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We first study a $d$-dimensional branching Brownian motion (BBM) among mildPoissonian obstacles, where a random trap field in $\mathbb{R}^d$ is createdvia a Poisson point process. The trap field consists of balls of fixed radiuscentered at the atoms of the Poisson point process. The mild obstacle rule isthat when particles are inside traps, they branch at a lower rate, which isallowed to be zero, whereas when outside traps they branch at the normal rate.We prove upper bounds on the large-deviation probabilities for the total massof BBM among mild obstacles, which we then use along with the Borel-Cantellilemma to prove the corresponding strong law of large numbers. Our results arequenched, that is, they hold in almost every environment with respect to thePoisson point process. Our strong law improves on the existing correspondingweak law in [6]. Then, we study a $d$-dimensional BBM inside subdiffusivelyexpanding balls, where the boundary of the ball is deactivating in the sensethat once a particle of the BBM hits the moving boundary, it is instantlydeactivated but can be reactivated at a later time provided its ancestral lineis fully inside the expanding ball at that later time. We obtain alarge-deviation result as time tends to infinity on the probability that themass inside the ball is aytpically small. An essential ingredient in the proofsof the mild obstacle problem turns out to be the large-deviation result on themass of BBM inside expanding balls.

Branching Brownian motion in an expanding ball and application to the mild obstacle problem