Mathematics Dept.
  Events
Technical University of Berlin
Wed 18 Oct 2017, 3:00pm
Probability Seminar
ESB 2012
Harnack inequality for degenerate balanced random walks
ESB 2012
Wed 18 Oct 2017, 3:00pm-4:00pm

Abstract


We consider an i.i.d. balanced environment omega(x,e)=omega(x,-e), genuinely d dimensional on the lattice and show that there exist a positive constant C and a random radius R(omega) with streched exponential tail such that every non negative omega harmonic function u on the ball B_{2r} of radius 2r>R(omega), we have max_{B_r} u <= C min_{B_r} u. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in  balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow's Harnack's inequality for i.i.d. bond percolation to the directed case. This is joint work with N. Berger, M. Cohen, and X. Guo.
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University of Alberta at Edmonton
Wed 1 Nov 2017, 3:00pm
Probability Seminar
ESB 2012
Feynman-Kac formula for the stochastic heat equation driven by general Gaussian noises
ESB 2012
Wed 1 Nov 2017, 3:00pm-4:00pm

Abstract


In this talk  I will  present some results on stochastic heat equations driven by a Gaussian noises. I will focus on Feynman-Kac representation of the solution and the moments of the solution. Both lower and upper bounds for the L^p moments of the solution are obtained which is relevant to the so-called intermittency. The Driving Gaussian noises include fractional Brownian fields of Hurst parameters greater or smaller than 1/2.
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