Eventual Absolute Continuity

• Eventual absolute continuity is a phenomenon where absolute continuity emerges in the limit after sufficient time shifts, scaling, or integration, leading to dissipation of singularities.
• In stochastic processes and flows, properties such as invertibility, Jacobian estimates, and integrability conditions are key to establishing explicit Radon–Nikodym derivatives.
• This concept unifies approaches in measure theory, geometric analysis, and SDEs by providing quantitative criteria to separate singular from absolutely continuous components.

Eventual absolute continuity is a phenomenon observed in the context of probability measures, functions, or stochastic processes where absolute continuity does not necessarily hold globally or instantaneously, but emerges in the limit—as the effect of time, flow, scaling, or integration accumulates. The rigorous characterization of eventual absolute continuity varies across measure theory, stochastic analysis, and real analysis, but in all cases it delineates when singularities dissipate under certain operations and when Radon–Nikodym derivatives exist or can be explicitly constructed.

1. Foundational Notions and Definitions

Absolute continuity is a classical concept in analysis: for functions, $f \in AC(I)$ on interval $I$ if for every $\varepsilon > 0$ there exists $\delta > 0$ so that any finite collection of disjoint intervals $\{(x_k, y_k)\}$ with $\sum_k (y_k - x_k) < \delta$ satisfies $\sum_k |f(y_k) - f(x_k)| < \varepsilon$. For measures $\mu \ll \nu$, $\mu$ is absolutely continuous with respect to $\nu$ if $\nu(A) = 0$ implies $\mu(A) = 0$ for all measurable $A$.

Eventual absolute continuity generalizes this to situations such as:

• Time shifts in stochastic processes, where the law of a process under large temporal translations becomes absolutely continuous with respect to its original law (Löbus, 2013).
• Stochastic flows, in which pushforwards of a reference measure under the evolution defined by a SDE become absolutely continuous for all positive times (Luo, 2010).
• Measures or functions on unbounded domains, where local absolute continuity may not promote to global unless integrability constraints ("eventual" conditions) are met (Banerjee, 20 Oct 2025).
• Measures on the real line, where square-function criteria involving quantitative Wasserstein deviations ("$\alpha$-numbers") characterize the Lebesgue decomposition pointwise and yield a dichotomy between (eventual) absolute continuity or singularity (Orponen, 2017).

2. Eventual Absolute Continuity for Stochastic Processes Under Time-Shift

A paradigmatic result is established for two-sided stochastic processes $X = W + A$, where $W$ is a two-sided Brownian motion with random initial law of density $m$ and $A$ is a possibly jump-valued functional of $W$, subject to temporal homogeneity conditions. Under suitable regularity and invertibility assumptions (e.g., $m$ is $C^1$, $A_0 = 0$, mapping $x \mapsto X_{-t}(W^x)$ is a $C^1$-diffeomorphism), the following holds (Löbus, 2013):

• For all sufficiently large $t \geq T_0$, the pushforward law of $X_{\cdot-t}$ is absolutely continuous with respect to that of $X$, and the Radon–Nikodym derivative is

$$\frac{dP_\nu(X_{\cdot-t} \in \cdot)}{dP_\nu(X_\cdot \in \cdot)}(X) = \frac{m(X_{-t})}{m(X_0)} \cdot \prod_{i=1}^d |\nabla_{W_0^i} X_{-t}|.$$

Here the product is over spatial coordinates and applies to the Jacobian determinant of the initial-position map.

• This "eventual" absolute continuity manifests only for large enough shift $t$ (as opposed to all $t > 0$), a consequence of the dependence on the lower bounds of the derivative (Jacobian) for the invertibility of the time-reversal mapping.
• The asymptotic property underpins ergodic theory, integration by parts, and the investigation of invariant measures for such classes of Markov processes.

3. Stochastic Flows and Absolute Continuity of Pushforwards

In the framework of Itô SDEs with non-degenerate diffusion and potentially merely measurable drift, absolute continuity of the induced stochastic flow is analyzed. Let $X_{s,t}(x)$ solve

$$dX_{s,t} = \sigma_t(X_{s,t})\,dW_t + b_t(X_{s,t})\,dt, \quad X_{s,s}=x,$$

and let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$. Under strong non-degeneracy, Sobolev regularity, and exponential integrability of $\sigma$ and the divergences of $\sigma$ and $b$, it is established that for every $0 \leq s < t \leq T$, the flow $(X_{s,t})_\# \gamma \ll \gamma$ with density $\rho_{s,t}$ given by an explicit exponential formula (Kunita–Stratonovich) (Luo, 2010).

This result is referred to as "eventual" absolute continuity in the sense that:

• For any positive time increment, regardless of how small, the pushforward of $\gamma$ is absolutely continuous with respect to $\gamma$—the property emerges immediately for $t > s$.
• The main technical advance is passing the result from regularized (smooth) coefficients to merely measurable coefficients with distributional divergence, relying on entropy estimates and tightness arguments.

4. Characterization of Eventual Absolute Continuity for Functions on Unbounded Domains

The classic absolute continuity result—on bounded intervals, $f$ is absolutely continuous iff $f' \in L^1$—fails on unbounded intervals. For functions $f : \mathbb{R} \to \mathbb{R}$, a sharp criterion is (Banerjee, 20 Oct 2025):

• $f \in AC(\mathbb{R})$ if and only if $f \in AC_{loc}(\mathbb{R})$ and $f' \in L^1_G(\mathbb{R})$, where

$$L^1_G(\mathbb{R}) := \left\{g \,:\, \int_K |g| < \infty\, \text{for all measurable } K \subset \mathbb{R} \text{ with } m(K) < \infty \right\}.$$

• Thus, "eventual" absolute continuity is determined by global integrability over all finite-measure sets—not merely local integrability—of the derivative, supplementing local absolute continuity.
• The paper also introduces the larger space $L^1_H(\mathbb{R})$, where $g \in L^1_H$ if the super-level set $\{x : |g(x)| \geq M\}$ has finite measure for some $M > 0$, which is sufficient (but not necessary) for $f \in AC(\mathbb{R})$.

A canonical example demonstrates that $f$ may be locally absolutely continuous but globally fails eventual absolute continuity due to a non-globally integrable derivative concentrated on a thin unbounded set.

5. Square Function and $\alpha$-Number Characterization

In geometric measure theory, Orponen characterized the dichotomy between the absolutely continuous and singular parts of a measure $\mu$ with respect to a doubling measure $\nu$ using square functions involving $\alpha$-numbers (Orponen, 2017). For Radon measures on $\mathbb{R}$:

• The $\alpha$-number for interval $I$,

$\alpha_{\mu,\nu}(I) := W_1(\mu_I, \nu_I),$

is the $L^1$-Wasserstein distance of the normalized restrictions to $I$.

• The dyadic (or continuous/smooth) square function,

$$S_\nu^2(\mu)(x) = \sum_{I \ni x} \alpha_{\mu,\nu}^2(I), \quad \text{or} \quad S_\nu^2(\mu)(x) = \int_0^1 \alpha_{s,\mu,\nu}^2(B(x, r))\,\frac{dr}{r}$$

(with $\alpha_{s,\mu,\nu}$ the smoothed variant).

• Main result: $S_\nu(\mu)(x) < \infty$ for $\mu_a$-a.e. $x$ (absolutely continuous part), and $S_\nu(\mu)(x) = \infty$ for $\mu_s$-a.e. $x$ (singular part). That is, control of the square function at a point precisely identifies the "eventual" (pointwise) absolute continuity or singularity of $\mu$ with respect to $\nu$.

The method provides a quantitative, local-to-global mechanism: absolute continuity emerges precisely where the square function remains finite.

6. Eventual Absolute Continuity of Exponential Functionals and Stochastic Integrals

Behme, Lindner, Rekert, and Rivero (Behme et al., 2019) investigate exponential integrals of Lévy processes and delineate sufficient and necessary criteria for absolute continuity in various regimes:

• For the "killed" exponential functional $V_{q,\xi,\eta} = \int_0^T e^{-\xi_{s-}}\,d\eta_s$ ($T$ exponential, $\xi, \eta$ independent Lévy processes), explicit SDE invariant measures correspond to their law.
• The support and structure of the law of $V_{q,\xi,\eta}$ are characterized across a range of cases—deterministic, finite/infinite variation, compound Poisson, and symmetric or asymmetric jump structures—with precise criteria as to when support is full $\mathbb{R}$, half-line, or single-point.
• Sufficient conditions for absolute continuity (for $q>0$, $q=0$, or finite horizon $t>0$) include criteria such as infinite absolute continuity or Gaussian part for the Lévy measure, potential kernel integrability, or specific Lusin conditions. Key smoothing mechanisms are also identified: pathwise conditioning, infinite divisibility and characteristic exponent growth, potential kernel smoothing, and convolution decompositions.
• A fundamental dichotomy is observed: in all regimes, given sufficient regularity (notably, when at least one process has sufficiently "rich" jump or Gaussian components), eventual absolute continuity of the law is achieved for all large times or in the stationary regime.

7. Connections, Implications, and Applications

Eventual absolute continuity emerges as a unifying lens for:

• The asymptotic equivalence of stochastic process laws under time shifts or large deviations, essential for ergodic theory and spectral analysis (Löbus, 2013).
• Structural stability and quasi-invariant properties of SDE flows and invariant measures, with crucial implications for Fokker–Planck and invariant measure regularity (Luo, 2010).
• Pointwise criteria for the Lebesgue decomposition of measures, with the square-function and $\alpha$-number analysis providing quantitative, scale-sensitive diagnostics (Orponen, 2017).
• A sharp, operationally verifiable criterion for the global absolute continuity of functions in real analysis on unbounded domains, crucial for understanding where local regularity fails to yield global properties (Banerjee, 20 Oct 2025).
• Fine-grained understanding of the density and regularity of stochastic integrals and perpetuity-type distributions in probability theory (Behme et al., 2019).

These results underscore the importance of "eventual" properties—that is, properties that may only arise in the limiting regime, under significant transformation, or at the right scale/time—across contemporary stochastic analysis, geometric measure theory, and real analysis.