Evolution Systems of Measures

• Evolution systems of measures are families of probability measures indexed by time, generalizing invariant measures for nonautonomous systems.
• They are constructed using techniques such as asymptotic compactness, measurable selection, and martingale convergence in stochastic flow settings.
• Applications include stochastic Navier–Stokes equations, diffusions on evolving manifolds, and time-dependent lattice models in statistical mechanics.

Searching arXiv for the core paper and related developments on evolution systems of measures. An evolution system of measures is a family of probability measures indexed by time and consistent with an evolution family, and it serves as the time-inhomogeneous analogue of an invariant measure. In the stochastic-flow setting, the notion was introduced for two-parameter stochastic flows on Polish spaces, where it generalizes invariant measures for random dynamical systems and, in the white-noise case, corresponds one-to-one to evolution systems of measures for the associated Markov transition family (Chen et al., 2010). In later work, closely related formulations were developed for time-inhomogeneous Markov operators on Polish spaces, diffusions on evolving manifolds, causal measure evolutions in globally hyperbolic spacetimes, nonsmooth dynamics governed by continuity equations, and several model-specific constructions in statistical mechanics, SPDEs, and measure-valued particle systems (Wang et al., 2022).

1. Formal definition and basic setting

The classical random-dynamical-systems framework starts from a metric dynamical system $$(\Omega,\mathcal{F},\mathbb{P},(\theta_t)_{t\in\mathbb{R}})$$ and a cocycle $\phi(t,\omega,x)$ on a Polish state space $(X,d)$ satisfying

$$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$

An invariant measure for the cocycle is a probability measure $\rho\in\mathcal{P}(\Omega\times X)$ with marginal $\mathbb{P}$ on $\Omega$ that is invariant under the skew-product flow

$$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$

This is the autonomous or stationary paradigm associated with Arnold and Crauel–Flandoli (Chen et al., 2010).

The stochastic-flow formulation replaces the cocycle by a two-time family

$$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$

defined on a Polish space $(X,d)$. The defining conditions are: the flow property

$\phi(t,\omega,x)$0

continuity in the state variable $\phi(t,\omega,x)$1, joint measurability in $\phi(t,\omega,x)$2, and the identity property $\phi(t,\omega,x)$3 (Chen et al., 2010). No Feller or strong Markov regularity is assumed for the flow itself; continuity in $\phi(t,\omega,x)$4 and joint measurability in $\phi(t,\omega,x)$5 suffice.

For such a stochastic flow, an evolution system of measures is defined on the extended space $\phi(t,\omega,x)$6. Writing

$\phi(t,\omega,x)$7

an evolution system of measures for $\phi(t,\omega,x)$8 is a family $\phi(t,\omega,x)$9, each with marginal $(X,d)$0, such that for all $(X,d)$1,

$(X,d)$2

Equivalently, in disintegrated form,

$(X,d)$3

For a time-inhomogeneous Markov transition family $(X,d)$4, the corresponding definition is the family $(X,d)$5 satisfying

$(X,d)$6

In the autonomous case, $(X,d)$7 recovers the usual invariant measure (Chen et al., 2010).

2. Relation to invariant measures and two-parameter semigroups

The conceptual role of an evolution system of measures is to replace stationarity by consistency across time. If the system is autonomous or stationary under the base flow $(X,d)$8, the family $(X,d)$9 becomes time-independent and reduces to a single invariant measure. If the stochastic flow is induced by a cocycle, the evolution-system viewpoint corresponds to invariant measures of the skew product; for genuinely nonautonomous two-time flows, the indexed family $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$0 is the appropriate substitute for a stationary law (Chen et al., 2010).

For white-noise stochastic flows, the associated transition family is defined by

$$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$1

and satisfies the composition rule $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$2 for $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$3. The white-noise condition is expressed through the independence of the completed sigma-algebras

$$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$4

and it is the structural assumption behind the flow/semigroup correspondence (Chen et al., 2010).

An analogous consistency relation appears in later nonautonomous Markov formulations. For a Polish space $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$5, time-inhomogeneous transition operators $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$6, and dual operators $$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$7, Da Prato–Röckner’s definition used in the nonautonomous stochastic-systems literature is

$$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$8

or equivalently

$$\phi(t+s,\omega,x)=\phi\bigl(t,\theta_s\omega,\phi(s,\omega,x)\bigr),\qquad \phi(0,\omega,x)=x.$$9

The same structure appears for diffusions generated by $\rho\in\mathcal{P}(\Omega\times X)$0 on evolving manifolds, where the evolution system $\rho\in\mathcal{P}(\Omega\times X)$1 satisfies

$\rho\in\mathcal{P}(\Omega\times X)$2

and acts as the natural time-dependent reference family for $\rho\in\mathcal{P}(\Omega\times X)$3-based semigroup estimates (Wang et al., 2022, Cheng et al., 2017).

3. Existence via asymptotic compactness and random attractors

A central existence mechanism for stochastic flows is asymptotic compactness. A flow $\rho\in\mathcal{P}(\Omega\times X)$4 is asymptotically compact if, on a full-measure set $\rho\in\mathcal{P}(\Omega\times X)$5, for every $\rho\in\mathcal{P}(\Omega\times X)$6 and $\rho\in\mathcal{P}(\Omega\times X)$7 there exists a compact attracting set $\rho\in\mathcal{P}(\Omega\times X)$8 such that

$\rho\in\mathcal{P}(\Omega\times X)$9

for every nonempty bounded $\mathbb{P}$0, with

$\mathbb{P}$1

From this one defines the random omega-limit and pullback attractor

$\mathbb{P}$2

Under asymptotic compactness, $\mathbb{P}$3 is a nonempty compact attracting set, minimal with this property, forward invariant,

$\mathbb{P}$4

and measurable with respect to $\mathbb{P}$5 (Chen et al., 2010).

The existence theorem is formulated for compact forward invariant families. If $\mathbb{P}$6 is measurable, compact on a full-measure set, and satisfies

$\mathbb{P}$7

then there exists an evolution system of measures $\mathbb{P}$8 such that

$\mathbb{P}$9

The construction uses measurable selection on the compact sets

$\Omega$0

to build a random trajectory $\Omega$1 with

$\Omega$2

and then sets $\Omega$3 (Chen et al., 2010).

A corollary is that asymptotically compact flows admit evolution systems of measures supported on the pullback attractor. At the same time, the support relation is not exhaustive. A deterministic example is $\Omega$4 on $\Omega$5, where the attractor $\Omega$6 does not support the evolution system $\Omega$7. Another deterministic example, the translation flow $\Omega$8 on $\Omega$9, admits no evolution system of measures at all. These examples preclude the common misconception that every evolution system of measures must be attractor-supported, or that existence follows from the flow property alone (Chen et al., 2010).

4. White-noise correspondence and principal applications

For white-noise stochastic flows, the correspondence between evolution systems of measures for the flow and for the semigroup is based on a martingale construction. Given an evolution system $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$0 for $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$1, fixing $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$2 and $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$3, one defines

$$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$4

With respect to the filtration $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$5, $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$6 is a bounded martingale, hence converges almost surely. The resulting random measures

$$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$7

exist in the sense of weak convergence along any deterministic sequence and satisfy the evolution property

$$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$8

adaptedness with respect to $$\Theta_t(\omega,x)=\bigl(\theta_t\omega,\phi(t,\omega,x)\bigr),\qquad \rho\circ\Theta_t^{-1}=\rho.$$9, and the averaging identity

$$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$0

Conversely, an adapted evolution system $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$1 for the flow produces an evolution system for the semigroup by

$$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$2

The correspondence is one-to-one only after this adaptedness restriction; nonuniqueness persists at the flow level in general (Chen et al., 2010).

The principal application in the original development is the two-dimensional stochastic Navier–Stokes equation with time-periodic forcing on a bounded domain $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$3: $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$4 with $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$5. After the Ornstein–Uhlenbeck transform

$$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$6

and the substitution $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$7, one obtains a random-coefficient equation in $$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$8. Galerkin approximations yield existence and uniqueness of weak solutions, continuity in the initial data, and measurability, so the solution map defines a stochastic flow. Energy inequalities produce compact attracting balls

$$S(t,s;\omega):X\to X,\qquad t\ge s,\ \omega\in\Omega,$$9

hence asymptotic compactness and a pullback attractor $(X,d)$0. By the existence theorem, the attractor supports an evolution system of measures for the flow; by the white-noise correspondence, there is an associated evolution system for the Markov transition operators. Because the forcing is $(X,d)$1-periodic and

$(X,d)$2

one obtains a $(X,d)$3-periodic evolution system,

$(X,d)$4

which the paper identifies as the probabilistic analogue of random periodic measures (Chen et al., 2010).

Later applications retain the same consistency principle while changing the analytic setting. On evolving Riemannian manifolds, space-time Lyapunov conditions and curvature assumptions imply non-explosion, existence, and under a Bakry–Émery-type lower bound

$(X,d)$5

uniqueness of the evolution system, together with gradient estimates, logarithmic Sobolev inequalities, hypercontractivity, supercontractivity, and ultraboundedness for the time-inhomogeneous semigroup (Cheng et al., 2017). For stochastic lattice reaction–diffusion equations with time-dependent nonlinear noise, a Krylov–Bogolyubov time-average construction, a Feller assumption at the limiting parameter, and convergence in probability on compact sets lead to existence and limiting stability of evolution systems of probability measures (Wang et al., 2022).

5. Broader variants and analogues across disciplines

The terminology of evolution systems of measures has been used well beyond stochastic flows. In deterministic lattice mechanics, time evolution of Gibbs measures can be written as a nonautonomous evolution family $(X,d)$6 acting by pushforward under a piecewise-defined flow $(X,d)$7, so that

$(X,d)$8

In the framework of Lefevere and Sasa, macroscopic mixing and transitive mixing are formulated through cumulant generating functions and macroscopic averages, and the family $(X,d)$9 plays the role of a macroscopic evolution system of measures. The emphasis is not on exact microscopic invariance but on macroscopic equivalence to an appropriate Gibbs measure after large-$\phi(t,\omega,x)$00 and long-time limits (Lefevere et al., 2021).

In globally hyperbolic spacetimes, a $\phi(t,\omega,x)$01-evolution is a family $\phi(t,\omega,x)$02 with $\phi(t,\omega,x)$03. Causality is expressed by monotonicity in the measure-valued causal order,

$\phi(t,\omega,x)$04

and is proved equivalent to the existence of a probability measure $\phi(t,\omega,x)$05 on the space of future-directed continuous causal curves with

$\phi(t,\omega,x)$06

and to a distributional continuity equation with a future-directed causal vector field $\phi(t,\omega,x)$07 satisfying $\phi(t,\omega,x)$08. This is an evolution-of-measures theory in which time slices replace the parameter set of a Markov family (Miller, 2021).

For nonsmooth dynamical systems described by an evolution variational inequality,

$\phi(t,\omega,x)$09

the measure-valued evolution is treated through three complementary formalisms: a superposition principle on path space, a Moreau–Yosida regularization leading to the classical continuity equation for

$\phi(t,\omega,x)$10

and a time-stepping scheme

$\phi(t,\omega,x)$11

Under the standing assumptions, the family $\phi(t,\omega,x)$12 is consistent with the EVI flow,

$\phi(t,\omega,x)$13

and thus forms an evolution system of measures in the pushforward sense (Chhatoi et al., 2024).

Additional instances show how broadly the idea extends. In continuum particle systems with random jumps and coalescence, the evolution of states $\phi(t,\omega,x)$14 is defined by duality,

$\phi(t,\omega,x)$15

for a measure-defining class of observables, and the main result is existence of such an evolution on a bounded time horizon (Kozitsky et al., 2018). For the one-dimensional nonlinear Schrödinger equation with Gaussian initial data, one defines

$\phi(t,\omega,x)$16

obtaining a genuinely time-dependent family: for each fixed $\phi(t,\omega,x)$17, $\phi(t,\omega,x)$18 is equivalent to the linearly evolved Gaussian measure, while $\phi(t,\omega,x)$19 and $\phi(t,\omega,x)$20 are mutually singular for $\phi(t,\omega,x)$21 (Thomann et al., 2023). In a different direction, the family of sum-of-digits measures $\phi(t,\omega,x)$22 is reorganized as a nonautonomous dynamics on pairs of $\phi(t,\omega,x)$23-valued probability measures driven by the binary-tree operators $\phi(t,\omega,x)$24 and $\phi(t,\omega,x)$25; the associated laws arise from stopped random walks and are analyzed through support, symmetry, variance, and asymptotic properties (Tarłowski, 9 May 2026).

6. Structural properties, limitations, and open directions

Several structural features recur across the literature. In time-periodic settings, evolution systems of measures inherit periodicity: $\phi(t,\omega,x)$26 In white-noise stochastic flows, tightness, martingale convergence, and the independence structure between $\phi(t,\omega,x)$27 and $\phi(t,\omega,x)$28 are the key tools. In attractor-based constructions, measurable selection and Prokhorov compactness are decisive. In nonautonomous stochastic systems on Polish spaces, convergence in probability on compact sets and a Feller property at the limiting parameter supply a limiting stability principle for weak limits of evolution systems [(Chen et al., 2010); (Wang et al., 2022)].

Uniqueness is a recurrent limitation. The stochastic-flow theory states explicitly that uniqueness of evolution systems may fail even in simple examples, and that the correspondence between flow-level and semigroup-level evolution systems is not one-to-one unless one restricts to adapted systems (Chen et al., 2010). The nonautonomous lattice reaction–diffusion work establishes tightness and subsequential convergence but does not impose ergodicity or uniqueness conditions. On evolving manifolds, uniqueness requires curvature-drift conditions of the form $\phi(t,\omega,x)$29; without such assumptions, existence does not by itself imply a canonical reference family (Wang et al., 2022, Cheng et al., 2017).

Support properties are also subtler than the attractor picture alone suggests. Pullback attractors support at least one evolution system of measures for asymptotically compact flows, but not necessarily all of them. This suggests that attractor geometry and the set of all admissible evolution systems need not coincide. The original stochastic-flow paper explicitly lists as open directions criteria ensuring uniqueness, ergodic properties, characterization of all evolution systems relative to a random attractor, and extensions to non-white-noise flows where the independence structure fails (Chen et al., 2010).

A final point is that evolution systems of measures are not synonymous with invariance. In compact or autonomous settings, invariant measures may exist and dominate the discussion. In nonautonomous settings, or in models where invariant measures are trivial or absent, the indexed family is the primary object. The nonlinear Schrödinger example on $\phi(t,\omega,x)$30 is explicit on this point: the relevant family is time-dependent, comparable at each fixed time to a linear Gaussian reference measure, yet mutually singular across distinct times (Thomann et al., 2023). This is the characteristic role of the concept: to encode temporal consistency when exact stationarity is unavailable or inappropriate.