Probability Seminar Essen
Summerterm 2021: The seminar will start on April 13th in form of video conferences.
Apr 20 | Mark C. Veraar (TU Delft) |
May 11 | |
May 18 | |
Jun 1 | |
Jun 8 | |
Jun 15 | |
Jun 22 | |
Jun 29 | |
Jul 6 | |
Jul 13 |
Abstracts:
Apr 20 |
Mark C. Veraar (TU Delft) In this talk I will present a new approach to maximal inequalities for stochastic convolutions both in discrete and continuous time. The proofs are based on extensions of the vector-valued Burkholder-Rosenthal inequalities to the non-martingale setting. Our results are new in both Hilbert spaces and 2-smooth Banach spaces. Applications to stochastic evolution equations will be discussed as well. Here we prove the existence of continuous modifications and convergence of discretization schemes in time. |
May 11 |
Stochastic partial differential equations (SPDEs) are widely used to model many real world phenomena such as stock market prices and fluid flows. Since explicit solutions of many SPDEs are unknown, developing numerical schemes is a good alternative to provide their approximations, and is therefore a hot topic. Numerical methods for autonomous SPDEs are thoroughly investigated in the literature, while to the best of our knowledge the non-autonomous cases are not yet well understood. In this talk, we propose a Magnus-type integrator for time-dependent stochastic advection-reaction-diffusion equation, which is based on an approximation of the Magnus series [1]. We use finite element method for the approximation in space and provide the strong convergence error of the fully discrete scheme toward the mild solution. |
Jun 1 |
The uniform spanning tree (UST) on Z^3 is the infinite-volume limit of uniformly chosen spanning trees of large finite subgraphs of Z^3. The main theorem in this talk is the existence of subsequential scaling limits of the UST on Z^3. We get convergence over dyadic subsequences. An essential tool is Wilson’s algorithm, which samples uniform spanning trees by using loop-erased random walks (LERW). This strategy imposes a restriction: results for the scaling limit of 3D LERW constrain the corresponding results for the 3D UST. We will comment on work in progress for the LERW that leads to the full convergence to the scaling limit of the UST. This talk is based on joint work with Omer Angel, David Croydon, and Daisuke Shiraishi; and work in progress with Xinyi Li and Daisuke Shiraishi. |
Jun 8 |
We analyze stochastic partial di˙erential equations (SPDEs) driven by an infinite-dimensional fractional Brownian motion using rough paths techniques. Since the breakthrough in the rough paths theory there has been a huge interest in investigating SPDEs with rough noise. We con-tribute to this aspect and develop a solution theory, which can further be applied to dynamical systems generated by such SPDEs. This talk is based on joint works with Robert Hesse and Christian Kuehn. |
Jun 15 |
Preferential attachment networks form a popular class of evolving random graph models that share many features with real-life networks. The basic mechanism is that newly incoming nodes connect preferably to old vertices with high degree. We consider a perturbation of these networks, where the attractiveness of nodes is randomly perturbed. We can identify two different phases: if the perturbation is small, then the model behaves as if the perturbation is replaced by its mean, while if the perturbation is strong then the system is essentially driven by the extremes of the perturbation. In both cases, we have a detailed understanding of the behaviour of the degree of a typical vertex as well as the largest degree in the system. In particular, we show that for small perturbations `the old get richer' phenomenon is true, while in the other case younger nodes can compete. We will also compare these results to a class of evolving random graphs, where the preferential attachment mechanism is `switched off' that can be seen as a version of the well-known random recursive tree, but now in a random environment. |
Jun 22 | On the Liouville Property for Generators of Lévy Processes We show necessary and sufficient conditions for the Liouville property and the strong Liouville property to hold for generators of Lévy processes. This extends the classical Liouville property known for Brownian motion, Random walks and the (discrete) Laplacian. |
Jun 29 |
One-dimensional Brownian SDEs have been studied for decades, many techniques were developed to give a full picture of existence/uniqueness/properties. The situation is different for SDEs driven by stable processes, only for $\alpha>1$ the classical local time tricks can be extended to prove Engelbert-Schmidt type results. We discuss recent results for $\alpha<1$. |
Jul 6 |
We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with with semi-martingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature as well as a microscopic foundation for the use of semi-martingale liquidation strategies. Our results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. The talk is based on joint work with Evgueni Kivman. |
Jul 13 |
We define a graph process $G(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on these graphs. However, it is left open whether the survival probability of $G(p,q)$ is monotone in $p$ or $q$. |
Talks of previous terms.