I don't see how it helps. It might be that $X$ is alpha Holder on any interval $[\epsilon,1]$ (with its Holder norm going to infinity as $\epsilon$ approaches $0$) and still not alpha Holder on $[0,1]$. Think about $\sqrt x$. Its Lipschitz on any interval $[\epsilon,1]$, but not Lipschitz on $[0,1]$. Maybe I am missing something here, but the proof definitely does not follow immediately from your comment. Do you have a formal proof or could you please expand your ideas? | |
| Jul 26, 2020 at 18:52 | comment | added | fedja | Just notice that your first inequality and the condition $Y\in Hol_\alpha$ forces $X\in Hol_\alpha$ as long as $Y$ is separated from $0$. | |
| Jul 26, 2020 at 15:29 | history | edited | Oleg | CC BY-SA 4.0 |
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| S Jul 26, 2020 at 15:28 | history | bounty started | Oleg | ||
| S Jul 26, 2020 at 15:28 | history | notice added | Oleg | Draw attention | |
| Jul 23, 2020 at 20:57 | history | edited | Oleg | CC BY-SA 4.0 |
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| Jul 23, 2020 at 19:16 | history | edited | Oleg | CC BY-SA 4.0 |
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| Jul 23, 2020 at 15:37 | history | edited | Oleg | CC BY-SA 4.0 |
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| Jul 23, 2020 at 15:04 | history | asked | Oleg | CC BY-SA 4.0 |