Probability theory

Many natural and important constructions in fractal geometry are random. Key examples include Mandelbrot or fractal percolation, first introduced as a model for fluid turbulence, Brownian motion (graphs, images etc), and various random variants of iterated function systems. I am interested in all of these models and more and have various results in these directions. With Tuomas Sahlsten I proved that the (almost sure) Fourier dimension of the graph of classical Brownian motion is 1:

► On the Fourier analytic structure of the Brownian graph (with T. Sahlsten), arXiv
        Analysis & PDE, 11, (2018), 115-132.

and considered some extensions of this with Ana de Orellana here:

► Fourier restriction for the additive Brownian sheet (with A. E. de Orellana), arXiv
        submitted.

I have studied Mandelbrot and fractal percolation, including proving that the Assouad dimension is typically as large as possible in certain settings, and also studying the Assouad spectrum:

► The Assouad spectrum of random self-affine carpets (with S. Troscheit), arXiv
        Ergodic Theory and Dynamical Systems, 41, (2021), 2927-2945.

► The Assouad dimension of randomly generated fractals (with J. J. Miao & S. Troscheit), arXiv
        Ergodic Theory and Dynamical Systems, 38, (2018), 982-1011.

I have also studied random IFS constructions, including a topological interpetation here:

► Dimension and measure for typical random fractals, arXiv
        Ergodic Theory and Dynamical Systems, 35, (2015), 854-882.

Further work in probabilistic directions includes:

► Multifractal spectra of random self-affine multifractal Sierpiński sponges in ℝ^d (with L. Olsen)
        Indiana University Mathematics Journal, 60, (2011), 937-984.

I gave a talk about my work with Tuomas Sahlsten at Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism at ICERM (Brown University) in March 2016. Here is the video and here are the slides.

I gave a talk about my work with Orponen and Sahlsten on Kahane's problem at an ETDS meeting in Jagiellonian University in 2014. Here are the slides.



Left: two realisations of Mandelbrot percolation. Right: one realisation of the graph of Brownian motion.