Extremal geometry of a Brownian porous medium

The path W [ 0 , t ] of a Brownian motion on a d -dimensional torus T d run for time t is a random compact subset of T d . We study the geometric properties of the complement T d \ W [ 0 , t ] as t → ∞ for d ≥ 3 . In particular, we show that the largest regions in T d \ W [ 0 , t ] have a linear scale φ d ( t ) = [ ( d log t ) / ( d - 2 ) κ d t ] 1 / ( d - 2 ) , where κ d is the capacity of the unit ball. More specifically, we identify the sets E for which T d \ W [ 0 , t ] contains a translate of φ d ( t ) E , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T d \ W [ 0 , t ] as t → ∞ and the ε -cover time of T d as ε ↓ 0 . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003 ), are based on a large deviation estimate for the shape of the component with largest capacity in T d \ W ρ ( t ) [ 0 , t ] , where W ρ ( t ) [ 0 , t ] is the Wiener sausage of radius ρ ( t ) , with ρ ( t ) chosen much smaller than φ d ( t ) but not too small. The idea behind this choice is that T d \ W [ 0 , t ] consists of “lakes”, whose linear size is of order φ d ( t ) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of T d \ W ρ ( t ) [ 0 , t ] as t → ∞ . Our results give a complete picture of the extremal geometry of T d \ W [ 0 , t ] and of the optimal strategy for W [ 0 , t ] to realise extreme events.