In the first part of the talk, I introduce sharp gradient bounds for the perturbed diffusion semigroup. In contrast with existing results, the perturbation studied here is random and the bounds obtained are pathwise. The approach builds on the classical work of Kusuoka and Stroock. It extends their program developed for the heat semi-group to solutions of stochastic partial differential equatio...

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as “lamplighter groups”. We prove that zero speed for random walk on a lamplig...

Advances in the technology of measurement and informatics have potential for supporting the clinical management of individual cases of psychotherapy. The Polaris-MH system described by Grissom and Lyons (2006) is one example of these advances. The potential utility of this and similar systems is discussed in light of research on feedback to therapists and of prediction of treatment outcomes.

We show how to generalize Lyons’ rough paths theory in order to give a pathwise meaning to some non-linear infinite-dimensional evolution equations associated to an analytic semigroup and driven by an irregular noise. As an illustration, we apply the theory to a class of 1d SPDEs driven by a space-time fractional Brownian motion.

Following Lyons (1990, [4]) we define a periodic tree, restate its branching number and consider a biased random walk on it. In the case of a transient walk, we describe the walk-invariant random periodic tree and calculate the asymptotic rate of escape (speed) of the walk. This is achieved by exploiting the connections between random walks and electric networks.

We study Girsanov’s theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of “gradient type”. We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the cas...

VanderWeele et al.'s paper is a useful contribution to the on-going scientific conversation about the detection of contagion from purely observational data. It is especially helpful as a corrective to some of the more extreme statements of Lyons (2011). Unfortunately, this paper, too, goes too far in some places, and so needs some correction itself.