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• Wave propagation for reaction-diffusion equations on infinite random trees

  Speaker: Wenqing Hu (Missouri University of Science and Technology) - http://web.mst.edu/~huwen/
  

  When: Fri, August 30, 2019 - 2:00pm
  Where: Kirwan Hall 1313
  

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  Abstract: The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees \vec{d} and the random branch lengths \vec{\ell} of the tree. This speed is slower than that of the same equation on the real line, and we estimate this slow down in terms of \vec{d} and \vec{\ell}. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure (\vec{d}, \vec{\ell}) of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of 2 by 2 random matrices parametrized by (\vec{d}, \vec{\ell}) and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.
  
  
• Spectral properties of reversible indel models

  Speaker: Linden Yuan (UMD) -
  

  When: Mon, October 7, 2019 - 2:00pm
  Where: Kirwan Hall 1313
  

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  Abstract: For modeling evolution of DNA sequences (string of letters A, C, T and G, such as "GATTACA"), Markov chains have become the computational biologist's preferred tool. The object of our study a reversible, infinite-state, continuous-time Markov chain introduced by Thorne, Kishino and Felsenstein in 1991, brings to the forefront three possible modes of evolution -- insertion, deletion and letter-switch (e.g., "A" -> "T"). Working with a simplified version of this model and drawing inspiration from Aldous's work on the hypercube graph, we use coupling to prove exponential ergodicity. This research has applications to reconstructing phylogenies, where, for Markov chain models, convergence rate and reconstructibility seem to be intimately linked (cf. Kesten--Stigum threshold). There are two main lines of future research: first, to obtain stronger results, such as L2 exponential convergence (i.e., a Poincare inequality) and a spectral decomposition for the transition function (cf. Karlin--McGregor theorem for birth-death processes); and second, to consider other models, such as the Poisson indel model introduced by Bouchard-Cote and Jordan. Joint work with Wai-Tong Fan (IU Bloomington) and Graham White (IU Bloomington).
  
• On stochastic equations with drift in $L_{d}$

  Speaker: Nikolai Krylov (University of Minnesota) - http://www-users.math.umn.edu/~nkrylov/
  

  When: Wed, October 23, 2019 - 2:00pm
  Where: Kirwan Hall 1313
  

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  Abstract: For Ito stochastic equations
  in $\bR^{d}$ with drift in $L_{d}$ several results
  are discussed such as the existence of weak solutions,
  Aleksandrov type estimates of their Green's functions,
  which yield their summability to power $d/(d-1)$,
  Fabes-Stroock type estimates which show that
  Green's functions are summable to a higher degree,
  Fang-Hua Lin type estimates, which is one of the
  main tools in the $W^{2}_{p}$-theory of fully nonlinear
  elliptic equations, and a few other results.
  
• Conditioned limit theorems for products of random matrices and Markov chains with applications to branching processes.

  Speaker: Ion Grama (Université de Bretagne-Sud) - http://web.univ-ubs.fr/lmba/grama/
  

  When: Wed, October 30, 2019 - 2:00pm
  Where: Kirwan Hall 1313
  

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  Abstract: Consider a random walk defined by the consecutive action of independent identically distributed random matrices on a starting point outside the unit ball in the d dimensional Euclidean space. We study the first moment when the walk enters the unit ball. We study the exact behaviour of this time and prove conditioned limit theorems for the associated Markov walk. This extends to the case of walks on group GL(d,R) the well known results by Spitzer. The existence of the harmonic function related to the Markov walk turns out to be crucial point of the proof. We have extended these results to general Markov chains and applied them to study the branching processes in Markov environment.
  
• Product of random matrices depending on a parameter.

  Speaker: Ilya Goldsheid (Queen Mary, University of London) - http://www.maths.qmul.ac.uk/~ig/
  

  When: Wed, November 20, 2019 - 2:00pm
  Where: Kirwan Hall 1313
  

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  Abstract: In this talk, I shall consider products of i.i.d. matrices $g_j(t),\ j\ge 1,$ where $t$ is a parameter, $\ t\in T,$ and $T$ is a compact metric space. Matrices $g(\cdot)$ are continuous functions of $t$. I shall discuss necessary and sufficient conditions under which with probability 1
  \[
  \frac1n \ln\| g_n(t)\ldots g_1(t)\| \lambda(t)\ \ \text{ uniformly in $t\in T$,}
  \]
  where $\lambda(t)$ is the corresponding Lyapunov exponent.
  I shall then explain what happens when $g_j$ are matrices corresponding to the Anderson model and the parameter is the energy $E$ and, time permitting, shall discuss some open problems.