Tsiry Randrianasolo (Montan University Leoben, Austria)

Travelling wave solutions to the KPP equation with branching noise

The one-dimensional KPP-equation driven by space-time white noise, $$ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 $$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. In a next step, I explain how to obtain a travelling wave solution to this SPDE. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions.

Metastability in diffusion processes and synchronization

Non-standard time & location: 17:15 @ WSC-S-U-4.02

The phase diagram of the complex branching Brownian motion energy model

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy. This is joint work with Lisa Hartung (Courant Institute, NYU)

Time-discretization of stochastic 2-D Navier-Stokes equations by a penalty-projection method

In this talk, I will present a time-discretization method of the stochastic incompressible Navier--Stokes problem using a penalty-projection method. Basically, the talk will consist of three parts. A brief introduction of the mathematical problem. Then an overview of the main computational issue that shares the stochastic and the deterministic form of Navier--Stokes. Different algorithm will be introduced: a main algorithm and in order to treat the nonlinear character of the equation two auxiliary algorithms. At the end of the day we will arrive at the convergence with rate in probability and a strong convergence of the main algorithm.

Convex Hulls of Random Walks: Expected Number of Faces

Consider a random walk $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(\xi_1,\dots,\xi_n)$ is exchangeable and some general position condition holds, we derive an explicit formula for the expected number of $k$-dimensional faces of $C_n$ in terms of the Stirling numbers of the first and second kind. Generalizing the classical discrete arcsine law for the position of the maximum due to E. Sparre Andersen, we compute explicitly the probability that for given indices $0\leq i_1 < \dots < i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of a direct product of finitely many reflection groups of types $A_{n-1}$ and $B_n$. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. All formulae are distribution-free, that is do not depend on the distribution of the $\xi_k$'s. (Joint work with Vladislav Vysotsky and Dmitry Zaporozhets.)

Continuous spin models on annealed generalized random graphs

We study spin models where the spins take values in a general compact Polish space and interact via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. We prove a variational formula for the corresponding annealed pressure.

We furthermore study classes of models with second order phase transitions which include models on an interval and rotation-invariant models on spheres, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model, but we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

This is joint work with Christof Külske and Philipp Schriever.