Hierarchical percolation is a toy model for percolation on Z^d that, much like the Euclidean model, is expected to exhibit mean-field behavior in high dimensions, non-mean-field behavior in low dimensions, and mean-field behavior with logarithmic corrections at the upper-critical dimension. Building on Hutchcroft’s work on cluster volumes in hierarchical percolation, we examine the distribution of the chemical distance in high dimensions and the upper-critical dimension. In this talk, I will explain the renormalization group—style analysis we use to obtain precise estimates on the moments of the chemical distance and discuss our attempt to generalize this approach to long-range percolation on the Euclidean lattice. Joint work with Tom Hutchcroft.

We study a version of De Finetti’s optimal dividend problem driven by a diffusion, where the control strategies are assumed to be absolutely continuous strategies which are bounded above by an increasing, concave function.

We provide sufficient conditions to show that an optimal strategy exists and lies within the set of generalized refraction strategies. In addition, we are able to characterize the optimal refraction threshold in our setting.

This is joint work with Hélène Guérin, Jean-François Renaud and Alexandre Roch.

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We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values on a grid of [0, 1], with mutation and selection. We study of the dynamics of the population in logarithm size and time scales, under a large population assumption. In the first part of the talk, individual mutations are rare but the global mutation rate tends to infinity. Then negligible sub-populations may have a strong contribution to evolution. The traits can also be horizontally transferred, leading to a trade-off between natural evolution to higher birth rates and transfer which drives the population towards lower birth rates. We prove that the stochastic discrete exponent process converges to a piecewise affine continuous function, which can be described along successive phases determined by dominant traits. In the second part of the talk, the individual mutations are small but not rare, we don’t have any transfer and we assume the grid mesh for the trait values becoming smaller and smaller. We establish that under our rescaling, the stochastic discrete exponent process converges to the viscosity solution of a Hamilton-Jacobi equation, filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations.

Joint works with N. Champagnat and V.C. Tran, and S. Mirrahimi for the second part.

Explicit quantitative bounds in the hyper-rectangle distance $d_{HR}$ in terms of a Berry-Esseen type bound are derived, in the setting of high-dimensional, non-linear functionals of Gaussian processes while allowing for a strong dependence structure. Under some smoothness and regularity assumptions, the rate under $d_{HR}$ is determined to be sub-polynomial in dimension $d$ and, in the case where the underlying Gaussian process has short-range dependence, the dependence on the number of observations $n$ is found to be $n^{-1/2} log(n)$. Many statistical applications arise, where important examples covered in the paper are; method of moments, empirical characteristic functions, empirical moment-generating functions and a functional Breuer-Major