Merken

# Long time dynamics of random data NLS and invariant measures

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00:05

time and again around 2 2 all right so let me 1st and in very much in front and in Germany and demands for the invitation to speak at this conference but also to the organizers for them organizers such a great program and it is great to be here so the only thing that I went to talk about this kind of them they will be cleared the planners say and I apologize as some of you who love her at least once a month I see Alex experience so like a and Hyacinthie is free to prevent 1 when I speak on your work so anyway so I would like to lose because of the makes audiences the 1st maybe and review beach and some of the work began which is in the background of 1 of these when is a talk about some recent work I would you guess the land from the Q makes armed aggression into the and the queen bee focusing Monday and then sort of like talk about some Will move more recent work which we still at riding a wave of suffer honey and John Madden evil look really engineering and I made it perhaps between 1 and 2 when the and I want you to think about that but it's kind of like talking about some of it in the 1st 2 ways like the the notion of gives measures which is the sort of like a equilibrium states that mean and when the 2nd part is about what sort of like the the the non equilibrium the the sole of the Sony quick also summoned before the addition integration just for the purpose of this document but most of the time but of all the debris of the case and wondered about the square Doris Monday direction 1 and set up for the purpose of the study redundant appropriately guidance to between the that also accepted the kind always mean relative to scaling assess everybody knows here because I don't know what we have people here who have demanded that that the work it has been a lot of progress and studying these equations and most of the fans the ambiance of and the beneath the custody of the of the phenomena but there's still some questions that remain this essentially in supercritical case which unity focus case people the nerve you have blowup on video so that drops Our indicating that would pose a serious problem we'll see some of the here's just quickly some review that of some of the work that you want honoring the and for because genes here the work of began to work within the system and that was about and then we know also for critical equations lending and I'm not putting all all names but there has been a lot of flak from domestic work and also some conditions results of Soviet critical and so I'm so because they said there is not much that is not thing only In default OK so on the Braley case and all this situation is a much more limited than the results and fewer and even or even a problem at the local would poles in theory right so also will again prove back 93 that actually this version in the period is Indian weaker and so for example and 1 B the L 6 Soviet foreign to the actually do have entered in the river is this is actually he constructed the examples as a word that means is that in the local would pose a theory you cannot handle these are the question for example the you to music when taking 1 B R L 2 create because you cannot close and there is no local fury at the level of it at all you this is bigger than 0 there are no results In a little look at OK and so in the same neighborhood the subcritical reduce of again the group as I said at the local where busses are just some examples and of course you have I'm delighted in the energy so how mobile and would be a new house the for so they were worked for for small they covet simple that barrel and then they sort of day-to-day break through the work for the 1st of lending over the last thing not 1 of our units composed of there but who actually did the last therefore the Quintaglie focusing in the evening I'm a student of mine did after liberation war actually of within the matter of the street as he did perspective of he sees he CU focusing a lesson for the and as said it was a small data brokers said before and as many as should measure here that this work of Cuban descent was to have also had to provide useful data and new process for the question please like I said I want to focus on the Russian space and the reason is because the nondeterministic setting you still relying heavily when leading number theory and so we don't know I feel how to treat many of the things in Russia so the point is that in India due later this month is not even now sold the the point of view that you want to take is that then maybe what you should do is used to study these equations from a domestic point of view and the spring just the 2 of them were again which is the starting point in

05:33

94 96 where he studied long-pending makes for periodic equations in the almost sure sense of I show that I you have we know what on the set of the Gulf War Peace measure so the news measures for an alleged and have been constructed for example for the pubic analysts in 1 into the and the focusing of reconstructed Viking Jaffe in the context of the quantum field theory later she formal the islands because the concurrent and in the context of 109 assistance by Lewis Rosenstiel who also triggered focus increasing when they mention that here you haven't you don't even know that's just a modest is that you have his measure that he take a sip of initial because flow at the percentage of women and so on and so why is it that you worked shining because usually what happens is that when you are sitting on the financial unit when you are sitting in the in the statistical sample and the support of the measure you're irregularity which is very rough and there are no closer quantities that live so why is it that you can get lulled into so the point is that assuming that you have a look at what Bozeman said he doesn't have to be that the meeting will be nondeterministic then the environs of the much-awaited after-tax articles and quantity that allows you to to continue the result from local I did little regularity wouldn't nobles of quantity so that's that's that's the key the day and the ritual abuse of doing this is that somehow it captures engineering behavior will say something more in a minute as I know this is our lives and so so on idea of what would and so and ingenue Guinness's appropriately as well and came back and I will in the Infiniti measures of the place or sell it so actually some are of course many dangers and challenges what is actually for the specifically the theatrical structures of this all these measures are there not that easy but they're easy to construct a furor over find among the domain securing the standard measure within volumes but there are some of the but if they if it even he is not 101 and is not always clear how to do it and I said into the and the need to be focusing on your but you cannot construct directly was proved on that actually would you have to do is you have to normalized in 1990 and removes some sort of like diverges along the easing of further measures to make sense there was known in the focusing case for it to be the bridges and Slate said that there was no obvious mentioned the measure is not having a wee mentions like for I'm in any the the evenings there to make your queen then is also know that if you try to do the same realization procedure you know what to something trivial to the Gulf measures of measures OK I and II for example if you did something like that focusing Quinn into the liquidating in the for which there is a measure the problem and I'm going to show you where is that you know the brewing bias but you'll you'll you of course have to cover a deal was flowing you can have actually which solution but if you want to show the glow with Paulson then you're going to see the single-mindedness of you going to see the referee roughest spaces are essentially not clear how to to to this challenging and some of them but but I know of sold so the focus and will begin to might be within reach things to the Soviet Union techniques from some sort of like by these arrangements but control distributions will be Nadia regularity structures but this is useful to talk about when they saw quickly how business should be fine because I'm going to be this meeting this of course is think of Europe equation of scientific information on him and insisting on the request vision and let's say 1 dimension the stages one-dimensional needed the people and finally also Spear said well you know you you starting his measure the foreman looks like this but I so this is like the McMansion this is kind of going in and isn't enough of you look at these cases but announces because every guest the the terms here that the living mentioned he had taken the potential energy and everything will fight infinity so this makes no sense but it's just above what is it that you you you you want to do defense also you cannot define the monitoring mission but what you do is you you take a new constructed using adoption measures reference national and then you construct the biggest measures awaited a measure interstate it's also the gumption measure I own some sort of like him space where it was constructed as the weakening the final emission measures of PAC is making it a part of your view 197 and this is day limit measure which 100 going and this year is the year of dozens born serving on the advent these is seasoning is the sort of like you can be a dozen years probably Nasher under this smart a they so would be suggests from now on is always the IED covers about non-viable sparing with means you and so all the

11:06

blind now most important point now is that you want to understand the gumption restaurants weakening of these final emission approximations of the season to see that their identity but then come the reality as so the point is that if you want to become the reality and the a theorem betrays the that those who wonder about the reality of the point is that of NHS their comfortably added the only farce is less than half day but not his beer equalled in half there into the dark Ontario behavior below is too I would probably be 0 or more in the if you are below minus half-brothers because you hear about this so what that means is that if I want to do this is this is exactly what guys you what is the support of the measure our Where is the date that willing to me that's the regularity of the day that you want to take someone the year looking at a big is measuring half to be sort of a nature half miners if you are and what do you mean you have to be just below and tool and so on so you see a single find that measures if it very very rough and so are once you you you don't then you know that these reduction measures just that all 4 of these are unreliable and so the Senate can began elements in support of the motion is holiday look fans of this is a functional emotionally 1 even distribution in Hialeah and you have this industry so as I said before was have constructed the notion show them the gives measure is just I awaken masher work were served this is than owning a part of a of the 100 billion is a renormalization 21 is the story of the rebels including the people I want this to be a new 1 really to to to role as a the focus in case you have nothing for free because you still need not to be too low the realization for it to be prominent initially you have to pull out deal to cut off from I so the same cut off that you see in the than music is comes here on Tuesday sectors as the which beef which cost and this is larger and actually is when the small so it's you don't win but anyway I so it is in powers is undefined you just need some kind of and the powers 5 which is the critical demonstrated that there has to be sufficient all right so what is

13:35

Morgan these like Morgan proving 1 day this is his results are just I'm just taking it because I wanted to get used to call rates in the you think the equation would that they doubted the date in support of the measure and lessening the blue almost surer means that you can find a set in here if you're focusing on you have the condition everything to have a smaller pool of Murcia 1 of the of the EU's mission 1 such a forum Saturday that initially problem can be continued abandoned then you also prove that these measures will survive but so why this important 1 this recalled for a 2nd that the mystical you have locally physicist strictly Maryland 0 I only viewing each 1 and and and and that's a result would the buyers mentioned this as I said before easing global resulted in nature Hoffman's where there is nothing there is no was appointed as a viable into it but that's that's what the dimensions week and so how does he proved this Limoges quickly with movies because we're going to need it is like what we have to do is you have to approximate the equation by finding the emission of oxidation and as I'm actually a crucial fight because where I want to bring home here is is how do you use the values of the measure and what is important languages in a bit of a of the other conditions that would you would you really use is the bias of the final dimension and therefore not the numbing buyers of the infinite miserly isn't sometimes comes in parallel to the to the global boulders and so so why is it that defined international measure is is is invariant 1 because this equation when you take these things you stay hadn't onion so the space 110 which is not always true when you take equation annually to provide local researchers have to be H I mean what happens is that you might think I'm going in but the problem is that you will know My notes and this is actually a very delicate point out what happens with Ag measures for measures and the sort of these unknown canonical transformations I that they have said before you leave fear and then tells you that it is the 100 billion volume serving this is conserved so this is a local services these missionaries is environment and I wouldn't do it you use the variants of the final Commission masher to continue so you know where they might look as security

16:01

you have diminished the look of fury for various local and so now using bias softly-spoken the solution to to the final animation approximation To a by 109 provided that you have unanimous said that would take us with and that in

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this process you have to select a perceptible only with broad the 1 that I believe that your speech then you can continue

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that process as once you have these then how do you continue you believe this approximation lamented tells you that once you have a date finite emissions at which time the you take you that you know that is local and I knew what the you in it did at the state stated a method side-by-side with the UN and is not a trivial problem because what happens is that you see after the nation did not die of existence is somehow spoken become really far apart and you have not .period Mountain you so this nontrivial limits to what you have to do some the data as a stepping stone and Intermediate System which is the evening dimension creation with a projected at the time and then when you do is you you use that as a stepping stone and you compare I yield to you Prime and the new bride to you and step by step and then that's that's how it as always have that now you have now been this of the globe you and so the icing on the cake is that you get out there but you prove that you have this converges to come to New England premiere giving buyers with reading buyers of all of but they still have to the

17:41

would-be wouldn't and here's but would becomes interesting what we would want to Is that s owing to the frequency of the molecule which is a steel to predict but now as I said before the gives measure which was known to exist and is need some sort of library normalization in order to exist again and you can actually see these women isolation also in trying to prove this theorem because it's actually the same thing and to remove what to do with residents and so what you do is

18:12

you have to treat what is called the week or the millionaire I think so you have to subtract these terms from the humane I'm a Hebrew exactly the same fearing that then you're going to win both the pool and and the associated measure use environment so here but now this is the quality of their indicating in their potential energy disassembled almost surely so you have to sort of weak or the end of the year re-normalised if fine Indonesian approximation of and that removes the storms and where we like to say is that actually this is the 1st result ever that is globalized and supercritical so indignities person this is the 1st results are very appropriate because until this you yet it is staking a of from etc. chemical publicly because appropriate because he itself related the sky but I saw its own something nice that it's actually a distant observation is that you see that the minister so always charges opens its with positive measures so if you improve using this argument if you can prove that such a result would would probably would measure 1 then what that does to you is that you cannot have a lot blocking that apology basis you can have a lot but if you can economically stable and that the border 1 thing so that's actually found something nice OK because you know all the ones that have is the focal singularly on rather than on the street but there is a lot of that kind of SNL the additional difficulty here is that we did not have a look at what missing places he did in 1 OK so he can understand the argument that just described because there is no doubt that this is not a candidate could local with all this before the super so what he says is as well look at me I just want to blow emotional global results a wide that the music local it's enough to prove that I have a lot of good boldness for that is the support of a mention in other words on the news of abroad realistic local with pulses wisdom and so if you look at the staple from 96 which is a very long paper slid 56 pages like know 50 of those I just to prove the probabilistic local polls can for that's less does the court that's the heart of the matter in the paper and the rest as follows the same as before so it's worth the ink on his so this brings us to this notion of saying the case and now forget about it for a moment about the national for at the moment about the deal was let's think about just the look wood boldness and so what we're going to saying essentially is that if you if you if you give you think the ten-year undermines the date that then you're always going to be able to improve your local fears and so that they can weigh guarantees is actually not a foreign idea for form for us formal for anybody but especially from Harmony finally because this is exactly what what work with what we know what we use all the time when you prove you inequalities for example that surround series better and be in yesterday and so the randomization Dassin and improve the regularity with investors improves they did in the but once you have that the that in turn to get that this thing is that you without once the the as always is this is a strategy to to prove these and this is actually the same strategy that I don't know if at the same time but but said that under the same strategy but what is new is used by the breadth and the most for the for this across the island ,comma for the few formal and will use exactly the same idea what's the idiot is that now you look solutions of a particular form sordid Caesars say where you run data now and when those unwilling to look for solutions they look like a emotional the run make that blasts some other than new and instead of solely for you so for the difference equation pain and so as a consequence of these and what he can prove that if you suffer this there is a question I use so here in a smaller space that you start this is for a fixed in what everyone to do when you're going to find value in a small regularity done they initially as as a consequence he proves almost surely not only that but the nine-year Park always has moved from the park reduces India's 96 94 96 now I an

22:49

important remark here is that by doing this is not that you're just saying that if you randomized bring the problem from the lake supercritical critical to abuse of crude that's not true I mean what it is is another problem the wording the 1990 a hybrid right I mean you have supercritical terms which are random and you have a sort of like that damages which should see the music which I really but may be a small they're also created by you have to but you have to treat all interaction from the mine longer than anything that the ministry when the as a 1 of the things that you need place before you even start this thing is because you have to deal with the many things that the minister got a smaller space you actually have to have a place in the deadliest local would Paulson is that in a smoother expects to stop at least In this approach alright so I'll images quickly the White works of works so I saw everybody I'm sure here just stated by sort of because of the last creationist theories of think OK here been comedy and the company random terms you're dealing with neuron United whatever that is as I hear you have just a civilian communication from the morals so as envisioned boasted that the knowing only that sort of thing they did the norm is controlled by the program has so what's the point of links to the point of this is that there are arbitrary journalists said were the absolute value of this function is bigger than 1 that is the case explanations so I'll I'll take it any any better than the land that here to be about the miners fear were work 2 times the normal here it's all about those issues that did exist it said but the which is the complement of these studies on the matter is the complement of these so the probability of the continent Army is the probability of this which if you put this down here said that but doesn't believe you're outside that said right from the outside is said to insecure here then you can replace the absolute value of by the other 2 square and that's fundamentally still initially 1 of best the forwards white

25:04

Bill is 1 example some of the major events like these babies is what Morgan was highly view are interlinked mentions just below do they would look like this but the and as in in doing your your around the rumbling directions will you have to do is you have to have the makings of this fall go when this is just that the set of and 1 and 2 and 3 each of them in the pool such that the ladies now what do we call and ask for you which is the day that which is actually very interesting because a different way of understanding is that it tells you that anyone in the league the school are never too which means that you never lose the independence of the runway for every time they 1002 orders becomes absolute disgrace was independence so that we quarantine would thus which is the same as they admit that there was as it does at this stage you hope you always have in the past I was so I'm not if you just want to do she sure here if you want to isolate the absolute value of these ozone the air anomalies and you look at these I'm using the word tool

26:13

to use and now because of the wind and the 1 you sit you she shots then you pick up you lose because you pick up the commonality of upset you pick up the commonality of this set which is big and then being designed integer company at this point and serious or intersections as fears or whatever and that and and then use them without loss means that you cannot use it even if you cannot close as all the last mediation the it tells you is that you are replacing us they said a minute ago more than a said the warning us about the absolute value would be to as would you are looking at this and this we know that and that's enough to close In 1st how it works and that's how it works in now I know I'm I want to say that something just because surgeons you have things isn't making factory at Wembley with a minute and then actually moved a couple of years so so so so they take away from here is that the randomization I would ask is actually allows it to improve the local would also miss the irrigation view the randomization as a separate issue from the environs of dementia and once you have improved Eurocard would pose the question is how do you go to the wall and then I the Bears which wishing you have which Regina you are but you have to make sure you don't have a measure of you can actually do a variety of things or you may not be able to do anything these with with Quintana the nothing that is conditioned the 1 thing that I should say here may be going back is that 1 problem that you have is that you see even if you if you have conserved quantities or original consequently is for you in a way that is that brothers when you study the difference equations you have no consequently you lose the analysis that in anything so it said it is a big

28:07

problem but OK so all

28:10

of us they said then that hardly passed so separated 2 issues randomization improves local and then you have a measure than newcomer from looking at the moment you don't have a match maybe you can do energy is the music you can do something has been region or your student you may still working with because think as I said there was this has been a lot of work after then so nothing happened between women's work and rounds of 7 and there's a lot of activity I'm not being comprehensive here this long before the whole work for a lot of equation and not just this person all sorts of electrons to integration and goes to save Jews here from windows and is kind of like this is actually part of a long-term project but you know and this interstate bear that was of consumers surveyed in the new where question with the dynamite former acuity and actually add this latest thing this question for it's a funny situation for dramatic purposes and for trying to understand these methods beyond that date is Regine which would you do we you look around the league senior in Europe pollution will want to look around special solutions and this is geometric equations have its ground states we're interested in and trying to understand how to 207 just cut out the property can be better but understand the transport of the of the randomization and there but by what I want to say that the nose is actually his is a beautiful recast of this new forms that actually you eat that exactly right there like a gift that people giving because exactly what you need when you have quadratic derivatives but to preserve their the independence very shortly before so I tell you that if you have random random interactions you have to do something that way calling to make sure that you use you keep independent the models and then and there and with the news from that is exactly that when I was in the probabilistic and so it so it's actually quite interesting as only quickly about this these were so this is actually to do things in which the results which are still separate the coverage but the reason why we're interested users because they're actually closing an important gap and there's some critical theory the candidate that the fury and they and they almost sure the award would bolster someone is saying to the yet there including the falls equation as I said it minute ago go up so where would improve walls that you're below was to you have a little short blowing and as for they look for the globalized that there needs using the iron method the best result known about 200 pictured here is about two-thirds you can't as a wouldn't provide here and if you're here you have globally but as I will explain in a minute if you have a global bourses that is produced the you can I use just the deterministic preparation regularity methods to to to filling this gap as so we were interested in understanding how global In his rage and for the one-day which is focusing on you have the same problem they would this is still a gap between us so that they almost surely always is just below half and as then of course you have the focus you have about which 1 and the focus seems to have about 4 months but in the focusing you have nothing and we also wanted to close the I'm although the results follow from a method that we would like to call approbation and produced a brief of regularity so enabled BC here because this is this just and just said on this is that we just don't want it to be why is this not the result is not training because if I give you the down which is a small but that on them that is seeking in each minus epsilon died 8 days with brought is 0 I don't priorities 0 is in support of so I cannot just take but assume that they can lodge the gate and is good and it is aware you have nothing the so also so you usually easy when you prove these results and this is what is it that the coming in disbursing having more regularity there helps you usually would you do it say when you need concert wanted to prove something Glover you proving that the would you have a conserved quantities and then you use probation regularity but that the smoother solutions I'm here but the bank and so on what we do instead is weak after they did here is that we want to do is we wanted to compose the data In due into into a term that is close to the support of I'm going to be made up assessed goes to support the motion and thereafter and rest the ball Regina Raffarin topology that where we are that's it's also forcibly intervene want to do it for as being of 0 so the referent apology would be the age minds and this was a reminder and the question is how to do this so that things work and then we want to do sort of like a sort of innovative but duration argument to conclude that and on the island and evaluate them when the presented here is is actually we do we can use the bases but is very generous so every time that you have real results are and that is almost sure using a measure and and there is a God above you you can use it also every time they have been mentioned in the news this every single so that is needed is examples of this is a statement of the fear sorry for it I have to state something so you I find his means is just to remind you that on the debate that I'm taking

33:55

hazardous form and so the from you that this is the date that the when I find 0 you are intertwined sense alone and this is with prominent is 0 in the support of amateur so your I

34:07

knew that that that they can use all the human aggression right now ,comma and so I don't know if you normalize so we normalize their ability to so you can always pull the recording which is finally coming to the meeting of the spokeswoman .period were about as other affairs and anything in his any lasting any other football stadium on some some abstinence then that it is the set all of a sudden the gumption so there's no this measurement you always have a ghost mission associated to the 1 blast of fire everything I am which is spending that your forming these with that anybody to leave to the Alpha square instead of some great enabled agree 1 square and the support this year as it said with with measure ones such that you can continue the student would lower solution and half this fall when Sylvester the result as onward

35:04

Romania's you'd think that they don't and that is the steps to say the 1st step or we want the school to do we really want to show that is they don't gives rise to 2 would lower solution in some rough at the border of the UK and the ones that do we will we will optimize the ring Soliday that you'd think the back I knew you do you decomposes into it that something that is going to be determined later I you break it into and less than him and then you take a sitcom Bob where you put this like it if you wish you can think of this and acid it as a high frequency high hypotheses they let you alleges business high-frequency spot and that's why I call again and you write the South's ancient blasts sigh 1 prominent as the storm and so would you know you know this is just the low frequencies than in the in the HSC on 1 the regular then this is uniformly bound for the and I and these aliens else are obvious sponsors it is really quite simple and as inside there was this week with the 1 personality noise you use Morgan's result in Asia Minor said to to claim that there exist a certain brought in 1 such that if you look at me now then why these way In this alone there is no way out but then I knew I had to learn that the storm but so this is a small the soap in the liberal-minded but this measure since this will probably his usual the support of the measure the diet just 1 room so there is a Morgan's measurement you cannot anything small to that because this is not seen by the membership so this is a statement the support of the motion and so this is 1 of the elements for which have that kind of continued lower against all right that has your wife personal sigh as of now you know that the symbols of my Bible symbols of to some solution you have to have this form when you have these involved because we proved that this is always a smoother than you start here In that so more actually part of the argument and I want to get into the details with that is news the situation arguing this approximation this nation argument that I mentioned before but they you get the is of byproduct of that you get an answer these are just the X being norms we CNN be so you also get these uniform of bounce on on the storms in the Sixers the norms 1 each it that at the time step going forward and the best part of this approximation duration argument of all will before so now how would you do East you used is that this is the role by the the UN is no way to prove the existence and uniqueness of the you Wi-Fi you want and as thinking the reason that the population argument and the real 1 corporation argument works it is because there is this norms are less than 1 around their small so the of small becoming a more as a when endorses there solely for you I'll find you look again I added the phrase as a suitable difference equations so you concede that Bryant had to be and that had evolution of 41 minus the solution this is that this is what you want to solve 2 wonderful for that but instead of looking at that you look at the difference between the regained 1 and that 1 that the preparation that looks like is these missiles this problem and this is an McDermott of the character and the fact that these 2 things are small is what allows you to to 2 to finish with a contraction in duration and indicated that little things are smaller his uniform bound allows you to prove that indeed this exists and that the solution to the gulf of after is simply this is prime and so you found you find a solution of the evolution of all that data last day as I was this what I no but the whole no if is changing this should all of among far-right so I

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I want to have anything to do with the is there is there is the recovery of regularity is the recovery of regularity which actually I'm going to skip since since now I'm being pressed so a parties is what I want the dog is so out of this was about the size of schools so that 1st fight is unveiled Monday a cosmic censorship worries about the developers would you like some of these measures which can of this kind later the day then it will even 1 as wanted quickly now is about this set result which is actually and the transfer of energy is related to and at the Local for Patrick may be the 1st day for the halfway but also to some alleges that suffer means that the other side of what Tucker with describing this week as a candidate the decomposition of the transfer of energy of the of William beyond makes for a less and I saw everybody knows that if you are 1 of the EU and he should have started in the nose of the state and then bombard them is used as the basis of all the question was how to hold who hope to capture that and this is what also go with him with prominent among approach was that what you want to do is you want to study the growth of oversold norms and to see how these rules hello this is transferred from basic is feasible he said actually he said conjecture on hidden in southern was it whether the Diaz with that of this village solutions to the cubic equations whose ages norms and indefinitely and that is where schools the interest of the country and so

41:07

I am willing himself from Newt as some promise and that he constructed some special but when the new wave equation somewhat affected which in which each exhibit that there's someone who's been another fundamental progress being made in this paper of Williams is the philosophical style in which they actually construct the larger about finally grow from Seoul of norms and you haven't seen the result I would be here is also worked behind the major and believe were at an ocean research and that and more recently actually it in real estate of 100 so there's of of here so they prove that if instead of the Taurus is David the problem in this city there's than actually they the result of the conjecture and they said they probe yes and they also gives them some and on Monday the 1st stop of a of this conference with whistle Patrick talking about similar things for me for the have waived

42:02

and sell it now I don't understand Morgan's Conjecture for the kill again and this is actually a very very ,comma very of course but the problem remains unsolved but something which is kind of like being between again and when describing the 1st part of the stoppages is this this this environment and its measures is everything really that the study all 4 of their existence and uniqueness of non-equity Ramadan's machines but they saw this that allowed inspect matter that is kind of like this is requirement very poorly understood that this is the existence of stationary non-exclusive states and unfortunately this is trying to alternating gives some sort of justification some sort of a rigorous sofa framework for for the statistical description of this sort of of and now this prize very harmonies kind forced plastic 34 systems but in which it is actually is actually very hard but there has been progress in these sort of like uh chains of oscillators of statement be leery military-related Thomason and more recently of her mockingly in which they look at some of the collection of I'm harmonic oscillators with with nears Nunavut copies so you have the handling of the concede I'm there are this is the 1st 2 papers and this is the relationship between the beginning of the coupling and this is is much harder case in which the relationships like this and there already they cannot solve this problem unless they only have the negotiators here there's no no no way sort of problem with the end here they cannot do more than they did that what you do is you think this and put them into contact with 2 he got a different temperatures so you have your system and what you do is you want to inject energy into the 1st modern dissipated into the last time and you want to try to see whether the system will relax into into a stationary in new state and so I to forecasts also supports the starting point is actually started modeling the work of quality and that is that we can provoke belt was 100 billion has this forms and and it is not that so here it is much harder than the other 1 because the directions don't depend just on the relative differences but also the moment of particle and neighbors and is not that there is a finite said for which of the resonant unless his skills and collapses due to the storm the 1 this until we do the same thing we attach to him but at the low walls C 1 and CNN and that would use of it is under way of so far by the anticipating energy Idaho expect system to do ,comma went on a minimum environment show which has some sort of flats from low frequencies to power and we also want some sort of like rate of converge and so that's that's what we want to do and so I have

45:04

just the graphical meanwhile looks like this for that age so this is seat he boss here we also understands that and there's a will leaving the new universe of the molten alone we just do this and as I saw you you look at this just a wrote just for convenience status this some them the degeneration of the of the transition similar to the Fokker Planck operator is here

45:31

and so when you know the the 1st thing you come true is that if you are equal temperatures of the 1 and then this is an

45:38

Indian relaxes inviting gives motion a ban on all of that follows from the fact that you look at the object of the of of of the operator of the plant and this is as always seem just facing the 98 million case that 6 of the Williams is what happened at different temperatures and I you want to prove whether the existing unique smoother going on at the Ramada Inn measure in the House this transfer of power energy and the answer is that again but it will have the molds some of the worst we only have 1 2 3 jails

46:12

then I actually come through that I may be to finish because I want to assure patients had from within 30 seconds of the 1st on the cell phone if that what I want to explain is that he approve the use swings hard it's the only thing that is really hard as exist everything else kind officers stand the but the existence of the measure is very very hard to understand Islam issued to transparency is 1st of for some going to movie and show you what we improved so I would also add that some rates and that the problem is that we're finishing this 1 region which is sedated so we don't know we're going to the explanation reporting on so also won 1 useful seems to change course then are you have to change course means more than once but you can change corners this is the adjacent and the sum of the squares and this is the the didn't president 1st Prime would have like previous works is is that is that we lose the high point of the city of free space so we we are not hyperbole take everywhere but would have to remove these and they have to win will have an agenda line that would have to remove the good thing is that if

47:30

you start would seek to equals 0 then you would remain at sea to close to 0 but not in the

47:36

habit of this city which is what gives you that if something exists is going to be swallowed up up then I would have an hour boundary of his and so I think this is a measure of photos from what is it that you want to do what you want to use it would be constructed at a function that somehow penalize in his region where this is a smaller onset of high frequencies and somehow but would want to do it is a once-over 2nd and that construction will give you an upper bound on the heating time of of the good region which should be a compact said television quickly picture because maybe the

48:12

reason this is actually worse handled the problem will have to chop the space which which which but you have to do before and so on how is that this is the good region where you want all your dynamics to happen as a what you want to do is you want to prove you want to construct happened function as a here that does about as soon as you enter any of these regions you you working but interview Duncan by using you know usually when indices are very hard to do when you should people what they do is they say Well you know your guess your best guess is that they have

48:44

a function should be the but none of them

48:47

work everywhere he knew how to actually understand very well aware of the dynamics of the deterministic system due to understand what waterworks where and in the regions in which you have more than 1 is bound to set them limit the temperature and so on but in the

49:04

regions in which you have a lot of

49:06

these numbers that have contact they talk to each other but you can get someone on on the way to solving the way to

49:13

solvency bank is 1 of them and they would have to do is you have to solve some of the it's sonic equation with a rebound of ideas and some comments the conditions and sold for all the other was flow at all those together an end in some of the region's to get days the problem is that you have to really understand did this to you really have to understand how the faces you need the need to understand that the fees Diamond the phases with look but at some point and so is actually quite there and so and so this is where they live in the heart of the matter is that as I said I want to have that then the rest the uniqueness and releasing the photos from Soviet control arguments will contribute them that those who that the domestic system sort of can access any region in face face-to-face so so if you give me 2 points the confined enacting a neighborhood of of 177 here in the Indian 1 anyway and then there is suspend their fear of stroke Maryland that those are the ones who have the authority to diminish system that the distant 1 would hit it and I think even fugitive considered the founder thank you questions the directions to the next this was the candidate had to support you the difference is that the assembly cannot do you see the problem is that this will have to break will be only with the amount you have 4 4 that the prevalence of brother Brian is a bit higher energies you can you can do with the promise that at lower energies like that when when you're 0 so you're stuck in the middle of the year energy doesn't really work right to the floor of formal to review your stock and unsold and will happy 1st with the news that I mean they here the paper of Janet than women with marking their also on 3 becomes the kind of 40 there's so so it's not is not clear will be much more complicated than the way you will have to chop and understanding is the construction of the skipper function is it's "quotation mark if you have more than sufficient to our the the John onions here I don't know good question the the thing that and Rob now

00:00

Quelle <Physik>

Resultante

Nachbarschaft <Mathematik>

Stellenring

Prozess <Physik>

Punkt

t-Test

Gruppenkeim

Regulärer Graph

Gleichungssystem

Raum-Zeit

Computeranimation

Eins

Richtung

Übergang

Einheit <Mathematik>

Gruppe <Mathematik>

Gradientenverfahren

Tonnelierter Raum

Einflussgröße

Deterministischer Prozess

Addition

Torus

Frequenz

Gesetz <Physik>

Spannweite <Stochastik>

Polstelle

Menge

Rechter Winkel

Sortierte Logik

Konditionszahl

Erhaltungssatz

Kubischer Graph

Wellenlehre

Physikalische Theorie

Arithmetische Folge

Fächer <Mathematik>

Perspektive

Optimierung

Beobachtungsstudie

Thermodynamisches Gleichgewicht

Physikalisches System

Fokalpunkt

Integral

Energiedichte

Elementare Zahlentheorie

Quadratzahl

Mereologie

Ruhmasse

Energiedichte

Quadratzahl

Einfügungsdämpfung

05:32

Distributionstheorie

Stellenring

Punkt

Relativistische Quantenfeldtheorie

Natürliche Zahl

Gleichungssystem

Zeitbereich

Element <Mathematik>

Massestrom

Raum-Zeit

Computeranimation

Potenzielle Energie

Hausdorff-Dimension

Einheit <Mathematik>

Regulärer Graph

Gruppe <Mathematik>

Theorem

Nichtunterscheidbarkeit

Schnitt <Graphentheorie>

Einflussgröße

Divergenz <Vektoranalysis>

Distributionstheorie

Approximation

Flüssiger Zustand

Frequenz

Variable

Invariante

Arithmetisches Mittel

Reihe

Funktion <Mathematik>

Menge

Verbandstheorie

Sortierte Logik

Derivation <Algebra>

Wiener-Hopf-Gleichung

Hyperbolischer Differentialoperator

Kubischer Graph

Standardabweichung

Renormierung

Renormierung

Hausdorff-Dimension

Fächer <Mathematik>

Vektorraum

Term

Unendlichkeit

Physikalisches System

Algebraische Struktur

Zufallszahlen

Koeffizient

Fächer <Mathematik>

Kommunalität

Stichprobenumfang

Hamilton-Operator

Freie Gruppe

Inverser Limes

Spezifisches Volumen

Gleichungssystem

Leistung <Physik>

Thermodynamisches System

Raum-Zeit

Zeitbereich

Gasströmung

Gibbs-sampling

Frequenz

Fokalpunkt

Ordnungsreduktion

Menge

Inverser Limes

Unendlichkeit

Existenzsatz

Mereologie

Energiedichte

Dimensionsanalyse

13:33

Resultante

Theorem

Stellenring

Punkt

Physiker

Sterbeziffer

Hausdorff-Dimension

Natürliche Zahl

IRIS-T

Gleichungssystem

Raum-Zeit

Computeranimation

Unendlichkeit

Physikalisches System

Zufallszahlen

Hamilton-Operator

Spezifisches Volumen

Stochastische Abhängigkeit

Einflussgröße

Deterministischer Prozess

Kanonische Transformation

Thermodynamisches System

Approximation

Axonometrie

Stellenring

Frequenz

Approximation

Unendlichkeit

Invariante

Gesetz <Physik>

Sortierte Logik

Physikalische Theorie

Konditionszahl

Erhaltungssatz

Surjektivität

Beweistheorie

16:15

Stellenring

Prozess <Physik>

Hausdorff-Dimension

Computeranimation

Unendlichkeit

Physikalisches System

Existenzsatz

Hamilton-Operator

Inverser Limes

Stochastische Abhängigkeit

Gleichungssystem

Deterministischer Prozess

Approximation

Finitismus

Axonometrie

Primideal

Physikalisches System

Frequenz

Approximation

Invariante

Lemma <Logik>

Physikalische Theorie

Parametersystem

Surjektivität

Aggregatzustand

17:39

Resultante

Länge

Momentenproblem

Mengentheoretische Topologie

Gewichtete Summe

Regulärer Graph

Raum-Zeit

Resonanz

Computeranimation

Strategisches Spiel

Eins

Potenzielle Energie

Puls <Technik>

Regulärer Graph

Theorem

Gruppe <Mathematik>

Randomisierung

Punkt

Einflussgröße

Deterministischer Prozess

Gebundener Zustand

Parametersystem

Approximation

Güte der Anpassung

Reihe

Stellenring

Element <Gruppentheorie>

Vorzeichen <Mathematik>

Frequenz

Invariante

Reihe

Sortierte Logik

Strategisches Spiel

Derivation <Algebra>

Ordnung <Mathematik>

Beweistheorie

Kubischer Graph

Lineare Abbildung

Theorem

Renormierung

Bilinearform

Term

Differenzengleichung

Ungleichung

Zufallszahlen

Hamilton-Operator

Gleichungssystem

Randomisierung

Thermodynamisches System

Raum-Zeit

Gibbs-sampling

Menge

Basisvektor

Normalvektor

Term

22:49

Lineare Abbildung

Subtraktion

Stellenring

Multiplikation

Punkt

Gewichtete Summe

Term

Physikalische Theorie

Raum-Zeit

Computeranimation

Richtung

Zufallszahlen

Standardabweichung

Gruppe <Mathematik>

Schätzung

Randomisierung

Optimierung

Gleichungssystem

Deterministischer Prozess

Beobachtungsstudie

Lineares Funktional

Verschlingung

Zeitabhängigkeit

Raum-Zeit

Stochastische Abhängigkeit

Gasströmung

Ereignishorizont

Invariante

Quadratzahl

Betrag <Mathematik>

Sortierte Logik

Rechter Winkel

Physikalische Theorie

Normalvektor

Ordnung <Mathematik>

Term

26:11

Lineare Abbildung

Einfügungsdämpfung

Punkt

Gewichtete Summe

Familie <Mathematik>

Regulärer Graph

Computeranimation

Differenzengleichung

Strategisches Spiel

Zufallszahlen

Standardabweichung

Schätzung

Randomisierung

Einflussgröße

Stochastische Abhängigkeit

Gleichungssystem

Deterministischer Prozess

Analysis

Inklusion <Mathematik>

Thermodynamisches System

Raum-Zeit

Gasströmung

Variable

Invariante

Betrag <Mathematik>

Menge

Ganze Zahl

Derivation <Algebra>

Faktor <Algebra>

Varietät <Mathematik>

28:09

Resultante

Momentenproblem

Mengentheoretische Topologie

t-Test

Regulärer Graph

Mathematik

Gleichungssystem

Stokes-Integralsatz

Fortsetzung <Mathematik>

Computeranimation

Regulärer Graph

Gruppe <Mathematik>

Randomisierung

Stützpunkt <Mathematik>

Einflussgröße

Deterministischer Prozess

Folge <Mathematik>

Parametersystem

Kategorie <Mathematik>

Variable

Invariante

Arithmetisches Mittel

Restglied

Reihe

Sortierte Logik

Beweistheorie

Projektive Ebene

Beweistheorie

Aggregatzustand

Kubischer Graph

Theorem

Unrundheit

Derivation <Algebra>

Bilinearform

Term

Physikalische Theorie

Topologie

Hamilton-Operator

Gleichungssystem

Randomisierung

Glattheit <Mathematik>

Matching <Graphentheorie>

Raum-Zeit

Stochastische Abhängigkeit

Multiplikationssatz

Gibbs-sampling

Fokalpunkt

Integral

Deterministischer Prozess

Energiedichte

Anfangswertproblem

Mereologie

Parametersystem

Ruhmasse

Grad 5

Numerisches Modell

34:05

Resultante

Theorem

Subtraktion

Kontraktion <Mathematik>

Gewichtete Summe

t-Test

Regulärer Graph

Geräusch

Bilinearform

Element <Mathematik>

Analysis

Statistische Hypothese

Computeranimation

Eins

Differenzengleichung

Iteration

Zufallszahlen

Unterring

Existenzsatz

Schätzung

Uniforme Struktur

Jensen-Maß

Kontraktion <Mathematik>

Störungstheorie

Gleichungssystem

Einflussgröße

Gleichverteilung

Parametersystem

Approximation

Raum-Zeit

Eindeutigkeit

Länge

Gasströmung

Frequenz

Approximation

Multilineare Abbildung

Quadratzahl

Existenzsatz

Menge

Verbandstheorie

Rechter Winkel

Parametersystem

Anfangswertproblem

Mereologie

Evolute

Eindeutigkeit

Normalvektor

Beweistheorie

39:28

Resultante

Regulärer Graph

Wärmeübergang

Zeitbereich

Analysis

Computeranimation

Unendlichkeit

Stabilitätstheorie <Logik>

Kubische Gleichung

Zufallszahlen

Regulärer Graph

Arithmetische Folge

Konstante

Wärmeübergang

Schätzung

Bestimmtheitsmaß

Wellengleichung

Kompakter Raum

Vorlesung/Konferenz

Biprodukt

Asymptote

Einflussgröße

Normalvektor

Sterbeziffer

Folge <Mathematik>

Thermodynamisches Gleichgewicht

Prozess <Physik>

Vorzeichen <Mathematik>

Gasströmung

Schlussregel

Frequenz

Helmholtz-Zerlegung

Energiedichte

Fundamentalsatz der Algebra

Turbulenztheorie

Orbit <Mathematik>

Menge

Basisvektor

Ruhmasse

Energiedichte

Derivation <Algebra>

Fourier-Entwicklung

Normalvektor

Kubischer Graph

Aggregatzustand

42:02

Punkt

Momentenproblem

Dichte <Physik>

Extrempunkt

Impuls

Aggregatzustand

Resonanz

Computeranimation

Richtung

Gewöhnliche Differentialgleichung

Numerisches Modell

Standardabweichung

Ausdruck <Logik>

Existenzsatz

Statistische Analyse

Einflussgröße

Kette <Mathematik>

Kartesische Koordinaten

Sterbeziffer

Thermodynamisches Gleichgewicht

Nichtlinearer Operator

Krümmung

Kardinalzahl

Bellmansches Optimalitätsprinzip

Mechanismus-Design-Theorie

Invariante

Orbit <Mathematik>

Sortierte Logik

Plancksches Wirkungsquantum

Hyperbolischer Differentialoperator

Messprozess

Pendelschwingung

Eindeutigkeit

Aggregatzustand

Stochastik

Subtraktion

Fächer <Mathematik>

Gruppenoperation

Nichtlinearer Operator

Fluss <Mathematik>

Unendlichkeit

Physikalisches System

Harmonischer Oszillator

Arithmetische Folge

Hamilton-Operator

Grundraum

Leistung <Physik>

Beobachtungsstudie

Brownsche Bewegung

Halbgruppe

Thermodynamisches System

Schmelze

Eindeutigkeit

Statistische Analyse

Gibbs-sampling

Gasströmung

Physikalisches System

Frequenz

Kette <Mathematik>

Energiedichte

Schwingung

Existenzsatz

Mereologie

Energiedichte

Konvergenzgeschwindigkeit

Beobachtungsstudie

Numerisches Modell

Einstein-Podolsky-Rosen-Paradox

45:31

Stochastik

Dichte <Physik>

Mathematik

Wärmeübergang

Nichtlinearer Operator

Computeranimation

Gewöhnliche Differentialgleichung

Physikalisches System

Numerisches Modell

Ausdruck <Logik>

Einflussgröße

Leistung <Physik>

Thermodynamisches Gleichgewicht

Nichtlinearer Operator

Distributionstheorie

Brownsche Bewegung

Halbgruppe

Koordinaten

Gibbs-sampling

E-Funktion

Invariante

Objekt <Kategorie>

Energiedichte

Energiedichte

Plancksches Wirkungsquantum

Eindeutigkeit

Beweistheorie

Ergodentheorie

46:11

Stochastik

Gewichtete Summe

Punkt

Dichte <Physik>

Randwert

Natürliche Zahl

Mathematik

Nichtlinearer Operator

Extrempunkt

Computeranimation

Gewöhnliche Differentialgleichung

Physikalisches System

Numerisches Modell

Zufallszahlen

Ausdruck <Logik>

Existenzsatz

Kompakter Raum

Gleichungssystem

Einflussgröße

Gerade

Thermodynamisches Gleichgewicht

Distributionstheorie

Poisson-Prozess

Brownsche Bewegung

Halbgruppe

Raum-Zeit

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Konvexer Körper

Gasströmung

Primideal

Menge

Invariante

Quadratzahl

Phasenumwandlung

Plancksches Wirkungsquantum

Eindeutigkeit

Explosion <Stochastik>

Beobachtungsstudie

Ergodentheorie

47:35

Lineares Funktional

Poisson-Prozess

Konvexe Hülle

Randwert

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Konvexer Körper

Gasströmung

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Raum-Zeit

Computeranimation

Randwert

Dynamisches System

Physikalisches System

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Phasenumwandlung

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Indexberechnung

Beobachtungsstudie

Gleichungssystem

Einflussgröße

48:43

Mittelwert

Stellenring

Punkt

Randwert

Familie <Mathematik>

Natürliche Zahl

Ljapunov-Funktion

Gleichungssystem

Massestrom

Computeranimation

Eins

Richtung

Dynamisches System

Kompakter Raum

Phasenumwandlung

Deterministischer Prozess

Thermodynamisches Gleichgewicht

Parametersystem

Lineares Funktional

Distributionstheorie

Invariante

Rhombus <Mathematik>

Lemma <Logik>

Funktion <Mathematik>

Sortierte Logik

Konditionszahl

Phasenumwandlung

Messprozess

Eindeutigkeit

Beweistheorie

Subtraktion

Theorem

Überschall

Zahlenbereich

Physikalisches System

Gleichungssystem

Poisson-Prozess

Raum-Zeit

Eindeutigkeit

Gasströmung

Konvexer Körper

Gibbs-sampling

Physikalisches System

Menge

Energiedichte

Existenzsatz

Energiedichte

Beobachtungsstudie

Ergodentheorie

### Metadaten

#### Formale Metadaten

Titel | Long time dynamics of random data NLS and invariant measures |

Serientitel | Trimestre Ondes Non linéaires - June Conference |

Teil | 09 |

Anzahl der Teile | 23 |

Autor | Nahmod, Andrea |

Lizenz |
CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. |

DOI | 10.5446/20808 |

Herausgeber | Institut des Hautes Études Scientifiques (IHÉS) |

Erscheinungsjahr | 2016 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Mathematik |

Abstract | In this talk we show how certain well posedness results that are not available using only deterministic techniques (eg. Fourier and harmonic analysis) can be obtained when introducing randomization in the set of initial data and using powerful but still classical tools from probability. These ideas go back to seminal work by J. Bourgain on the almost sure global well posedness and invariance of Gibbs measures for NLS and other dispersive PDE. After explaining some of these ideas , we describe in more detail some further ideas in recent probabilistic well posedness results for NLS (joint with G. Staffilani) and new work on the existence and uniqueness 3 of non-equilibrium in variant measures associated to resonant NLS (joint with Z. Hani, J. Mattingly, L. Rey-Bellet and G. Staffilani) |