Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a rigorous treatment of this object. Through my previous question, I notice that Colombeau's theory might help. Thank you in advance for any points. :)
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2$\begingroup$ Maybe at the same time keep in mind the possibility to define multiplication provided the wave front sets are transversal (which is not the case for deltas being singular in every direction). In this way you do not go out of the standard theory. Colombeau's theory seems slightly too large to be really useful (my personal opinion) $\endgroup$– Piero D'AnconaCommented Dec 6, 2010 at 10:19

$\begingroup$ @Prof. D'Ancona, thank you very much for your comments. I am considering the case $\delta_0^2(x)$ on the real line $R$. In this case, do you think that the wave front sets are transversal? Sorry for my unfamiliarity of $\endgroup$– AnandCommented Dec 6, 2010 at 10:31

1$\begingroup$ Unfortunately not, on $R$ my remark is useless. $\endgroup$– Piero D'AnconaCommented Dec 6, 2010 at 10:45

$\begingroup$ @Prof. D'Ancona, thank you very much. That's why I ask this question. :) $\endgroup$– AnandCommented Dec 6, 2010 at 10:49

1$\begingroup$ How is this related to algebraic geometry? $\endgroup$– AngeloCommented Dec 6, 2010 at 14:53

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Sorry for repeating myself, but as you can see on the nLab here, Colombeau himself has written an elementary introduction to his theory, mainly for people who are interested in applications:
Jean François Colombeau: "Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics."
See this review in the Zentralblatt Mathematik. This will also give you a hint if this approach is suitable to your problem (BTW: what is the application you are thinking about?).

$\begingroup$ @Tim van Beek,thank you very much for your comments. I browsed nLab, it is very informative and helpful. But I still could get an easier introduction for this theory. I will have a look of JF Colombeau's book that you recommended. As for my application, consider the heat equation with random noise. I am calculating the second moments of the solution. If the initial data is a Delta function, then the initial data for the second moment will be the square of delta. That's where my problem comes from. :) $\endgroup$– AnandCommented Dec 6, 2010 at 11:26

3$\begingroup$ Alright, but in this case I'd recommend to ask an expert of stochastic differential equations if this approach is valid, or if a different approach using e.g. Wick products would be more appropriate (see for example Oksendahl et. alt., "Stochastic Parital Differential Equations"). What you are dealing with here is a well known problem, both from quantum field theory and from stochastic PDE, with different solutions  and the approach of Colombeau is IMHO not the best one :) (But, again, this depends on the problem you try to solve.) $\endgroup$ Commented Dec 6, 2010 at 12:16

2$\begingroup$ @Anand: for your application, you are much better off just taking the second moment to be infinite on the initial slice. Heat equation is infinitely smoothing, so at any later time the solution is Schwartz, random noise will not change the integrability by that much. My hunch is that for any reasonable definition, the limit $t\to 0^+$ of your second moment will necessarily blowup. That said, you should also look at some of the literature on the Bohmian methods in Quantum Mechanics. I vaguely remember there being some similar issues, unfortunately I don't recall the reference. $\endgroup$ Commented Dec 6, 2010 at 12:17

$\begingroup$ @Willie Wong, thank you very much for your comment. What I derived (second moment) is something like $\frac{e^{x^2/t}}{2\pi t}$, while another known result is $\frac{e^{x^2/t}}{2\pi t}+c \frac{e^{x^2/(2t)}}{\sqrt{2\pi t}}$ with some constant $c$. It is true that in either case, the results blow up at $t=0$. But which one is right? Does the system has multiple second moments? Our result looks more like $\delta_0^2(x)$ for $t=0$. I will have a look of Bohmian's method. Thank you very much! $\endgroup$– AnandCommented Dec 6, 2010 at 12:37

1$\begingroup$ Anand, could you say what you mean by "second moment"? I am under the impression that the second moment of the delta function is zero. $\endgroup$ Commented Dec 6, 2010 at 14:43