  Causal inference in longitudinal studies is often hampered by
treatment-confounder feedback. Existing methods typically assume discrete time
steps or step-like data changes, which we term ``regular and irregular
functional studies,'' limiting their applicability to studies with continuous
monitoring data, like intensive care units or continuous glucose monitoring.
These studies, which we formally term ``functional longitudinal studies,''
require new approaches. Moreover, existing methods tailored for ``functional
longitudinal studies'' can only investigate static treatment regimes, which are
independent of historical covariates or treatments, leading to either stringent
parametric assumptions or strong positivity assumptions. This restriction has
limited the range of causal questions these methods can answer and their
practicality. We address these limitations by developing a nonparametric
framework for functional longitudinal data, accommodating dynamic treatment
regimes that depend on historical covariates or treatments, and may or may not
depend on the actual treatment administered. To build intuition and explain our
approach, we provide a comprehensive review of existing methods for regular and
irregular longitudinal studies. We then formally define the potential outcomes
and causal effects of interest, develop identification assumptions, and derive
g-computation and inverse probability weighting formulas through novel
applications of stochastic process and measure theory. Additionally, we compute
the efficient influence curve using semiparametric theory. Our framework
generalizes existing literature, and achieves double robustness under specific
conditions. Finally, to aid interpretation, we provide sufficient and intuitive
conditions for our identification assumptions, enhancing the applicability of
our methodology to real-world scenarios.
