A global view of Brownian penalisations

TL;DR

This paper constructs a sigma-finite measure related to penalizations of Brownian motion and extends it to other stochastic processes, providing a new framework for understanding these measures.

Contribution

It introduces a novel sigma-finite measure connected to Brownian penalizations and generalizes it to multiple stochastic processes.

Findings

Constructed a sigma-finite measure linked to Brownian penalizations

Extended the measure to 2D Brownian motion, recurrent diffusions, and Markov chains

Provides a unified framework for penalization measures across processes

Abstract

In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are absolutely continuous with respect to Wiener measure. This remarkable sigma-finite measure can be generalized in three other cases: one can start from a two-dimensional Brownian motion, from a recurrent diffusion with values in R_+, and from a discrete, recurrent Markov chain.