Explicit limiting measures for higher-dimensional penalisation

Identify explicit limiting measures governing penalisation limits for diffusion processes and discrete-time Markov chains in dimensions two and higher, extending the explicit two-dimensional Brownian-motion construction.

Background

The paper establishes penalisation limits for a class of nonnegative, adapted, decreasing functionals of two-dimensional Brownian motion that become constant after the path exits a bounded region. Its analysis relies on a specific path decomposition at the last exit from a circle and on the associated universal sigma-finite measure $\mathbf{W}^{(2)}$.

The authors note that, although the general penalisation framework of Najnudel, Roynette, and Yor applies broadly, explicit identification of the limiting measures remains unresolved for diffusion processes and discrete-time Markov chains in dimensions two and higher. Solving this problem would extend the explicit planar Brownian-motion theory to substantially broader stochastic-process settings.