Hi Eric!
I thought I might just as well post a reply here.
You wrote
[quote]
We know that
dW dW = dt
[/quote]
Ok.
[quote]
so that means that W' ~ 1/sqrt(dt) since
dW dW = (W')^2 dt dt = dt.
[/quote]
This is making me a little nervous. But I realize my stochastic calculus is rusty.
[quote]
If we wave the wand of NCG
[/quote]
Now I feel more at home...
[quote]
[dW, W]
= dW W - W dW
= W' dt W - W W' dt
= W' W dt - W W' dt (W and dt commute)
= dt.
[/quote]
I understand
[dW,W] = dt .
But does it make sense to write dW = W' dt ??
Is there any discrete 0-form W' with this property?
I wouldn't think so, but it has been a while since I thought about this stuff.
But consider a 2D diamond graph. Introduce "lightcone" variables A and B such that
t = A + B
x = A - B .
Then
dt = dA + dB
dx = dA - dB
and there is manifestly no 0-form f such that
dx = f dt .
But in your example above W=x, if I understood correctly.
Do you agree? Maybe I am missing your point.
On the other hand, maybe you are right and it is useful to *introduce* new formal expressions like W' that are defined to satisfy relations like dW = W' dt. Haven't thought enough about that.
White Noise and Wiener Processes
-
IAmEric
- Posts: 0
- Joined: Thu Jan 01, 2004 12:00 am
White Noise and Wiener Processes
Hi Urs!
Welcome to NP! Beer Party
I'll take the liberty for a little introductory comment...
I had the great pleasure to coauthor a paper with Urs (the highlight of my academic life actually). After six years of flailing around in a corner on my own, I finally somehow finished my PhD and decided that reaching out to others wouldn't be such a bad thing. I knew Urs through the newsgroup sci.physics.research (of which he eventually became a moderator along with John Baez and others last time I checked). Urs had some interest in stochastic formulations of QM (by Nelson if memory serves) and some comments I made relating NCG and SC must have attracted some of his attention.
The basic idea is that any directed graph gives rise to a special algebra/calculus that is unique to that graph. I had also (re)discovered that a certain algebra leads to stochastic calculus. Since every directed graph gives rise to an algebra and a particular algebra gives rise to stochastic calculus, the question we asked ourselves one day is kind of an inverse problem, "Which directed graph gives rise to the algebra from which SC derives." Urs subsequently left for a two week bike ride across Europe and when he came back we had both answered the question independently. In a flash it all became almost obvious. (particularly in hind sight). The binary tree is the unique directed graph that gives rise to the algebra that leads to SC. That was enough to ignite an intense collaboration that was very exciting (for me at least) that lasted several months and culminated in this paper
Discrete Differential Geometry on Causal Graphs
Anyway, it is always a great pleasure to discuss things with Urs so I'm glad he made an appearance. So now back to the topic at hand! (which I think has some profound (although perhaps too academic) significance). That is the possible fact that Brownian motion and white noise are conjugate variables in the sense that position and momentum are in QM.
[quote]
[quote]
so that means that W' ~ 1/sqrt(dt) since
dW dW = (W')^2 dt dt = dt.
[/quote]
This is making me a little nervous. But I realize my stochastic calculus is rusty.
[/quote]
I don't blame you, it makes me nervous too Smiley Without making too much out of it, we should probably think of that as no more than dimensional analysis, i.e. getting the units to work out right. Although if anything I am saying has any validity, it must be more than that. That, or what I think is more likely, writing things like dW dW = dt is not really valid in the first place and the NCG approach of commutators is the proper way to proceed.
[quote]
I understand
[dW,W] = dt .
But does it make sense to write dW = W' dt ??
[/quote]
I don't know Smiley Not sure if it helps, but it has been pointed out that W' should be thought of in the distributional sense. The reason for me feeling reasonably comfortable writing that down in the first place is the (almost) definition of white noise
W(t) = int_0^t W'(tau) dtau.
Since W(0) = 0, we can rewriting this as
W(t) - W(0) = int_0^t W'(tau) dtau
which can be rewritten
int_0^t dW = int_0^t W'(tau) dtau,
which I thought would allow us to write
dW = W' dt.
[quote]
Is there any discrete 0-form W' with this property?
[/quote]
The commutative relation
W W' - W' W = 1
would seem to indicate that perhaps neither W nor W' are discrete 0' forms (in the obvious sense) and maybe should be thought of as matrices or something. Not sure.
It might help to make the analogy with QM more clear. Since it makes me so happy, I will repeat the little computation
(using x in place of W now)
[dx,x]
= dx x - x dx
= x' dt x - x x' dt
= x' x dt - x x' dt
= dt
which implies
[x',x] = 1.
The QM version of this is
[dx,x]
= dx x - x dx
= x' dt x - x x' dt
= x' x dt - x x' dt
= i*hbar*dt
which implies
[x',x] = i*hbar.
(maybe should replace hbar with hbar/m)
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
It is then somewhat tempting to write something like
dW = W' dt = d/dW dt
so that
[W', W] = [d/dW, W]
and
[d/dW, W] psi
= (d/dW W psi - W d/dW psi)
= (psi + W d/dW psi - W d/dW psi)
= 1 psi
so that
[d/dW, W] = 1.
Is it possible that if W is a Wiener process then white noise can be thought of as W' = d/dW?
That would be kind of wild and, if true, I'm pretty sure no one has ever observed that. We could maybe write up a page or two and throw it on the arxives Smiley
What do you think?
Cheers Beer
Eric
Welcome to NP! Beer Party
I'll take the liberty for a little introductory comment...
I had the great pleasure to coauthor a paper with Urs (the highlight of my academic life actually). After six years of flailing around in a corner on my own, I finally somehow finished my PhD and decided that reaching out to others wouldn't be such a bad thing. I knew Urs through the newsgroup sci.physics.research (of which he eventually became a moderator along with John Baez and others last time I checked). Urs had some interest in stochastic formulations of QM (by Nelson if memory serves) and some comments I made relating NCG and SC must have attracted some of his attention.
The basic idea is that any directed graph gives rise to a special algebra/calculus that is unique to that graph. I had also (re)discovered that a certain algebra leads to stochastic calculus. Since every directed graph gives rise to an algebra and a particular algebra gives rise to stochastic calculus, the question we asked ourselves one day is kind of an inverse problem, "Which directed graph gives rise to the algebra from which SC derives." Urs subsequently left for a two week bike ride across Europe and when he came back we had both answered the question independently. In a flash it all became almost obvious. (particularly in hind sight). The binary tree is the unique directed graph that gives rise to the algebra that leads to SC. That was enough to ignite an intense collaboration that was very exciting (for me at least) that lasted several months and culminated in this paper
Discrete Differential Geometry on Causal Graphs
Anyway, it is always a great pleasure to discuss things with Urs so I'm glad he made an appearance. So now back to the topic at hand! (which I think has some profound (although perhaps too academic) significance). That is the possible fact that Brownian motion and white noise are conjugate variables in the sense that position and momentum are in QM.
[quote]
[quote]
so that means that W' ~ 1/sqrt(dt) since
dW dW = (W')^2 dt dt = dt.
[/quote]
This is making me a little nervous. But I realize my stochastic calculus is rusty.
[/quote]
I don't blame you, it makes me nervous too Smiley Without making too much out of it, we should probably think of that as no more than dimensional analysis, i.e. getting the units to work out right. Although if anything I am saying has any validity, it must be more than that. That, or what I think is more likely, writing things like dW dW = dt is not really valid in the first place and the NCG approach of commutators is the proper way to proceed.
[quote]
I understand
[dW,W] = dt .
But does it make sense to write dW = W' dt ??
[/quote]
I don't know Smiley Not sure if it helps, but it has been pointed out that W' should be thought of in the distributional sense. The reason for me feeling reasonably comfortable writing that down in the first place is the (almost) definition of white noise
W(t) = int_0^t W'(tau) dtau.
Since W(0) = 0, we can rewriting this as
W(t) - W(0) = int_0^t W'(tau) dtau
which can be rewritten
int_0^t dW = int_0^t W'(tau) dtau,
which I thought would allow us to write
dW = W' dt.
[quote]
Is there any discrete 0-form W' with this property?
[/quote]
The commutative relation
W W' - W' W = 1
would seem to indicate that perhaps neither W nor W' are discrete 0' forms (in the obvious sense) and maybe should be thought of as matrices or something. Not sure.
It might help to make the analogy with QM more clear. Since it makes me so happy, I will repeat the little computation
(using x in place of W now)
[dx,x]
= dx x - x dx
= x' dt x - x x' dt
= x' x dt - x x' dt
= dt
which implies
[x',x] = 1.
The QM version of this is
[dx,x]
= dx x - x dx
= x' dt x - x x' dt
= x' x dt - x x' dt
= i*hbar*dt
which implies
[x',x] = i*hbar.
(maybe should replace hbar with hbar/m)
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
It is then somewhat tempting to write something like
dW = W' dt = d/dW dt
so that
[W', W] = [d/dW, W]
and
[d/dW, W] psi
= (d/dW W psi - W d/dW psi)
= (psi + W d/dW psi - W d/dW psi)
= 1 psi
so that
[d/dW, W] = 1.
Is it possible that if W is a Wiener process then white noise can be thought of as W' = d/dW?
That would be kind of wild and, if true, I'm pretty sure no one has ever observed that. We could maybe write up a page or two and throw it on the arxives Smiley
What do you think?
Cheers Beer
Eric
One day, in the midst of another one of his increasingly frequent homicidal fantasies, Croke noticed a new member had invaded his favorite forum. It was an obnoxious coed (or so he thought) who went by the nickname "Lilly". At first, all Croke could think about was strangling the life out of this giddy new member. Her insistent flirting with everyone was disgusting to Croke and he began a merciless vendetta against her.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.
-
urs
- Posts: 1
- Joined: Thu Jan 01, 2004 12:00 am
White Noise and Wiener Processes
I perfectly agree that the equation
[dx,x] = dt
or
[dx,x] = i hbar dt
morally encodes the commutator which you have in mind. What you point out is that if we could somehow divide both sides by dt, we would get something like a commutator of dx/dt with x.
This is certainly a good way to think about it. Personally, however, instead of wanting to rewrite this in the form [x',x] = 1, I would regard [dx,x]=dt as the better and more robust formula.
I also agree that there is a close relation to how white noise is morally like the derivative of the Wiener process. After all, one way to derive the commutation relation [x',x] = k 1 is to use the path integral formulation of the free particle. The rigorous version of this is nothing but an integral over paths using the Wiener measure.
Hence, many technically quite sophisticated facts are somehow quite neatly and very simply already encoded in an equation as simple as [dx,x]=dt .
[quote]
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
[/quote]
Wait. Is that what you want to say?
I think the Wiener process is like the position observable in QM. White noise is like the velocity (~momentum) of the particle.
[quote]
dW = W' dt = d/dW dt
[/quote]
The second equation looks a little strange. Wouldn't you want
W' dt = (dW/dt) dt ?
[dx,x] = dt
or
[dx,x] = i hbar dt
morally encodes the commutator which you have in mind. What you point out is that if we could somehow divide both sides by dt, we would get something like a commutator of dx/dt with x.
This is certainly a good way to think about it. Personally, however, instead of wanting to rewrite this in the form [x',x] = 1, I would regard [dx,x]=dt as the better and more robust formula.
I also agree that there is a close relation to how white noise is morally like the derivative of the Wiener process. After all, one way to derive the commutation relation [x',x] = k 1 is to use the path integral formulation of the free particle. The rigorous version of this is nothing but an integral over paths using the Wiener measure.
Hence, many technically quite sophisticated facts are somehow quite neatly and very simply already encoded in an equation as simple as [dx,x]=dt .
[quote]
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
[/quote]
Wait. Is that what you want to say?
I think the Wiener process is like the position observable in QM. White noise is like the velocity (~momentum) of the particle.
[quote]
dW = W' dt = d/dW dt
[/quote]
The second equation looks a little strange. Wouldn't you want
W' dt = (dW/dt) dt ?
-
IAmEric
- Posts: 0
- Joined: Thu Jan 01, 2004 12:00 am
White Noise and Wiener Processes
[quote]
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
Wait. Is that what you want to say?
[/quote]
I guess before we can settle this question, we'd have to settle the question of whether we can write
dW = W' dt.
I think we can, but there are enough moving pieces that I could be wrong about that. If we can write this, then
[dW,W] = dt
leads to
[W',W] = 1.
Looking at the first expression (dW = W' dt) makes you want to say
W' = dW/dt,
but looking at [W',W] = 1 makes me want to say
W' = d/dW
unless there is another representation of W' that satisfies that commutative relation. By a quirk in dimensions, at least the two have the same units, i.e.
[d/dW] = [dW/dt]
since
[W] = [sqrt(t)],
where I'm using brackets to denotes "units of", i.e. [W] = "Units of W".
I think there is something neat here, but as usually, I'm only smart enough to "sense" it and am struggling to dig it out Smiley
Oh! Let me clarify one point because it is important. The expression
[W',W] = 1
does not come from "dividing by dt". Rather, it comes from plugging
dW = W' dt
into
[dW,W] = dt
and using the fact that dW and dt commute. I carried out that computation (since it is so neat) a couple of times Smiley
Looking at this, we see that a Wiener process (Brownian motion) is analogous to momentum in QM.
Wait. Is that what you want to say?
[/quote]
I guess before we can settle this question, we'd have to settle the question of whether we can write
dW = W' dt.
I think we can, but there are enough moving pieces that I could be wrong about that. If we can write this, then
[dW,W] = dt
leads to
[W',W] = 1.
Looking at the first expression (dW = W' dt) makes you want to say
W' = dW/dt,
but looking at [W',W] = 1 makes me want to say
W' = d/dW
unless there is another representation of W' that satisfies that commutative relation. By a quirk in dimensions, at least the two have the same units, i.e.
[d/dW] = [dW/dt]
since
[W] = [sqrt(t)],
where I'm using brackets to denotes "units of", i.e. [W] = "Units of W".
I think there is something neat here, but as usually, I'm only smart enough to "sense" it and am struggling to dig it out Smiley
Oh! Let me clarify one point because it is important. The expression
[W',W] = 1
does not come from "dividing by dt". Rather, it comes from plugging
dW = W' dt
into
[dW,W] = dt
and using the fact that dW and dt commute. I carried out that computation (since it is so neat) a couple of times Smiley
One day, in the midst of another one of his increasingly frequent homicidal fantasies, Croke noticed a new member had invaded his favorite forum. It was an obnoxious coed (or so he thought) who went by the nickname "Lilly". At first, all Croke could think about was strangling the life out of this giddy new member. Her insistent flirting with everyone was disgusting to Croke and he began a merciless vendetta against her.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.
-
IAmEric
- Posts: 0
- Joined: Thu Jan 01, 2004 12:00 am
White Noise and Wiener Processes
One more thing...
You said
[quote]
I think the Wiener process is like the position observable in QM. White noise is like the velocity (~momentum) of the particle.
[/quote]
Yes. This is basically what I'm trying to say. I was being a little cavalier about constants and not making much effort to distinguish between velocity and momentum (although I made a comment about perhaps writing hbar/m at one point).
I would say it this way...
If we can write
dW = W' dt,
then the commutative relation
[dW,W] = dt
implies the Wiener process is like the position [b]operator[/b] in QM and white noise is like the velocity (~momentum) [b]operator[/b] in QM.
You said
[quote]
I think the Wiener process is like the position observable in QM. White noise is like the velocity (~momentum) of the particle.
[/quote]
Yes. This is basically what I'm trying to say. I was being a little cavalier about constants and not making much effort to distinguish between velocity and momentum (although I made a comment about perhaps writing hbar/m at one point).
I would say it this way...
If we can write
dW = W' dt,
then the commutative relation
[dW,W] = dt
implies the Wiener process is like the position [b]operator[/b] in QM and white noise is like the velocity (~momentum) [b]operator[/b] in QM.
One day, in the midst of another one of his increasingly frequent homicidal fantasies, Croke noticed a new member had invaded his favorite forum. It was an obnoxious coed (or so he thought) who went by the nickname "Lilly". At first, all Croke could think about was strangling the life out of this giddy new member. Her insistent flirting with everyone was disgusting to Croke and he began a merciless vendetta against her.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.
He was sure that his prominent status would cause the other "regulars" to outcast the newcomer as he wished. On the contrary, everyone dug Lilly and even Croke's most vehement beratings fell on def ears. This infuriated Croke even more.