Title: A sublinear bound for the variance of solutions for random Hamilton-Jacobi equations
Speaker: Ivan Matic, Baruch College - CUNY
Seminar Type: Combinatorics
Abstract: The central limit theorem states that if a fair coin is tossed n times, the number of heads will be around \(n/2\) with an error of order \(n^{1/2}\). A more complicated problem is that of finding the cheapest flight between two cities that are so far away that multiple connections are necessary. If all airports choose their landing fees at random, and if the two cities are \(n\) units apart, then the expected minimum price is of order \(n\), but the error is conjectured to be of order \(n^{1/3}\). We provide an argument that bounds this error by \((n/log n)^{1/2}\). This problem is related to studying solutions of random Hamilton-Jacobi equations with Hamiltonian that is stationary and ergodic with respect to translations. In dimensions two and higher we prove that as the time \(t\) increases, the variance of the solution increases at a rate slower than \(t/log t\).
