Lyapunov Criteria for Positive Recurrence and Ergodicity

From Local Drift to Long-Run Behavior

A diffusion can have random local fluctuations and still spend most of its long-term evolution in a bounded region. A Lyapunov function makes this tendency quantitative: outside a chosen compact set, its expected instantaneous change is negative. To turn that drift statement into recurrence and ergodicity, one also needs the process to be able to communicate between regions of the state space and to have enough regularity on the compact set.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a Brownian motion $B_t$. Consider a time-homogeneous diffusion on $\mathbb{R}^d$, $dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t$, with initial state $X_0=x$. Here $b$ is the drift and $\sigma$ is the diffusion coefficient. Assume the coefficients are locally Lipschitz and that the solution is nonexplosive, meaning it is defined for every finite time almost surely. The solution may be considered strong or weak; the recurrence criterion below concerns its transition probabilities and does not, by itself, assert pathwise uniqueness or uniqueness in law.

For a twice continuously differentiable function $f$, the generator is $\mathcal{L}f=b\cdot\nabla f+\tfrac12\operatorname{tr}(\sigma\sigma^\top D^2f)$. Ito's formula has the correction term $\tfrac12\operatorname{tr}(\sigma\sigma^\top D^2f)\,dt$; this term comes from $dB_t\,dB_t=dt$, while $dB_t\,dt=0$ and $dt\,dt=0$. The Ito integrals in this formula are defined for progressively measurable integrands in $\mathcal{L}^2$, that is, with $\mathbb{E}\int_0^T |H_s|^2ds<\infty$, on each finite interval where that condition holds. In local arguments one first stops the process so the integrand is square-integrable. A stopped Ito integral is then a martingale; without an integrability or localization argument, an Ito integral should only be called a local martingale.

Definition: A Lyapunov function for this criterion is a function $V\in C^2(\mathbb{R}^d)$ with $V(x)\ge1$, used to control the process at large distances. It is called coercive if $V(x)\to\infty$ as $|x|\to\infty$; equivalently, each sublevel set $\{x:V(x)\le R\}$ is compact.

The Foster–Lyapunov Criterion

Theorem (Foster–Lyapunov criterion, standard Harris recurrence theorem): Suppose the diffusion is nonexplosive and Feller, and there are a coercive $V\in C^2(\mathbb{R}^d)$, a compact set $C$, and constants $c>0$, $b_0<\infty$ such that $$ \mathcal{L}V(x)\le -cV(x)+b_0\mathbf{1}_C(x),\qquad x\in\mathbb{R}^d. $$ Assume also that the diffusion is irreducible with respect to a nonzero measure $\psi$, and that $C$ is petite (a minorization condition for its transition probabilities). Under the standard aperiodicity condition, the diffusion is positive Harris recurrent: it has a unique invariant probability measure $\pi$, and its transition law converges to $\pi$ in total variation from every initial state. This is a standard Harris recurrence result; its proof uses technical recurrence and minorization arguments beyond this tutorial.

Here irreducibility means that every set $A$ with $\psi(A)>0$ is accessible: from every state $x$, the process has positive probability of entering $A$ at some positive time. A useful sufficient condition in many smooth, uniformly elliptic diffusions is positivity of transition densities. A compact set is small if, for some $t_0>0$, $\varepsilon>0$, and probability measure $\nu$, $P_{t_0}(x,A)\ge\varepsilon\nu(A)$ for all $x\in C$ and measurable $A$; small sets are petite. Compactness alone does not imply this minorization. Regularity and nondegeneracy conditions are commonly used to verify it.

The drift inequality separates two jobs. Outside $C$, the term involving $b_0$ vanishes, so $\mathcal{L}V\le-cV$: the process is pushed back toward lower values of $V$. Inside $C$, the generator may be positive, but the positive contribution is bounded. The compact region is therefore allowed to contain locally unfavorable behavior; its role is not to force negative drift everywhere. Irreducibility and the petite-set condition ensure that repeated returns do not leave the process trapped in disconnected or inaccessible parts of the state space.

Why the Drift Controls Return Times

Proposition: Under the hypotheses above, if $x\notin C$ and $\tau_C=\inf\{t\ge0:X_t\in C\}$, then $$ \mathbb{E}_x[e^{c\tau_C}]\le V(x), \qquad \mathbb{E}_x[\tau_C]\le \frac{\log V(x)}{c}. $$

Proof. Until $\tau_C$, the drift condition gives $\mathcal{L}V(X_s)\le-cV(X_s)$. Apply Ito's formula to $e^{cs}V(X_s)$, stopped at $t\wedge\tau_C$ and at the first exit $\rho_n$ from a ball of radius $n$. Its drift is $e^{cs}(cV+\mathcal{L}V)(X_s)\,ds$, which is nonpositive before $\tau_C$. On the stopped compact region and finite time interval, the stochastic integrand is square-integrable, so its Ito integral is a true martingale with expectation zero. Consequently, $$ \mathbb{E}_x\!\left[e^{c(t\wedge\tau_C\wedge\rho_n)} V(X_{t\wedge\tau_C\wedge\rho_n})\right]\le V(x). $$ Nonexplosion and Fatou's lemma let $n\to\infty$, giving the same inequality without $\rho_n$. Since $V\ge1$, this implies $\mathbb{E}_x[e^{c(t\wedge\tau_C)}]\le V(x)$. Let $t\to\infty$ and apply Fatou again to obtain $\mathbb{E}_x[e^{c\tau_C}]\le V(x)$; in particular $\tau_C<\infty$ almost surely. Jensen's inequality gives $e^{c\mathbb{E}_x[\tau_C]}\le\mathbb{E}_x[e^{c\tau_C}]\le V(x)$, which yields the claimed mean bound after taking logarithms. $\square$

This estimate explains the value of the negative drift: starting farther out may increase the return time, but the Lyapunov function quantifies that cost. The estimate alone does not prove ergodicity. It does not establish communication between different regions or a minorization on returns; those are separate hypotheses in the Harris theorem.

Lemma: If $V$ is coercive and a family of probability measures $\{\mu_\alpha\}$ satisfies $\sup_\alpha\int V\,d\mu_\alpha\le K<\infty$, then the family is tight.

Proof. For any $R\ge1$, Markov's inequality gives $\mu_\alpha(\{V>R\})\le R^{-1}\int V\,d\mu_\alpha\le K/R$. The sublevel set $\{V\le R\}$ is compact by coercivity, and its complement has probability at most $K/R$, uniformly in $\alpha$. Given any $\varepsilon>0$, choosing $R>K/\varepsilon$ supplies a compact set with mass at least $1-\varepsilon$ for every $\mu_\alpha$. This is tightness. $\square$

Coercive growth is important because a bound on $V(X_t)$ must control escape in the state variable. If $V$ stays bounded along a sequence with $|x|\to\infty$, its values cannot provide that control. In applications, drift estimates often yield bounds on expected $V(X_t)$; coercivity then turns those bounds into tightness estimates. Tightness is useful in existence arguments, but it is not a substitute for the irreducibility and petite-set assumptions needed for uniqueness and convergence.

Worked Drift Checks

Worked Example: The Ornstein–Uhlenbeck Drift

Consider $dX_t=-X_t\,dt+dB_t$ on $\mathbb{R}$, and take $V(x)=1+x^2$. The function is $C^2$, satisfies $V\ge1$, and is coercive. Its derivatives are $V'(x)=2x$ and $V''(x)=2$. Ito's formula gives $dV(X_t)=2X_t(-X_t\,dt+dB_t)+\tfrac12(2)\,dt$, hence $\mathcal{L}V(x)=-2x^2+1$.

Choose $c=1$. Then $\mathcal{L}V+V=(-2x^2+1)+(1+x^2)=2-x^2$. Let $C=[-\sqrt{2},\sqrt{2}]$. Outside $C$, $2-x^2\le0$; inside $C$, $2-x^2\le2$. Therefore $\mathcal{L}V\le -V+2\mathbf{1}_C$.

Check: at $x=0$, $\mathcal{L}V=1$ and $-V+2=1$, so equality holds in the proposed bound. At $x=2\notin C$, $\mathcal{L}V=-7$ while $-V=-5$, so $-7\le-5$, as required. The coefficients are globally Lipschitz, and the diffusion is nondegenerate, so the usual irreducibility and compact-set minorization conditions apply.

Worked Example: Correcting the Drift Bound for a Cubic Drift

Consider $dX_t=-X_t^3\,dt+\sqrt{2}\,dB_t$ and $V(x)=1+x^2$. The derivatives are $V'(x)=2x$ and $V''(x)=2$. Thus Ito's formula gives $dV(X_t)=2X_t(-X_t^3\,dt+\sqrt{2}\,dB_t)+\tfrac12(2)(\sqrt{2})^2dt$, and the generator is $\mathcal{L}V(x)=-2x^4+2$.

Set $y=x^2$, so $y\ge0$. For $c=1$, $\mathcal{L}V+V=-2y^2+y+3$. Choose $C=\{x:x^2\le3/2\}$. On this set, $0\le y\le3/2$. The concave quadratic $-2y^2+y+3$ has vertex at $y=1/4$, where its value is $-2(1/4)^2+1/4+3=-1/8+1/4+3=25/8$. Outside $C$, $y\ge3/2$; the derivative of the quadratic is $-4y+1\le-5$, so it is decreasing there, and its value at $y=3/2$ is $-2(3/2)^2+3/2+3=0$. Therefore $\mathcal{L}V+V\le(25/8)\mathbf{1}_C$, or $\mathcal{L}V\le -V+(25/8)\mathbf{1}_C$.

Check: at $x=1/2$, $y=1/4\in[0,3/2]$, and $\mathcal{L}V+V=25/8$. The bound with constant $3$ would fail there because $25/8>3$; the corrected constant $25/8$ gives equality. The drift is locally Lipschitz, and this Lyapunov estimate also gives $\mathcal{L}V\le2$, the usual Lyapunov bound used to rule out explosion. With the nondegenerate constant diffusion, the compact-set and irreducibility hypotheses hold.

Worked Example: Why Brownian Motion Fails the Criterion

For $dX_t=dB_t$ on $\mathbb{R}$, use $V(x)=1+x^2$. Its derivatives are $V'(x)=2x$, $V''(x)=2$, and Ito's formula gives $dV(X_t)=2X_t\,dB_t+\tfrac12(2)\,dt$, so $\mathcal{L}V(x)=1$. For any $c>0$, outside any compact set $C$, the desired inequality would require $1\le-c(1+x^2)$, which is impossible because the right side is negative.

Check: at $x=2$, if $2\notin C$, then $\mathcal{L}V(2)=1$, whereas $-cV(2)=-5c<0$. Thus this coercive $V$ cannot satisfy the required negative drift outside a compact set. Brownian motion is irreducible, but irreducibility cannot replace the missing inward drift.

Common Pitfalls and Interpretation

A compact set is not automatically a small set. The drift calculation only bounds behavior outside the compact region. To conclude positive Harris recurrence, the transition probabilities must also mix sufficiently on returns; this is why irreducibility and minorization appear as distinct hypotheses.

Do not omit the indicator. The inequality $\mathcal{L}V\le-cV+b_0\mathbf{1}_C$ permits positive generator values within $C$. Requiring $\mathcal{L}V\le-cV$ everywhere would be a stronger and often unnecessary condition. Conversely, a bound with an incorrectly small $b_0$ is invalid even if the inequality is correct outside $C$; the cubic example shows why the maximum on the whole compact set must be computed accurately.

Drift, tightness, and ergodicity are not synonyms. The drift condition controls returns; coercivity makes bounds on $V$ useful for tightness; irreducibility and minorization support uniqueness and long-run convergence. The Foster–Lyapunov theorem combines these ingredients. It concerns the law under $\mathbb{P}$ of the stated diffusion, not a change to a risk-neutral measure $\mathbb{Q}$.

Check Your Understanding

For each question, identify which part of the criterion is being tested.

For the Ornstein–Uhlenbeck example, compute $\mathcal{L}V+V$ at $x=1$ and decide whether $1\in C$.
In the cubic-drift example, evaluate $\mathcal{L}V+V$ at $x^2=3/2$. Is the point in $C$, and does the expression equal zero?
Explain why coercivity of $V$ turns a uniform bound on $\int V\,d\mu_\alpha$ into tightness.
Can irreducibility alone establish positive recurrence when the generator has no negative drift outside a compact set? Explain.
What extra role does a petite-set or minorization condition play beyond the drift estimate?