In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. Fundamental examples of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion and fractional kinetics (FK) processes, the introduction of which were partly motivated by the study of the localization and aging properties of physical spin systems, and the two- dimensional Liouville Brownian motion, which is the diffusion naturally associated with planar Liouville quantum gravity.
This FIN diffusions and FK processes are known to be the scaling limits of the Bouchaud trap models, and the two-dimensional Liouville Brownian motion is conjectured to be the scaling limit of simple random walks on random planar maps.
In the first part of my talk, I will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals.
In the second part of my talk, I will discuss heat kernel estimates for (generalized) FIN diffusions and FK processes on metric measure spaces.
This talk is based on joint works with D. Croydon (Warwick) and B.M. Hambly (Oxford) and with Z.-Q. Chen (Seattle), P. Kim (Seoul) and J. Wang (Fuzhou).[-]

I will first introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. I will then present quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e. where spins flip with Metropolis rates at inverse temperature $\beta $. The main result identifies conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. This result provides an extension of the results known for the Ising model on the the Erdos-Renyi random graph. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities.
Based on a joint work in collaboration with Anton Bovier, Frank den Hollander, Saeda Marello and Martin Slowik.[-]

We study an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. We analyze the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity. We derive a spectral representation for the free energy and use this to show that there is a critical curve in the (charge bias, inverse temperature)-plane separating a ballistic phase from a subballistic phase. We show that the phase transition is first order, identify the scaling behaviour of the critical curve for small and for large charge bias, and also identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. In addition, we prove a large deviation principle for the joint law of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate function. This in turn leads to a law of large numbers and a central limit theorem. What happens for the quenched free energy per monomer remains open. We state two modest results and raise a few questions. Joint work with F. Caravenna, N. Petrelis and J. Poisat[-]