The n-Category Café

Functional Equations VI: Using Probability Theory to Solve Functional Equations

Posted by Tom Leinster

A functional equation is an entirely deterministic thing, such as $$f(x + y) = f(x) + f(y)$$ or $$f(f(f(x))) = x$$ or $$f\Bigl(\cos\bigl(e^{f(x)}\bigr)\Bigr) + 2x = \sin\bigl(f(x+1)\bigr).$$ So it’s a genuine revelation that one can solve some functional equations using probability theory — more specifically, the theory of large deviations.

This week and next week, I’m explaining how. Today (pages 22-25 of these notes) was mainly background:

• an introduction to the theory of large deviations;

• an introduction to convex duality, which Simon has written about here before;

• how the two can be combined to get a nontrivial formula for sums of powers of real numbers.

Next time, I’ll explain how this technique produces a startlingly simple characterization of the $p$-norms.