Signatures of Gaussian processes and SLE curves - Boedihardjo, H (2014)
This thesis contains three main results. The first result states that, outside a slim set associated with a Gaussian process with long time memory, paths can be canonically enhanced to geometric rough paths. This allows us to apply the powerful Universal Limit Theorem in rough path theory to study the quasi-sure properties of the solutions of stochastic differential equations driven by Gaussian processes. The key idea is to use a norm, invented by B. Hambly and T.Lyons, which dominates the p-variation distance and the fact that the roughness of a Gaussian sample path is evenly distributed over time. The second result is the almost-sure uniqueness of the signatures of SLE kappa curves for kappa less than or equal to 4. We prove this by first expressing the Fourier transform of the winding angle of the SLE curve in terms of its signature. This formula also gives us a relation between the expected signature and the n-point functions studied in the SLE and Statistical Physics literature. It is important that the Chordal SLE measure in D is supported on simple curves from -1 to 1 for kappa between 0 and 4, and hence the image of the curve determines the curve up to reparametrisation. The third result is a formula for the expected signature of Gaussian processes generated by strictly regular kernels. The idea is to approximate the expected signature of this class of processes by the expected signature of their piecewise linear approximations. This reduces the problem to computing the moments of Gaussian random variables, which can be done using Wick’s formula.