  Dynamical systems are widely used in science and engineering to model systems
consisting of several interacting components. Often, they can be given a causal
interpretation in the sense that they not only model the evolution of the
states of the system's components over time, but also describe how their
evolution is affected by external interventions on the system that perturb the
dynamics. We introduce the formal framework of structural dynamical causal
models (SDCMs) that explicates the causal semantics of the system's components
as part of the model. SDCMs represent a dynamical system as a collection of
stochastic processes and specify the basic causal mechanisms that govern the
dynamics of each component as a structured system of random differential
equations of arbitrary order. SDCMs extend the versatile causal modeling
framework of structural causal models (SCMs), also known as structural equation
models (SEMs), by explicitly allowing for time-dependence. An SDCM can be
thought of as the stochastic-process version of an SCM, where the static random
variables of the SCM are replaced by dynamic stochastic processes and their
derivatives. We provide the foundations for a theory of SDCMs, by (i) formally
defining SDCMs, their solutions, stochastic interventions, and a graphical
representation; (ii) studying existence and uniqueness of the solutions for
given initial conditions; (iii) providing Markov properties for SDCMs with
initial conditions; (iv) discussing under which conditions SDCMs equilibrate to
SCMs as time tends to infinity; (v) relating the properties of the SDCM to
those of the equilibrium SCM. This correspondence enables one to leverage the
wealth of statistical tools and discovery methods available for SCMs when
studying the causal semantics of a large class of stochastic dynamical systems.
The theory is illustrated with examples from different scientific domains.
