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Langevin equation provides an example of a physical model which involves a differential equation with a stochastic term. Now, I wonder, how should one treat this?

When I studied stochastic processes, I learned about Itô and Stratonovich calculus*. Back then, I just saw these things as technical tools devised to define the stochastic integral in a meningful way. Yes, depending on which one you use, you would obtain some results or some others, but this didn't bother me, since I was looking at these objects as mere mathematical structures.

But, when faced to the Langevin equation, I wonder, how does one make a meaningful interpretation of it? Both Itô and Stratonovich calculus seem like reasonable alternatives, but I don't find any strong argument to prefer one over the other. The point is, there must be a way to decide which type of stochastic calculus we should use! Two different definitions of the stochastic integral will lead to contradictory physics law, which is troublesome.

I believe that the Fokker-Planck equation can be derived from the Langevin equation via Itô's formula, but I don't think there's any way to achieve this using Stratonovich's calculus... But of course, this would be a poor ad hoc argument in favor of Itô's calculus, so this is clearly unsatisfactory.

suming up: how would you proceed to find out which version of the stochastic integral is more convenient to deal with stochastic equations in physics? Is there any physical assumption which makes Itô's calculus more reasonable?

  • A note: Of course, there is a whole parametric family of possible "interpolating" calculus between both Itô and Stratonovich, but I believe these two are the only ones that are relevant: one of them gives rise to the martingale property, and the other one preserves the classical integration-by-parts rule
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  • $\begingroup$ You should see every model for what it is: a limited tool that is sometimes correct and sometimes not. When it is wrong and when it is right is something that nature decides for us. Every model invented to date is almost always wrong (and in most cases it can't even be calculated), but that does not matter. What matters are the cases in which they are correct. $\endgroup$ – CuriousOne Mar 26 '16 at 21:51
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    $\begingroup$ Still, it is natural to try and look for a deeper theoretical argument serving as motivation for a particular stochastic integral. I understand in what sense "every model is wrong", but given the model, one wishes to have a particular formal framework for that model. I am kind of asking what would be that particular formal framework in this case. $\endgroup$ – Qwertuy Mar 27 '16 at 0:40
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    $\begingroup$ Well, it is certanly not just mathematics, since I am asking specifically for a justification -on physical grounds- for one type or another of stochastic integration. Your answer is that there isn't any beyond "accordance with experiment", and maybe that's rigth, and maybe there is no profound justification for employing one stochastic integral or the other, but nevertheless I think the theoretical exercise of searching for a motivation which makes preferable (in some intuitive non-formal sense) either Itô or Stratonovich integral is very interesting. $\endgroup$ – Qwertuy Mar 27 '16 at 0:47
  • $\begingroup$ The only "physical justification" is that it agrees with an experiment. That's what we do in empirical science. We don't do axioms, we don't do proofs, we use math as a sledgehammer to drive a stake trough data from observations. As my first theoretical physics professor said (I am translating somewhat loosely to not appear like I don't like the philosophy department) in the very first lecture "If you don't like that, there is the door... you will not like what is about to follow.". $\endgroup$ – CuriousOne Mar 27 '16 at 0:53
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    $\begingroup$ I completely disagree. The ultimate justification is that it agrees with experiment, of course. But there are tons of ideas, heuristics and other models which shed ligth into which models are more reasonable or more likely. Langevin equation should be tested with experiments, of course, but it follows mathemayically from the Newton equations and some other considerations, so there you have an example of how to find motivation for proposing a model. I am not asking for proofs. I have not used that word. Just for physical insigth into which tool is more convenient for a concrete situation. $\endgroup$ – Qwertuy Mar 27 '16 at 1:30
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Both are relevant, and "the misconception that Langevin equation is the universal stochastic differential equation for all kinds of noisy systems is responsible for the difficulties mentioned"* in your post.

Take the SDE from Thomas' answer, $$\frac{dy}{dt} = A(y) + C(y)L(t)$$ where $L(t)$ is the noise term. Suppose we can turn the noise off, so we'd only be looking at the isolated, deterministic, system, and suppose that we got: $$\frac{dy}{dt} = A(y)$$ It is thus obvious that the noise term in this case must be interpreted as per Stratonovich, because an Itô interpretation would change the dynamics of the isolated system.

Above we had assumed, as one does, that $\langle L(t)L(t') \rangle = D \delta(t-t')$, but rarely are actual physical processes made out of Dirac deltas. Now if they are not, then by Wong-Zakai, Stratonovich is the only possible interpretation here (i.e. if $L(t)$ is more general, and as it approaches a delta, Stratonovich is the integration form that arises).

Now by turning the noise on and off we have arrived at the conclusion that Stratonovich is the only way forward, but in the very beginning I said that both formulations are relevant. Indeed, what if the noise cannot be separated out?

The important distinction is to make between external and internal sources of noise. We dealt with the external, but what if the noise is internal, and is impossible to turn off like say in chemical reactions? It's not that we have a deterministic process with a noise term slapped on top, we have a stochastic process and one might be able to argue that the averages behave in a deterministic manner, but by no means can then one then simply just add a noise term back on top without more justification: "For internal noise one cannot just postulate a nonlinear Langevin equation or a Fokker-Planck equation and then hope to determine its coefficients from macroscopic data. The more fundamental approach of [the expansion of the master equation] is indispensable."

* Any quotes in my answer have been taken from the definitive book on the matter: Stochastic Processes in Physics and Chemistry by van Kampen, which I strongly suggest you consult for more detailed explanations of the issues and how one goes about dealing with them.

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  • $\begingroup$ Ok, thanks for the answer and the reference. The title is very suggestive indeed. $\endgroup$ – Qwertuy Mar 29 '16 at 0:35
  • $\begingroup$ I am not sure that I completely understand. In the limit that I turn of the multiplicative noise, $b\to 0$ in my notation, Ito and Stratonovich processes are the same. For additive noise, they are the same anyway. $\endgroup$ – Thomas Mar 29 '16 at 13:49
  • $\begingroup$ By the way: In physical systems the noise and the drag are not independent. This is the content of fluctuation-dissipation (FD) relations. Physicists like Stra bettter, because FD relations take on a more transparent form, but its not obvious to me what is wring with Ito. $\endgroup$ – Thomas Mar 29 '16 at 13:52
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Both Ito and Stratonovich stochastic PDEs can be used to derive a Fokker-Planck equation. Indeed, for simple one-dimensional processes the Ito process $$ dx = a\, dt + b\, dW(t) $$ is equivalent to the Stratonovich process $$ dx =\left( a\,-\frac{1}{2}b\partial_x b\right) dt + b\, dW(t) $$ The answer is then that both are physically reasonable, for a given FP equation I can find both and Ito and a Stratonovich process that will realize it. These are physically equivalent.

Having said this, Physicists tend to think of Stratonovich as the more reasonable scheme, because the FP equation is the one that can be derived by taking moments and "naive" integration by parts.

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protected by Qmechanic Jul 17 '16 at 14:25

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