Peter Koepernik | Searcharxiv

arXiv2026

ARCANE: Scalable high-degree cubature formulae for simulating SDEs without Monte Carlo error

Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but accurate estimates require huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree $D$. They prove that cubature formulae exist for arbitrary $D$, but explicit constructions are difficult and have only reached $D=7$, too small for practical use. We present ARCANE, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree. It reproduces the state of the art in seconds and reaches $\boldsymbol{D=19}$ within hours on modest hardware. In simulations across multiple different SDEs and error metrics, our cubature formulae robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.

math.NA↗

arXiv2024

The Brownian Spatial Coalescent

We introduce a class of Markov coalescent processes on the continuous $d$-dimensional torus, in the most general setting of simultaneous multiple mergers, called the Brownian spatial coalescent. It is axiomatically defined through a property that is satisfied by the genealogies of any population model in which individuals follow independent Brownian motions forwards in time, regardless of the branching mechanism. We prove that a Brownian spatial coalescent is characterised by a set of "transition measures", reminiscent of the transition rates that characterise a non-spatial coalescent. We prove that it is sampling consistent in a suitable sense if and only if all transition measures are uniform with intensity given by the transition rates of a $\Xi$-coalescent. This defines the "Brownian spatial $\Xi$-coalescent", which we show describes the genealogies of neutral population models with Brownian movement in the limit of large population size, and in particular those of the $\Xi$-Fleming-Viot process - a generalisation of the well-known Fleming-Viot process - at stationarity. An important consequence of our results is that all spatial population models in which individuals follow independent Brownian motions and the branching mechanism is not neutral, that is, depends non-trivially on the spatial distribution, for example through local regulation, have non-Markovian genealogies. Byproducts of our results include explicit formulas for samples from the stationary distribution of a $\Xi$-Fleming-Viot process, and a representation of the backward dynamics of lineages in terms of Brownian motions with coupled drift. This includes calculations of the drift that leads to multiple or even simultaneous mergers in any dimension.

math.PR↗

arXiv2023

On a Repulsion-Diffusion Equation with Immigration

We study a repulsion-diffusion equation with immigration, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is $|x|^{1-d}$, that of the Newtonian potential.

math.AP↗