Conference Agenda

There is a duality between the extremum of space-heterogeneous Branching Brownian Motions, where branching rates are space dependent, and a certain semilinear partial differential equation, the so-called F-KPP equation. Given a space-dependent medium $\xi(x), x\in\mathbb{R}$, the associated PDE looks like this: $$u_t = \frac{1}{2} u_{xx} + \xi(x)u(1-u).$$ The solution to this equation exhibit a much richer behavior than in the homogeneous case, i.e. $\xi \equiv 1$, obviously depending on the choosing of the medium $\xi$.

In this talk, we look at the diameter of the so-called transition front, which is the length the solution needs to transition between the stable states $1$ and $0$. In the homogeneous case the diameter is uniformly bounded in time, while in the heterogeneous case we will see that the diameter is generally unbounded and, more specifically, for every $\alpha \in (0,1)$ one can construct a $\xi$ where the diameter grows like $t^\alpha$.

Our proofs are entirely probabilistic and revolve around the duality with Branching Brownian Motions. We will see that for our choosing of $\xi$ the key step is to analyze the minimal position of a heterogeneous BBM starting close to zero, where branchin rates are equal to $1$ to the left of $0$ and slightly larger than $2$ to right of $0$.

New selected values of odd random simplex volumetric moments (moments of the volume of a random simplex picked from a given body) are derived in an exact form in various bodies in dimensions three, four, five and six. In three dimensions, the well known Efron’s formula was used by Buchta & Reitzner and Zinani to deduce the mean volume of a random tetrahedron in a tetrahedron and a cube. However, for higher moments and/or in higher dimensions, the method fails. As it turned out, the same problem is also solvable using Blashke-Petkanchin formula in Cartesian parametrisation in the form of the Canonical Section Integral (Base-height splitting). In our presentation, we show how to derive the older results mentioned above using our base-height splitting method and also touch the essential steps how the method translates to higher dimensions and for higher moments.