--- title: Evolution Systems of Measures url: https://www.emergentmind.com/topics/evolution-system-of-measures type: topic --- # Evolution Systems of Measures 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 [1011.1689]. 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 [2203.13039]. ## 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 [1011.1689]. 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 \[ S(t,r;\omega)\,S(r,s;\omega)=S(t,s;\omega),\qquad s\le r\le t, \] continuity in the state variable \(x\), joint measurability in \((s,\omega)\), and the identity property \(S(t,t;\omega)=\mathrm{id}_X\) [1011.1689]. No Feller or strong Markov regularity is assumed for the flow itself; continuity in \(x\) and joint measurability in \((s,\omega)\) suffice. For such a stochastic flow, an evolution system of measures is defined on the extended space \(\Omega\times X\). Writing \[ \nu(\mathrm{d}\omega,\mathrm{d}x)=\mathbb{P}(\mathrm{d}\omega)\,\nu^\omega(\mathrm{d}x), \] an evolution system of measures for \(S\) is a family \(\{\nu_t\}_{t\in\mathbb{R}}\subset\mathcal{P}(\Omega\times X)\), each with marginal \(\mathbb{P}\), such that for all \(s\le t\), \[ S(t,s;\cdot)_*\nu_s=\nu_t. \] Equivalently, in disintegrated form, \[ \nu_t^\omega=\bigl(S(t,s;\omega)\bigr)_*\nu_s^\omega\qquad \text{for }\mathbb{P}\text{-a.e. }\omega,\ \forall s\le t. \] For a time-inhomogeneous Markov transition family \(\{P_{s,t}\}_{t\ge s}\), the corresponding definition is the family \(\{p_t\}_{t\in\mathbb{R}}\subset\mathcal{P}(X)\) satisfying \[ p_t=p_sP_{s,t},\qquad s\le t. \] In the autonomous case, \(p_t\equiv p\) recovers the usual invariant measure [1011.1689]. ## 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 \((\theta_t)\), the family \(\{\nu_t\}\) 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 \(\{\nu_t\}\) is the appropriate substitute for a stationary law [1011.1689]. For white-noise stochastic flows, the associated transition family is defined by \[ P_{s,t}f(x):=\mathbb{E}\bigl[f(S(t,s;\omega)x)\bigr],\qquad t\ge s, \] and satisfies the composition rule \(P_{s,u}=P_{t,u}\circ P_{s,t}\) for \(s\le t\le u\). The white-noise condition is expressed through the independence of the completed sigma-algebras \[ \mathcal{F}_{\ge s}:=\sigma\{S(u,t;\cdot)x:\ s\le t\le u,\ x\in X\},\qquad \mathcal{F}_{0. \] With respect to the filtration \(\mathcal{G}_s=\mathcal{F}_{\ge t-s}\), \((M_s)_{s>0}\) is a bounded martingale, hence converges almost surely. The resulting random measures \[ \nu_t^\omega:=\lim_{s\to\infty}\bigl(S(t,t-s;\omega)\bigr)_*p_{t-s} \] exist in the sense of weak convergence along any deterministic sequence and satisfy the evolution property \[ \nu_t^\omega=\bigl(S(t,s;\omega)\bigr)_*\nu_s^\omega\qquad \text{a.s.}, \] adaptedness with respect to \(\mathcal{F}_{\le t}\), and the averaging identity \[ \mathbb{E}\bigl[\nu_t^\omega(B)\bigr]=p_t(B). \] Conversely, an adapted evolution system \(\{\nu_t\}\) for the flow produces an evolution system for the semigroup by \[ p_t(B):=\int_\Omega \nu_t^\omega(B)\,\mathbb{P}(\mathrm{d}\omega). \] The correspondence is one-to-one only after this adaptedness restriction; nonuniqueness persists at the flow level in general [1011.1689]. The principal application in the original development is the two-dimensional stochastic Navier–Stokes equation with time-periodic forcing on a bounded domain \(D\subset\mathbb{R}^2\): \[ \partial_t u+(u\cdot\nabla)u-\nu\Delta u+\nabla p =f(t)+\sum_{j=1}^m \sigma_j\,\dot W_j(t),\qquad \nabla\cdot u=0,\quad u|_{\partial D}=0, \] with \(f(t+T)=f(t)\). After the Ornstein–Uhlenbeck transform \[ z(t)=\sum_{j=1}^m \sigma_j Z_j(t),\qquad Z_j(t)=\int_{-\infty}^t e^{-a(t-s)}\,\mathrm{d}W_j(s), \] and the substitution \(v(t)=u(t)-z(t)\), one obtains a random-coefficient equation in \(H\). 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 \[ K(t,\omega):=\{u\in V:\ \|u\|\le r(\omega)+|z(t,\omega)|\}, \] hence asymptotic compactness and a pullback attractor \(\mathcal{A}(t,\omega)\). 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 \(T\)-periodic and \[ P_{s+T,t+T}=P_{s,t}, \] one obtains a \(T\)-periodic evolution system, \[ p_{t+T}=p_t,\qquad \nu_{t+T}^\omega=\nu_t^\omega, \] which the paper identifies as the probabilistic analogue of random periodic measures [1011.1689]. 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 \[ \mathcal{R}_t^Z:=\mathrm{Ric}_t-h_t-\nabla^t Z_t\ge k(t), \] uniqueness of the evolution system, together with gradient estimates, logarithmic Sobolev inequalities, hypercontractivity, supercontractivity, and ultraboundedness for the time-inhomogeneous semigroup [1708.04951]. 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 [2203.13039]. ## 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 \(P_{s,t}\) acting by pushforward under a piecewise-defined flow \(\tilde U\), so that \[ \mu_sP_{s,t}=\mu_t. \] In the framework of Lefevere and Sasa, macroscopic mixing and transitive mixing are formulated through cumulant generating functions and macroscopic averages, and the family \(\{\rho^{m(t)}\}\) 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-\(N\) and long-time limits [2111.10348]. In globally hyperbolic spacetimes, a \( \mathcal T\)-evolution is a family \(t\mapsto\mu_t\) with \(\operatorname{supp}\mu_t\subset\mathcal{T}^{-1}(t)\). Causality is expressed by monotonicity in the measure-valued causal order, \[ s\le t\quad \Rightarrow \quad \mu_s\preceq \mu_t, \] and is proved equivalent to the existence of a probability measure \(\sigma\) on the space of future-directed continuous causal curves with \[ (\mathrm{ev}_t)_\#\sigma=\mu_t, \] and to a distributional continuity equation with a future-directed causal vector field \(X\) satisfying \(X\mathcal{T}=1\). This is an evolution-of-measures theory in which time slices replace the parameter set of a Markov family [2104.02552]. For nonsmooth dynamical systems described by an evolution variational inequality, \[ \dot x(t)\in f(t,x(t))-N_{S(t)}(x(t)), \] 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 \[ g_t^\lambda(x):=f(t,x)-\frac{1}{\lambda}\bigl(x-\operatorname{proj}_{S(t)}(x)\bigr), \] and a time-stepping scheme \[ \mu_{k+1}=(\Phi_h^k)_\#\mu_k,\qquad \Phi_h^k(x):=\operatorname{proj}_{S_{k+1}}\bigl(x+h\,f(t_k,x)\bigr). \] Under the standing assumptions, the family \(\{\mu_t\}_{t\in[0,T]}\) is consistent with the EVI flow, \[ P_{s,t}^*\mu_s=(X_{s,t})_\#\mu_s=\mu_t, \] and thus forms an evolution system of measures in the pushforward sense [2405.09189]. Additional instances show how broadly the idea extends. In continuum particle systems with random jumps and coalescence, the evolution of states \(\mu_0\to\mu_t\) is defined by duality, \[ \mu_0(F_t)=\mu_t(F_0), \] for a measure-defining class of observables, and the main result is existence of such an evolution on a bounded time horizon [1807.07310]. For the one-dimensional nonlinear Schrödinger equation with Gaussian initial data, one defines \[ \mu_s:=(\Psi(s,0))_\#\mu_0, \] obtaining a genuinely time-dependent family: for each fixed \(s\), \(\mu_s\) is equivalent to the linearly evolved Gaussian measure, while \(\mu_s\) and \(\mu_{s'}\) are mutually singular for \(s'\neq s\) [2303.06964]. In a different direction, the family of sum-of-digits measures \(\{\mu_t\}_{t\in\mathbb{N}}\) is reorganized as a nonautonomous dynamics on pairs of \(\mathbb{Z}\)-valued probability measures driven by the binary-tree operators \(\Phi_L\) and \(\Phi_R\); the associated laws arise from stopped random walks and are analyzed through support, symmetry, variance, and asymptotic properties [2605.08624]. ## 6. Structural properties, limitations, and open directions Several structural features recur across the literature. In time-periodic settings, evolution systems of measures inherit periodicity: \[ \mu_{t+T}=\mu_t. \] In white-noise stochastic flows, tightness, martingale convergence, and the independence structure between \(\mathcal{F}_{\ge s}\) and \(\mathcal{F}_{