Infinitesimal Generators and Lyapunov Drift Conditions

From an SDE to Its Generator

For a diffusion, the drift and diffusion coefficients describe its local change, but the generator packages their effect on smooth test functions. Applying it to a carefully chosen nonnegative function can reveal whether the process tends to move toward regions where that function is smaller. Such a function is called a Lyapunov function. The resulting drift inequality is a useful route to moment bounds; by itself, it does not establish every form of stability or a stationary distribution.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). Consider a continuous, adapted, nonexplosive solution of the one-dimensional Itô SDE \(dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t\), with initial value \(X_0\). Here \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient. Whenever an Itô integral \(\int_0^t H_s\,dB_s\) is used, its integrand \(H\) is progressively measurable and belongs to \(\mathcal{L}^2\) on the time interval in question: \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). We assume the solution and coefficients are regular enough for Itô's formula to apply.

To see the source of the formula, Itô's formula gives, for \(f\in C^2\), \(df(X_t)=f'(X_t)\,dX_t+\tfrac12 f''(X_t)\sigma(X_t)^2\,dt\). Substituting the SDE and using \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\) yields the generator term \(Lf(X_t)\,dt\). These multiplication rules are Itô rules; they are not ordinary differential identities.

The semigroup interpretation can be justified directly in a convenient bounded setting. Suppose the coefficients are bounded and continuous, \(f\in C^2\) has bounded derivatives, and \(Lf\) is bounded and continuous. Start the solution at a deterministic \(x\). Then the stochastic-integral integrand \(f'(X_s)\sigma(X_s)\) is bounded and progressively measurable, so it is in \(\mathcal{L}^2\) on every finite interval. Its Itô integral is a square-integrable martingale and has mean zero. Taking expectations in Itô's formula gives \(P_hf(x)-f(x)=\int_0^h\mathbb{E}_x[Lf(X_s)]\,ds\). Continuity of \(X\) at zero and continuity and boundedness of \(Lf\) imply \(\mathbb{E}_x[Lf(X_s)]\to Lf(x)\) as \(s\downarrow0\). Dividing by \(h\) and taking the limit proves the asserted semigroup-generator formula in this setting.

The Drift Inequality and an Expected-Value Bound

The drift inequality has a precise finite-time consequence, provided all terms are integrable. This qualification matters: a stochastic integral with a square-integrable integrand is a martingale, but adding a nonintegrable initial value does not produce an integrable martingale.

Proof. Itô's formula, with \(f=V\), \(f'=V'\), and \(f''=V''\), states \(V(X_t)=V(X_0)+\int_0^t LV(X_s)\,ds+\int_0^t V'(X_s)\sigma(X_s)\,dB_s\), where \(LV=\mu V'+\tfrac12\sigma^2V''\). The last integrand is progressively measurable and satisfies the stated \(\mathcal{L}^2\) condition. By the Itô isometry, its integral is square-integrable; its conditional expectation at a later time given \(\mathcal{F}_s\) equals its value at \(s\), so it is a true martingale. Also \(V(X_0)\) is integrable, and the assumed integrability of \(LV\) permits taking expectations in the identity. Thus \(m(t)=\mathbb{E}V(X_t)\) satisfies \(m(t)=m(0)+\int_0^t\mathbb{E}[LV(X_s)]\,ds\). The integrability assumptions make \(m\) absolutely continuous on finite intervals. The pointwise drift condition gives \(m'(t)\le c-\lambda m(t)\) for almost every \(t\). Multiplying by \(e^{\lambda t}\) gives \(\frac{d}{dt}(e^{\lambda t}m(t))\le ce^{\lambda t}\) almost everywhere. Integrating from \(0\) to \(t\), and using \(\int_0^t ce^{\lambda s}\,ds=\frac{c}{\lambda}(e^{\lambda t}-1)\), proves the displayed bound. Taking the limit superior proves the final assertion. \(\square\)

The estimate controls the expected value of \(V(X_t)\), not necessarily the whole distribution of \(X_t\). If \(V\) is coercive, large values of the state force large values of \(V\), so the bound limits the probability of being far away: for \(R>0\) with \(a_R=\inf_{|x|\ge R}V(x)>0\), Markov's inequality gives \(\mathbb{P}(|X_t|\ge R)\le \mathbb{E}V(X_t)/a_R\). This does not by itself prove convergence to an invariant distribution, uniqueness of such a distribution, or ergodicity.

Worked Generator Calculations

What the Condition Does—and Does Not—Say

A negative term \(-\lambda V\) in the generator offsets the constant input \(c\). The theorem quantifies this balance: the initial contribution is multiplied by \(e^{-\lambda t}\), while the long-time upper bound is \(c/\lambda\). A larger \(\lambda\) means stronger negative drift relative to the same \(c\); it does not automatically give a sharper bound if \(c\) changes too.

Common pitfall: omitting the Itô correction. In the Ornstein–Uhlenbeck example, an incorrect ordinary-chain-rule calculation might write \(LV=V'\mu=-4x^2\), forgetting the second derivative term. The correct calculation is \(V'\mu+\tfrac12V''\sigma^2=(-4x^2)+\tfrac12(2)(9)=-4x^2+9\). That missing \(9\) changes the drift inequality and the resulting bound. The diffusion coefficient contributes through its square, even when it is constant.

Another pitfall is overstating the conclusion. The inequality controls \(\mathbb{E}V(X_t)\) under explicit integrability assumptions. It is not, without additional arguments, a proof of pathwise convergence, an invariant-measure theorem, or ergodicity. Conversely, failure of this particular inequality for a chosen \(V\) does not prove instability: it only says that this test function and this drift condition do not furnish the stated bound.