Asymptotic Log-Harnack Inequality

• The asymptotic log-Harnack inequality is a semigroup inequality that compares log evaluations at different states using a persistent control term and a decaying remainder.
• It applies to systems with degenerate noise, infinite memory, or path dependence, restoring asymptotic strong Feller properties and the uniqueness of invariant measures.
• The approach employs asymptotic coupling by change of measure to yield sharp gradient estimates and heat kernel behaviors when classical Harnack inequalities fail.

The asymptotic log-Harnack inequality is a large-time semigroup inequality that compares $P_t \log f$ at two initial states by a time-independent control term plus a remainder that vanishes as $t\to\infty$. In one standard formulation on a metric space $(E,p)$, it has the form

$$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$

for $f\in B_+(E)$ with $\|\nabla \log f\|_\infty<\infty$ (Bao et al., 2017). Closely related variants replace the last term by $\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$ (Xu, 2011) or by $\Psi_t(x,y)\,\|\log f\|_\infty$ (Li et al., 3 Mar 2026). The inequality is designed for settings in which classical Wang-type log-Harnack estimates and the strong Feller property are unavailable, notably under degenerate noise, infinite memory, path dependence, or reflection, but where asymptotic smoothing, asymptotic irreducibility, and uniqueness of invariant measures can still be recovered (Liu, 2020, Hamaguchi, 2023, Wang et al., 19 Feb 2025, Zhao, 12 Jul 2025).

1. Formal structure

The asymptotic log-Harnack paradigm is characterized by a split between a persistent “entropy cost” and a decaying “smoothing defect.” In the path-space framework of Bao–Wang–Yuan, the basic definition is

$$P_t \log f(x) \le \log P_t f(y) + \Phi(x,y) + V_t(x,y)\, \| \nabla \log f \|_\infty,\qquad t>0,$$

with $V_t\to0$ as $t\to\infty$0 (Bao et al., 2017). Xu’s modified log-Harnack inequality is

$t\to\infty$1

where $t\to\infty$2, $t\to\infty$3, and $t\to\infty$4 (Xu, 2011). For degenerate SPDEs with reflection, the function class is

$t\to\infty$5

and the inequality becomes

$t\to\infty$6

(Li et al., 3 Mar 2026).

Across these formulations, the time-independent term $t\to\infty$7 measures the nontrivial cost of changing initial data, while the time-decaying term quantifies how far the semigroup remains from a classical log-Harnack regime. This structure is weaker than a classical log-Harnack inequality, but it is precisely adapted to degenerate or non-Markovian mechanisms in which immediate regularization is absent.

2. Development of the framework

Xu introduced the modified log-Harnack inequality in 2011 and proved that it implies the asymptotically strong Feller property; the motivating application was the 2D stochastic Navier–Stokes equation driven by highly degenerate but essentially elliptic noise (Xu, 2011). In that setting the inequality already had the now-standard architecture: a polynomial control in $t\to\infty$8 and a gradient-dependent remainder decaying exponentially in time.

Bao–Wang–Yuan extended the theory to stochastic systems of infinite memory. They established asymptotic log-Harnack inequalities for non-degenerate SDEs, neutral SDEs, semilinear SPDEs, and stochastic Hamiltonian systems on segment spaces $t\to\infty$9, and derived asymptotic heat kernel estimates, uniqueness of invariant probability measures, asymptotic gradient estimates, asymptotically strong Feller, and asymptotic irreducibility (Bao et al., 2017).

For nonlinear monotone SPDEs with multiplicative noise, Hong–Li–Liu derived an explicit asymptotic log-Harnack inequality by asymptotic coupling via change of measure. Their results cover both non-degenerate monotone SPDEs and highly degenerate finite- and infinite-dimensional diffusion systems, and they identify asymptotically strong Feller, asymptotic irreducibility, and unique ergodic invariant measures under dissipativity (Liu, 2020).

Subsequent work broadened the framework in several directions: Markovian lifts of stochastic Volterra integral equations with completely monotone kernels (Hamaguchi, 2023), path–distribution dependent SDEs with infinite memory and time-space singular or Dini drifts (Wang et al., 19 Feb 2025, Zhao, 12 Jul 2025), and degenerate SPDEs with reflection, including reflected stochastic Navier–Stokes dynamics (Li et al., 3 Mar 2026). The unifying theme is that asymptotic log-Harnack inequalities survive in regimes where either classical Harnack inequalities or strong Feller estimates fail.

3. Coupling by change of measure

The dominant proof strategy is asymptotic coupling by change of measure. In the monotone SPDE setting

$(E,p)$0

one constructs a controlled copy

$(E,p)$1

with control $(E,p)$2 and Girsanov density

$(E,p)$3

If $(E,p)$4, then

$(E,p)$5

and under the tilted measure $(E,p)$6,

$(E,p)$7

Combining Young’s inequality with the Lipschitz control of $(E,p)$8 yields

$(E,p)$9

(Liu, 2020).

The specific coupling mechanism depends on the model. Xu’s Navier–Stokes construction enforces exact synchronization of low modes in finite time by setting $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$0 for $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$1 and $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$2 for $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$3, while high modes decay exponentially under dissipation (Xu, 2011). In infinite-memory systems, the weighted norm

$$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$4

makes it possible to contract the whole segment asymptotically even though coupling in finite time is generally impossible (Bao et al., 2017). For path–distribution dependent SDEs, one Girsanov transform removes the law dependence and a second one drives pathwise contraction; in the Dini-drift case, this is combined with a Zvonkin transform (Wang et al., 19 Feb 2025, Zhao, 12 Jul 2025). In reflected SPDEs, the control acts only on a finite-dimensional subspace $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$5 through $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$6, and the reflection term contributes a nonpositive finite-variation term in the distance estimate (Li et al., 3 Mar 2026).

4. Semigroup consequences

The principal applications are semigroup regularization and ergodic consequences. In the abstract framework of Bao–Wang–Yuan, if

$$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$7

then for $$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$8,

$$P_t \log f(x)\le \log P_t f(y)+\Phi(x,y)+V_t(x,y)\,\|\nabla \log f\|_\infty,\qquad V_t\to0,$$9

If $f\in B_+(E)$0, the semigroup is asymptotically strong Feller. The same theorem yields an asymptotic heat kernel estimate, uniqueness of invariant probability, and asymptotic irreducibility (Bao et al., 2017).

Xu proved directly that the modified log-Harnack inequality implies the asymptotically strong Feller property (Xu, 2011). In monotone SPDEs with multiplicative noise, the asymptotic log-Harnack inequality also yields a dimension-free gradient estimate

$f\in B_+(E)$1

and hence asymptotically strong Feller; if $f\in B_+(E)$2, the semigroup admits a unique invariant probability measure and is ergodic (Liu, 2020). In the reflected SPDE framework, the asymptotic log-Harnack inequality implies the analogues of the gradient estimate, asymptotic heat kernel estimate, and vanishing of $f\in B_+(E)$3 for closed $f\in B_+(E)$4 with invariant measure zero, together with at most one invariant probability measure (Li et al., 3 Mar 2026).

5. Representative classes of stochastic systems

For monotone SPDEs with multiplicative noise, the canonical result is explicit. Under hemicontinuity, local monotonicity, coercivity, growth, and bounded invertible $f\in B_+(E)$5, one has

$f\in B_+(E)$6

and the degenerate product-space extension replaces $f\in B_+(E)$7 by $f\in B_+(E)$8 on the forced component (Liu, 2020). The examples include a degenerate SODE on $f\in B_+(E)$9 with $\|\nabla \log f\|_\infty<\infty$0, $\|\nabla \log f\|_\infty<\infty$1, a dissipative finite-dimensional system with degenerate $\|\nabla \log f\|_\infty<\infty$2, the stochastic $\|\nabla \log f\|_\infty<\infty$3-Laplacian, generalized porous media, and a coupled reaction–diffusion system with $\|\nabla \log f\|_\infty<\infty$4 (Liu, 2020).

For infinite-memory systems, the segment semigroup satisfies

$\|\nabla \log f\|_\infty<\infty$5

under uniform non-degeneracy of $\|\nabla \log f\|_\infty<\infty$6 and suitable monotonicity/Lipschitz assumptions (Bao et al., 2017). This same structure is proved for neutral SDEs, semilinear SPDEs, and stochastic Hamiltonian systems with memory, and the non-decaying term depends only on the present value while the decaying term depends on the full weighted path distance (Bao et al., 2017).

For stochastic Volterra integral equations with completely monotone kernels, Markovian lifting yields an infinite-dimensional SEE on $\|\nabla \log f\|_\infty<\infty$7 and, when $\|\nabla \log f\|_\infty<\infty$8,

$\|\nabla \log f\|_\infty<\infty$9

For fractional kernels, $\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$0, so the time-decaying term is no longer decaying (Hamaguchi, 2023).

For path–distribution dependent SDEs, one law-level result is

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$1

which extends the path-dependent distribution-independent inequality to the McKean–Vlasov setting (Wang et al., 19 Feb 2025). In the infinite-memory Dini-drift setting, Wang–Yuan–Zhao obtained asymptotic log-Harnack inequalities both on the segment space and at the level of initial laws, with exponential decay in the remainder term and corresponding Lipschitz continuity in law (Zhao, 12 Jul 2025).

Degenerate fluid models furnish further examples. Xu’s 2D stochastic Navier–Stokes result has the precise form

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$2

(Xu, 2011). For stochastic convective Brinkman–Forchheimer equations with degenerate noise and $\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$3,

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$4

in the additive case, and

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$5

in the multiplicative case (Mohan, 2020). For degenerate SPDEs with reflection,

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$6

so the remainder decays exponentially whenever $\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$7 (Li et al., 3 Mar 2026).

6. Relation to classical log-Harnack inequalities

Classical Wang-type Harnack inequalities and their log-Harnack limit have the form

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$8

typically under non-degenerate noise, and they usually imply the strong Feller property (Xu, 2011). Asymptotic log-Harnack inequalities are weaker: the decaying remainder explicitly records the failure of instantaneous smoothing. What survives is asymptotically strong Feller rather than strong Feller, together with asymptotic rather than uniform heat-kernel and irreducibility statements (Bao et al., 2017, Liu, 2020).

A common misconception is that asymptotic log-Harnack is merely a classical log-Harnack estimate with a looser constant. The literature shows a sharper distinction. In infinite-memory systems, the segment semigroup is not strong Feller because memory destroys instantaneous regularization on path space, and the weighted memory norm is needed to recover only large-time contraction (Bao et al., 2017). In degenerate finite-dimensional diffusion, strong Feller may fail completely: for

$\delta(t)\,\bar C\,|x-y|^\beta |D\log f|$9

the strong Feller property is known to fail, yet the asymptotic log-Harnack inequality still yields asymptotically strong Feller, asymptotic irreducibility, and a unique ergodic invariant measure (Liu, 2020). Another limitation is model-dependent time decay: in the Volterra setting the decay rate is governed by $\Psi_t(x,y)\,\|\log f\|_\infty$0, and for the fractional kernel $\Psi_t(x,y)\,\|\log f\|_\infty$1, the argument does not recover asymptotically strong Feller (Hamaguchi, 2023).

The modern role of the asymptotic log-Harnack inequality is therefore not to replace the classical theory, but to provide a semigroup regularization principle in regimes where non-degeneracy, finite-dimensional ellipticity, or Markovian locality are absent. In that sense it is a long-time functional inequality tailored to degenerate SPDEs, memory equations, Volterra lifts, and McKean–Vlasov path systems.