Within a contest there is some probability M_i(t) that contestant  i will be the winner, given information available at time t, and M_i(t) must be a martingale in t.  Assume continuous paths, to capture the idea that relevant information is…

We consider the long-time behaviour of a branching random walk in random environment on the lattice  $\mathbb{Z}^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of…

The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive…

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson…

The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group $S_\infty$. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz…

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobole spaces $\mathfrak{H}^{\gamma,q}_{p,\theta}(\mathcal{O},T)$. The summability…

We consider the tree-valued Fleming–Viot process, (Xt )t≥0 , with mutation and selection. This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely…

We study conservative particle systems on $W^S$, where $S$ is countable and $W = \{0, \dots, N\}$ or $W = \mathbb{N}$, where the generator reads\[Lf(\eta) = \sum_{x,y} p(x,y) b(\eta_x,\eta_y) (f(\eta - \delta_x + \delta_y) - f(\eta)).\]Under assumptions on $b$ and…

We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q Whittaker functions (or Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted algorithm when q=0. The q-insertion algorithm is `randomised', or `quantum',…

We study "tricolor percolation" on the regular tessellation of $\mathbb{R}^3$ by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector…