# An elementary introduction of Colombeau's generalized function theory

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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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) – Piero D'Ancona Dec 6 '10 at 10:19
@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 – Anand Dec 6 '10 at 10:31
Unfortunately not, on $R$ my remark is useless. – Piero D'Ancona Dec 6 '10 at 10:45
@Prof. D'Ancona, thank you very much. That's why I ask this question. :-) – Anand Dec 6 '10 at 10:49
How is this related to algebraic geometry? – Angelo Dec 6 '10 at 14:53

@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 blow-up. 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. – Willie Wong Dec 6 '10 at 12:17
@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! – Anand Dec 6 '10 at 12:37