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We study convolution groups generated by completely monotone sequences and completely monotone functions. Using a convolution group, we define a fractional calculus for a certain class of distributions. When acting on causal functions, this definition agrees with the traditional Riemann-Liouville definition for t>0 but includes some singularities at t=0 so that the group property holds. Using this group, we are able to extend the definition of Caputo derivatives of order in (0,1) to a certain class of locally integrable functions without using the first derivative. The group property allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gronwall inequality (or comparison principle) with the most general conditions. This then opens the door of a priori energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions. Besides, we also obtain some interesting results about completely monotone sequences.

We compute the almost-sure Hausdorff dimension of the double points of chordal SLE$_\kappa$ for $\kappa > 4$, confirming a prediction of Duplantier--Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE$_\kappa$ for $\kappa > 4$ as well as analogous dimensions for the radial and whole-plane SLE$_\kappa(\rho)$ processes for $\kappa > 0$. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field $e^{ih/\chi}$, where $h$ is a Gaussian free field and $\chi > 0$, of different angles with each other and with the domain boundary.

No abstract uploaded!