We use a long history of global temperature and atmospheric carbon dioxide (CO2) concentration to
estimate the conditional joint evolution of temperature and CO2 at a millennial frequency. We document
three basic facts. First, the temperature–CO2 dynamics are non-linear, so that large deviations in either
temperature or CO2 concentrations take a long time to correct–on the scale of multiple millennia. Second,
the joint dynamics of temperature and CO2 concentrations exhibit multimodality around historical turning
points in temperature and concentration cycles, so that prior to the start of cooling periods, there is a
noticeable probability that temperature and CO2 concentrations may continue to increase. Finally,
evaluating the future evolution of temperature and CO2 concentration conditional on alternative scenarios
realizing, we document that, even conditional on the net-zero 2050 scenario, there remains a significant
risk of elevated temperatures for at least a further five millennia.