Characterising the solutions to an integral equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:26:45Z http://mathoverflow.net/feeds/question/94006 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94006/characterising-the-solutions-to-an-integral-equation Characterising the solutions to an integral equation genneth 2012-04-14T01:55:53Z 2012-04-14T01:55:53Z <p>Consider the integral equation: $$\phi(u) = \int_0^\infty f[\phi(u e^{-\alpha y})]dG(y)$$ where $G(y)$ is a cumulative distribution function on the non-negative reals, $f$ is a generating function $f(x) = \sum_{k=0}^\infty p_k x^k$ with $1 &lt; f^\prime(1) &lt; \infty$ and $\phi$ is the characteristic function of a distribution with unit mean on the non-negative reals.</p> <p>Playing around with it it quickly becomes clear that the form for $\phi$ is quite restricted. For instance, one can show that all moments exist. From numerical evidence, I expect it to have an exponential tail.</p> <p><strong>Question: does anyone know a way, or even just the result from the top of their head, to characterise (necessary and sufficient conditions of) possible solutions $\phi$?</strong></p> <p>Background: the equation appear as the solution to an asymptotic limit theorem of super-critical branching processes. I'm actually more interested in finding $G$ given (a possibly experimentally, and therefore noisy) $\phi$; so in some sense I would like to know what is the "closest" allowed distribution to the experimental data.</p> <p>Also, I'm a physicist by training, so my real mathematics knowledge is quite weak; i.e. I know elementary things, but advanced functional analysis tends to slow me down quite a bit.</p>