In this talk, we look for a bounded function $w$ and a constant $\lambda$ such that, for a given bounded function g and a Markov transition operator $P(x,dy)$, the following multiplicative Poisson equation (MPE) is satisfied: \( w(x)=g(x)−\lambda +\ln(\int_E e^{w(y)}P(x,dy)). \) We first consider the case of a countable state space $E$, and then study the problem on locally compact separable metric spaces. In both cases, we formulate necessary and sufficient conditions for the existence of bounded solutions to the MPE.

The results are based on the following two papers, written jointly with Marcin Pitera of the Jagiellonian University:

• M. Pitera, Ł. Stettner, Bounded solutions to multiplicative Poisson equations on countable state spaces, submitted.
• M. Pitera, Ł. Stettner, Bounded continuous solutions to multiplicative Poisson equations on locally compact separable metric spaces, in preparation.