Events

Date: 1 April 2026   Time: 13:00 - 14:00

Location: MB 501 (hub)

We give a constructive characterisation of the least superharmonic majorant arising in the optimal stopping problem for $d$-dimensional Brownian motion ($d\ge 2$) absorbed at the boundary of the unit ball, with continuous gain function $g$. Classical potential theory identifies the value function as the least superharmonic majorant of $g$, but in dimensions $d\ge 2$ this characterisation is not constructive and, in general, cannot be realised via harmonic functions associated with single stopping times.

We show that the value function admits a representation as the pointwise infimum over a new class of objects, which we term branched harmonic majorants of $g$. These arise naturally by iterating harmonic extensions along successive stopping times, and yield a direct multidimensional analogue of the Dynkin–Yushkevich representation in terms of the smallest concave majorant.

The obstruction in higher dimensions is that Brownian paths are too thin to support classical (unbranched) harmonic constructions: high-gain sets of low Newtonian capacity cannot be captured by single-step harmonic majorisation. This is overcome by allowing finite compositions of stopping times, encoded by recursive harmonic-measure averages over subdomains. The resulting branched harmonic majorants yield a hierarchy indexed by branching depth, providing a monotone approximation scheme that converges pointwise to the least superharmonic majorant of $g$ and admits a direct probabilistic interpretation in terms of iterated stopping rules. The talk is based on doi.org/10.48550/arXiv.2505.09725 .