Definition Stochastic Dynamical Systems Theories [efr-HS7G]
Definition Stochastic Dynamical Systems Theories [efr-HS7G]
A stochastic dynamical systems theory consists of a stochastic module \mathcal {D} over \mathcal {C} equipped with a section T: \mathcal {C}_\mathrm {det} \to \mathcal {D}|_\mathrm {det} of the underlying fibration.
We adopt the notation \widetilde {\mathsf {\mathbb Arena}}(\mathcal {D}) for the double category which was denoted \widetilde {\mathsf {\mathbb Span}}(\mathcal {D})^\mathrm {lens} above. We use the symbol \leftrightarrows for the horizontal (span) morphisms and \rightrightarrows for the vertical morphisms (in \mathcal {D}).
Given a stochastic dynamical systems theory (\mathcal {D}, T), the double category of dynamical systems \mathsf {Sys}(\mathcal {D},T) has
- Objects triples S \in \mathcal {C}, \bar {A} \in \mathcal {D}, \xi : TS \leftrightarrows \bar {A} \in \widetilde {\mathsf {\mathbb Arena}}
- Horizontal morphisms given by a pair f: S \to S', g: \bar {A} \rightrightarrows \bar {B} \in \mathcal {D}, c where c is a square filling \xi , \xi ', Tf, g
- Vertical morphisms given by a pair f: S \to S', g: \bar {A} \leftrightarrows \bar {B} \in \overline {\mathcal {D}}, c where c is a square filling Tf, g \xi , \xi ', 1_{\bar {B}}