Ergodicity in Total Variation

• Ergodicity in total variation is defined as the property where a process converges to a unique invariant measure in the total variation norm.
• Lyapunov functions combined with drift and jump analyses quantify convergence rates and establish rigorous mixing criteria.
• Applications include SDEs, Lévy processes, and integro-differential systems, demonstrating robust methods for handling unbounded coefficients.

Ergodicity in total variation refers to the property of a stochastic process or dynamical system whereby its transition law, starting from any initial condition, converges to a unique invariant probability measure in the topology induced by the total variation (TV) norm. More generally, ergodicity in total variation provides a strong, quantitative measure of mixing and long-term behavior, applicable to a wide range of Markovian and non-Markovian models, including those defined by stochastic differential equations (SDEs), integro-differential operators, and processes with unbounded coefficients or degenerate noise. This article synthesizes the modern theory and methods for establishing total variation ergodicity, including both classical and non-classical approaches.

1. Definition and Framework of Total Variation Ergodicity

A Markov process $(X_t)_{t \geq 0}$ on $\mathbb{R}^d$ (or a more general state space) with transition kernel $P_t(x, dy)$ is ergodic in the total variation norm if there exists a unique invariant probability measure $\pi$ such that, for every initial condition $x$,

$$\lim_{t \to \infty} \| P_t(x, \cdot) - \pi(\cdot) \|_{TV} = 0,$$

where

$$\| \mu - \nu \|_{TV} = \sup_{A} | \mu(A) - \nu(A) | = \frac{1}{2} \int |d\mu - d\nu|.$$

This property is often referred to as strong ergodicity, indicating uniform mixing in the strongest classical probabilistic topology. In the context of stochastic processes governed by integro-differential operators with unbounded coefficients, as in the Lévy-type processes discussed in (Mokanu, 23 Sep 2025), the main question is to establish existence of $\pi$ and quantify the rate of convergence.

2. Integro-Differential Operators and Lévy-Type Dynamics

Let $L$ denote the generator for a process $X_t$ on $\mathbb{R}^d$ of the form

$$L f(x) = \ell(x) \cdot \nabla f(x) + \int_{\mathbb{R}^d \setminus \{ 0 \}} \left[ f(x + u) - f(x) - \mathbf{1}_{|u| < 1} \nabla f(x) \cdot u \right] \nu(x, du),$$

where $\ell(x)$ is a possibly unbounded drift, and $\nu(x, du)$ is a Lévy kernel that may also be unbounded in $x$. Typical examples include diffusion with jumps, stable-like processes, and processes with state-dependent jump rates and directions. The canonical problem is to identify conditions under which the process is strongly ergodic in TV, and, if so, to determine the rate of convergence to equilibrium.

3. Lyapunov Functions and Quantitative Rates

A standard tool is the method of Lyapunov functions. One constructs a function $V(x) \geq 1$, typically of the form $V(x) = 1 + \varphi(x)^p$ (where $p \in (0,1)$ and $\varphi(x)$ is often a norm-like function, e.g., $\varphi(x) = |x|$ for large $|x|$), and seeks to verify a drift inequality: $L V(x) \leq -f(V(x)) + C,$ for some increasing function $f: [1,\infty) \to (0,\infty)$ and constant $C > 0$. This inequality encapsulates a balance between stabilizing terms (the negative drift, possibly arising from $\ell(x)$ or favorable properties of $\nu(x, du)$) and destabilizing terms (which can arise from large jumps or insufficient dissipation at infinity).

If such an inequality holds outside a compact set, and certain accessibility or minorization conditions are satisfied on compact sets, then one obtains not only existence of an invariant measure $\pi$ but also quantitative convergence estimates: $$\| P_t(x, \cdot) - \pi(\cdot) \|_{TV} \leq \psi(t) V(x),$$ where the rate $\psi(t)$ depends on the function $f$ and is often given (up to constants and exponents) by

$$\psi(t) = \left[ \frac{1}{f( F^{-1}(\gamma t)) } \right]^{\delta}, \qquad F(t) = \int_1^t \frac{dw}{f(w)}, \qquad \gamma, \delta \in (0,1).$$

Thus, the explicit rate—polynomial, stretched exponential, or exponential—depends on the growth of $f$ at infinity.

4. Handling Unbounded Coefficients: Drift vs. Jump-Induced Ergodicity

When either $\ell(x)$ or $\nu(x, du)$ exhibits unbounded growth (e.g., $\ell(x) = |x|^\eta$ or $\nu(x, du) = |x|^\beta |u|^{-d-\alpha} du$ for some $\beta > 0, \alpha \in (0,2)$), verifying the Lyapunov drift condition demands careful analysis of the generator. The interplay involves:

• Drift-induced ergodicity: The term $\ell(x) \cdot \nabla V(x)$ provides the required negative drift for large $|x|$. If, for large $|x|$, this dominates any destabilizing effects of the jump mechanism, one obtains

$$L V(x) \leq -C_d \varphi(x)^{\eta + p - 1} + \text{lower order terms},$$

with convergence rate determined accordingly.

• Ball-induced (small jumps) ergodicity: If small-jump contributions (through the second-order Taylor expansion in the Lévy integral) sufficiently dominate, ergodicity can be established even in the absence of strong drift, particularly in symmetric settings ($d=1$ or isotropic kernels).
• Tail-induced (big jumps) effects: In higher-dimensional or stable-like settings, control over the "big jumps" (i.e., the behavior of $\nu(x, du)$ for large $|u|$) is crucial. Additional moment or "tail" conditions are required to ensure that large, infrequent jumps do not destabilize the process.

Inequalities such as (LI1), (LI2a), and (LI2b) (see (Mokanu, 23 Sep 2025)) formalize the comparisons needed between these contributions.

5. Examples and Regimes

Two instructive examples illustrate the application of these criteria:

• Symmetric, bounded-kernel, drift-free case (d=1): With $\ell(x) \equiv 0$ and symmetric, bounded $\nu(x, du)$, the balance between positive and negative small-jump contributions is exact, and ergodicity via the ball-induced mechanism is readily verified.
• Unbounded drift and stable-like jumps: For $\ell(x) = |x|^\eta$, $\nu(x, du) = |x|^\beta |u|^{-d-\alpha} du$, ergodicity is guaranteed if either drift or small-jump stabilization ($\eta < \beta$) dominates over the destabilizing jump tails and "annulus" terms; the Lyapunov inequalities can be explicitly calculated, and the rate is extracted from the dominant term $f$.

These cases demonstrate how either strong drift or sufficient “jump-induced” dissipation ensures total variation ergodicity—provided destabilizing effects are subdominant.

6. Quantitative Rates and Explicit Bounds

The speed of convergence to equilibrium in total variation explicitly depends on the choice of the function $f$ in the Lyapunov drift condition. For instance, if $f(w) = c w^{q}$ (power-law), then

$F(t) \sim \frac{1}{c(1-q)} t^{1-q},$

yielding a rate $\psi(t) \sim t^{-q/(1-q)}$. Under appropriate conditions, exponential ergodicity (i.e., $f(w) \gtrsim w$ for large $w$) is achievable, while subgeometric (e.g., polynomial) rates occur for $f$ slower than linear.

7. Conclusion and Significance

Ergodicity in total variation for Lévy-type processes with unbounded coefficients is established by synthesizing Lyapunov function techniques, explicit generator analysis, and careful balance of stabilizing vs. destabilizing terms. The main contributions are:

• Explicit, verifiable Lyapunov-type criteria ($L V(x) \leq -f(V(x)) + C$), accounting for both drift and jump effects, even in the presence of unbounded coefficients.
• Precise formulas for quantitative convergence rates derived from the growth of $f$.
• Illustration of two main stabilization regimes: drift-induced and jump-induced.
• Examples demonstrating the critical dependence of ergodicity on the relative sizes and signs of the drift, small-jump, and big-jump contributions.
• The framework extends to nonhomogeneous and high-dimensional settings by rigorous decomposition of the generator.

These advances provide robust tools for proving rigorous ergodic properties in complex stochastic models, clarify the impact of unbounded coefficients, and offer pathways to computing mixing rates for practical models in probability, mathematical physics, and applied domains involving Lévy-type processes (Mokanu, 23 Sep 2025).