Stationary Kolmogorov Equation

• The stationary Kolmogorov equation is a second-order elliptic or integro-differential PDE that characterizes invariant measures in Markov and diffusion systems.
• Methodologies such as Lyapunov functions, coupling techniques, and finite element analysis ensure existence, regularity, and uniqueness of stationary solutions.
• This framework underpins applications in invariant measure characterization, homogenization, nonlocal dynamics, and computational modeling in both continuous and discrete settings.

The stationary Kolmogorov equation is a central concept in the theory of stochastic processes, partial differential equations, and mathematical physics, describing the stationary distribution of diffusion, jump, or more general Markovian systems. It appears in both classical and modern probability, underlies the theory of invariant measures for Markov processes, and is foundational for the analysis of Fokker–Planck and related evolution equations in infinite-dimensional and nonlinear regimes.

1. Fundamental Formulations and Interpretation

The stationary Kolmogorov equation refers to a (typically second-order elliptic or integro-differential) equation that governs the invariant law of a Markov process. In the setting of diffusions, the stationary version of the forward Kolmogorov or Fokker–Planck equation is: $L^*\mu = 0,$ where $L^*$ is the formal adjoint of the infinitesimal generator $L$ acting on functions. In $\mathbb R^d$, for a diffusion process with drift $b(x)$ and diffusion matrix $A(x)$, the adjoint operator is typically: $$L^* \mu = \sum_{i,j=1}^d \partial^2_{ij}(a^{ij}(x)\mu) - \sum_{i=1}^d \partial_i(b^i(x)\mu) = 0.$$ Here $\mu$ is a probability measure describing the stationary distribution, and $L$ is the Kolmogorov forward operator: $$L\varphi(x) = \sum_{i,j=1}^d a^{ij}(x)\,\partial_{ij}\varphi(x) + \sum_{i=1}^d b^i(x)\,\partial_i\varphi(x).$$ Probabilistically, solutions $\mu$ characterize the long-time equilibrium or invariant states of the associated Markov process. The equation also extends to nonlocal problems—such as jump processes—where the Kolmogorov–Feller operator includes integral terms (Denisov et al., 2018), as well as to discrete settings like Markov chains, where the stationary equation is replaced by a flux-balance system (Hansen et al., 2023).

2. Existence, Structure, and Regularity of Stationary Solutions

The existence of a stationary solution $\mu$ for the Kolmogorov equation depends on both the drift and diffusion coefficients, as well as the domain and boundary conditions. Key mechanisms are:

• Lyapunov/coercivity conditions: Suitable Lyapunov functions $V$ ensure tightness and control of solutions at infinity, yielding existence of stationary measures under general growth conditions on the coefficients. For the nonlinear Fokker–Planck–Kolmogorov equation,

$$-\nabla \cdot(b(x,\mu)\,\mu) + \tfrac12\,\Delta(A(x,\mu)\,\mu) = 0,$$

existence and uniqueness follows if

$L_\mu V(x) \leq C_1 - C_2 V(x)$

for appropriate $C_1, C_2 > 0$ (Bogachev et al., 2018).

• Regularity and absolute continuity: For diffusions on $\mathbb R^d$ with drift $b(x) = -x + v(x)$, and $v \in L^1(\mu)$, the stationary solution $\mu$ is absolutely continuous with respect to the standard Gaussian measure $\gamma$, i.e., $\mu \ll \gamma$. The density $f = d\mu/d\gamma$ satisfies nontrivial integrability and Orlicz-type log-Sobolev inequalities

$$\int f(x) |\log(f(x)+1)|^\alpha d\gamma(x) \leq C(\alpha) [1 + \|v\|_{L^1(\mu)} (\log(1+\|v\|_{L^1(\mu)}))^\alpha]$$

for all $\alpha < 1/4$, under minimal regularity assumptions (no a priori Sobolev control) (Bogachev et al., 2018).

• Tail behavior and moment control: The Orlicz bound implies sub-Gaussian tail decay for the stationary density. Quantitative decay of level sets can be inferred: for $\lambda\gg 1$,

$$\gamma\{f > \lambda\} \rightarrow 0 \text{ rapidly}.$$

3. Uniqueness of Stationary Measures: Doubling Variables and Contractivity

The problem of uniqueness of stationary solutions—in degenerate or low-regularity settings—can be addressed via the coupling or "doubling variables" approach. For degenerate elliptic operators with drift $b(x)$ and possibly singular diffusion $A(x)$, one considers the squared distance functional

$$q(x, y) = \langle x-y,\, b(x)-b(y)\rangle + \operatorname{tr}[(\Sigma(x)-\Sigma(y))(\Sigma(x)-\Sigma(y))^T],$$

where $A(x) = \Sigma(x)\Sigma(x)^T$ (Bogachev et al., 11 Nov 2025). By constructing a coupling $\pi$ of two candidate stationary measures and analyzing $\int q(x, y) \, d\pi(x, y) = 0$, a strict sign condition (e.g., $q(x, y) > 0$ for $x\neq y$) implies uniqueness: all couplings concentrate on the diagonal, so the measures coincide.

In cases with weaker sign or nondegeneracy hypotheses, uniqueness persists provided certain control quantities (e.g., $$r(x, y) = |(\Sigma(x) - \Sigma(y))^T (x-y)/|x-y||^2$$) are positive and $q(x, y)$ can be bounded accordingly. This PDE-based argument is robust and does not require construction of SDE couplings on the same probability space.

4. Extensions: Nonlocal and Nonlinear Stationary Kolmogorov/FKPP Equations

The stationary Kolmogorov equation framework generalizes to several nonlinear and nonlocal contexts:

• Nonlocal elliptic (FKPP-type) equations: For stationary solutions of the nonlocal Fisher–KPP equation,

$$0 = \Delta u + u (1 - \phi_\sigma * u) \quad \text{on } \mathbb R^d,$$

with small nonlocality (small $\sigma$) and under minimal kernel regularity, only constant solutions $u\equiv 0, 1$ persist among bounded, non-negative stationary states. No additional branches emerge unless a secondary bifurcation threshold is crossed, confirmed via perturbation arguments in weighted Sobolev spaces (Achleitner et al., 2013).

• Nonlinear McKean–Vlasov/Kolmogorov systems: In the fully nonlinear setting,

$$-\nabla \cdot (b(x, \mu^*) \mu^*) + \frac12 \Delta (A(x, \mu^*) \mu^*) = 0,$$

existence, uniqueness, and exponential stability to the stationary state are ensured under smallness/Lipschitz bounds on the coefficients in measure and Lyapunov structure. In the critical regime (e.g., with drift $b(x, \mu) = -x + \varepsilon \int y \, d\mu(y)$), uniqueness breaks down, and basins of attraction correspond to measure invariants (e.g., means) (Bogachev et al., 2018).

• Jump processes and Kolmogorov–Feller equations: For processes with bounded jumps and saturation (e.g., bounded Poisson-driven jump processes), the stationary Kolmogorov–Feller equation is an integral equation for the stationary distribution, and can admit complex, explicitly computable multibranch solutions depending on the ratio of domain width to jump amplitude (Denisov et al., 2018).

5. Numerical Methods and Computational Analysis

Numerical analysis of the stationary Kolmogorov equation—particularly in degenerate or stiff settings—requires specialized discretization:

• Degenerate elliptic problems: For the Jacobi diffusion with unsteady drift, the stationary Kolmogorov equation is degenerate elliptic, requiring monotone finite difference stencils for convergence to the viscosity solution. The accuracy of such schemes is highly sensitive to regularity of the boundary data, with higher-order schemes only effective when the boundary values are sufficiently smooth (Yoshioka, 6 Jan 2025).
• Finite element approximations: For stationary Fokker–Planck–Kolmogorov problems with periodic or homogenization structure, conforming finite element schemes are developed for both divergence- and non-divergence-form equations. In settings with minimal regularity (e.g., coefficients only essentially bounded; Cordes-type conditions), unique $L^2$ stationary measures are obtainable and finite element error estimates can still be proven, with convergence rates depending on regularity and mesh structure (Sprekeler et al., 11 Sep 2024).

6. Discrete and Markov Chain Analogues

The stationary Kolmogorov equation also admits discrete analogues in the theory of continuous-time Markov chains, where the stationary measure $\pi$ for a process with finite jump set $\Omega$ and transition rates $\lambda_\omega(x)$ satisfies the flux-balance system: $$0 = \sum_{\omega \in \Omega} \lambda_\omega(x - \omega) \, \pi(x - \omega) - \sum_{\omega \in \Omega} \lambda_\omega(x) \, \pi(x).$$ For one-dimensional birth-death or finite jump processes, this structure allows explicit recurrence relations, low-dimensional generator-space representations, and systematic algorithms for numerical solution, uniqueness detection, and positivity enforcement (Hansen et al., 2023).

7. Applications and Special Contexts

The stationary Kolmogorov equation underpins a diverse set of applications:

• Invariant measures for diffusions: Characterization of equilibrium distributions for Langevin, Ornstein–Uhlenbeck, and generalized diffusions, with sharp regularity and integrability needed for ergodicity and long-time statistical properties (Bogachev et al., 2018).
• Homogenization theory: The invariant solution to the stationary Kolmogorov equation yields the effective coefficients in periodic homogenization and related upscaling of stochastic processes in random or periodic media (Sprekeler et al., 11 Sep 2024).
• Nontrivial stationary states in fluid mechanics: In the context of 2D Euler flows, a stationary Kolmogorov equation approach reveals the existence of nontrivial stationary structures near canonical flows when global degeneracies occur, e.g., on the square torus $\mathbb{T}^2$, in contrast to rigidity/uniqueness on rectangles (Zelati et al., 2020).

The theory is thus a cornerstone across probability, analysis, computational mathematics, and statistical physics, with ongoing advances in low regularity, degeneracy, and high-dimensional or nonlinear regimes.