University of Uppsala

We consider the two-phase Stefan problem with p-degenerate diffusion, p larger than two, and we prove continuity up to the boundary for weak solutions, providing a modulus of continuity which we conjecture to be optimal. Since our results are proven in the form of a priori estimates for appropriate regularized problems, as corollary we infer the existence of a globally continuous weak solution for continuous Cauchy-Dirichlet datum.

The reaction-diffusion master equation (RDME) is a popular model in systems biology. In the RDME, diffusion is modeled as discrete jumps between voxels in the computational domain. However, it has been demonstrated that a more fine-grained model is required to resolve all the dynamics of some highly diffusion-limited systems.

In Greenʼs Function Reaction Dynamics (GFRD), a method based on the Smoluchowski model, diffusion is modeled continuously in space.

This will be more accurate at fine scales, but also less efficient than a discrete-space model.

We have developed a hybrid method, combining the RDME and the GFRD method, making it possible to do the more expensive fine-grained simulations only for the species and in the parts of space where it is required in order to resolve all the dynamics, and more coarse-grained simulations where possible. We have applied this method to a model of a MAPK-pathway, and managed to reduce the number of molecules simulated with GFRD by orders of magnitude and without an appreciable loss of accuracy.

The Brownian snake (with lifetime given by a normalized

Brownian excursion) arises as a natural limit when studying random trees. This

may be used in both directions, i.e. to obtain asymptotic results for random

trees in terms of the Brownian snake, or, conversely, to deduce properties of

the Brownian snake from asymptotic properties of random trees. The arguments

are based on Aldous' theory of the continuum random tree.

I will discuss two such situations:

1. The Wiener index of random trees converges, after

suitable scaling, to the integral (=mean position) of the head of the Brownian

snake. This enables us to calculate the moments of this integral.

2. A branching random walk on a random tree converges, after

suitable scaling, to the Brownian snake, provided the distribution of the

increments does not have too large tails. For i.i.d increments Y with mean 0,

a necessary and sufficient condition is that the tails are o(y^{-4}); in

particular, a finite fourth moment is enough, but weaker moment conditions are

not.