Abstract
We obtain bounds for the expected loss of torsional rigidity of a cylinder CL of length L and planar crosssection Ω due to a Brownian fracture that starts at a random point in CL and runs until the first time it exits CL. These bounds are expressed in terms of the geometry of the crosssection Ω⊂R2. It is shown that if Ω is a disc with radius R, then in the limit as L→∞ the expected loss of torsional rigidity equals cR5 for some c∈(0,∞). We derive bounds for c in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in R3 with radius 1, and runs until the first time it exits this ball.
Original language  English 

Pages (fromto)  321339 
Number of pages  19 
Journal  Bulletin of the London Mathematical Society 
Volume  50 
Early online date  25 Jan 2018 
DOIs  
Publication status  Published  Apr 2018 
Keywords
 Brownian motion
 torsional rigidity
 heat kernel
 capacity
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Profiles

Professor Michiel Van Den Berg
 School of Mathematics  Emeritus Professor
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member, Honorary and Visiting Academic