Functionals of the path of a Brownian particle (an integral of a given function along the path) have many applications in various fields and have attracted considerable attention. Most remarkably, it was found by Kac in 1949 that the probability density function of one dimensional Brownian functionals satisfies a Schrodinger-like equation [1]. Over the past forty years or so, even though experimental and theoretical interest in anomalous, non-Brownian diffusion has exponentially increased [2], a general treatment of non-Brownian functionals was still missing. We derive a fractional Schrodinger-like equation for the distribution of functionals of non-Brownian paths, thus providing the key equation for any future investigation of non-Brownian functionals. In our equation, the ordinary integer derivative is replaced with a fractional derivative, resulting in a behavior vastly different than the normal diffusion case. We demonstrate the importance of our equation, methods of solution and applications. In particular, we show how our equation can be exploited to give insight into the problem of weak ergodicity breaking [3].

[1] S. N. Majumdar, Current Science 89, 2076 (2005).
[2] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000).
[3] L. Turgeman, S. Carmi, and E. Barkai. Phys. Rev. Lett. (in press)