Infinitesimal Generator Matching

• Infinitesimal generator matching is the explicit identification and parameterization of a Markov process generator to capture precise, infinitesimal dynamics.
• It integrates sublinear semigroup theory, neural network approximations, and numerical schemes to model nonlinear, nonlocal, or data-driven processes.
• This framework underpins applications in generative modeling, PDE analysis, and optimal control by ensuring consistency with stochastic evolution.

Infinitesimal generator matching refers to the explicit identification, parameterization, or approximation of the infinitesimal generator of a Markov process, semigroup, or evolution family, with the aim of characterizing, learning, or leveraging the dynamics through this generator in analytic, computational, or applied contexts. This paradigm enables the construction, analysis, and learning of Markovian models—including nonlinear, nonlocal, or data-driven processes—by matching a target generator with functional, symbolic, or learned representations, ensuring consistency with the desired stochastic evolution at an infinitesimal level.

1. Formal Definition and General Principles

Let $(X_t)_{t\in[0,1]}$ denote a (possibly time-inhomogeneous) Markov process on a Polish space $S$, with transition kernels $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$. The infinitesimal generator $\mathcal{L}_t$ is defined via its action on a suitable domain of test functions $f$:

$$(\mathcal{L}_t f)(x) = \lim_{h\to0}\frac{\mathbb{E}[f(X_{t+h})|X_t=x] - f(x)}{h}$$

with adjoint (forward Kolmogorov) form: $\partial_t\,p_t(x) = (\mathcal{L}_t^* p_t)(x)$ for any density $p_t$. Structural cases include:

• Diffusions: $$\mathcal{L}_t f = b_t(x)\cdot\nabla f + \frac{1}{2}\mathrm{Tr}[\sigma_t \sigma_t^T \nabla^2 f]$$
• Jump processes: $\mathcal{L}_t f(x) = \int (f(y)-f(x)) Q_t(dy;x)$

Infinitesimal generator matching refers to explicitly constructing, parameterizing, or approximating a generator $S$0 such that its associated Kolmogorov equation, semigroup, or evolution coincides with the dynamics of interest.

2. Generator Matching in Sublinear and Nonlinear Semigroup Theory

For sublinear Markov semigroups—families of sublinear, monotone operators $S$1 on a convex cone $S$2 (closed under nonnegative scaling and addition)—infinitesimal generator matching is formalized as follows (Kühn, 2019):

• Strong generator $S$3: domain $S$4 exists in norm}.
• Pointwise generator: existence of limit for each $S$5.

The positive maximum principle (PMP) characterizes operators $S$6 such that $S$7 if $S$8 attains a non-negative maximum at $S$9.

The sublinear extension of the Courrège–von Waldenfels theorem asserts that any sublinear operator $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$0 on $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$1 satisfying the PMP can be matched as

$k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$2

where for each $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$3, $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$4 is a negative-definite symbol (Lévy–Khintchine form). Hence, the generator is matched to a supremum of pseudo-differential operators, and applications include nonlinear HJB equations and sublinear Lévy processes.

The matching construction is algorithmic:

1. For each $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$5, define $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$6, extend via Hahn–Banach, and represent as a supremum over linear functionals.
2. Each linear functional inherits PMP, and thus a Lévy–Khintchine representation.
3. The index set $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$7 and family $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$8 are assembled to cover all pairs $k_{t+h\mid t}(A\mid x)=\Pr(X_{t+h}\in A\mid X_t=x)$9, ensuring uniform boundedness.

The construction yields generator representations underlying viscosity solutions to fully nonlinear PDEs and sublinear stochastic processes.

3. Generator Matching in Learning and Generative Modeling

Generator matching has been introduced as a unifying framework for learning generative models driven by arbitrary Markov processes (Holderrieth et al., 2024). Core steps:

• Specify conditional Markov paths $\mathcal{L}_t$0 (e.g., conditional SDEs) starting from a simple prior.
• Parameterize the corresponding conditional generators $\mathcal{L}_t$1, e.g., by drift, diffusion, or jump rates.
• The marginal generator for the process $\mathcal{L}_t$2 is

$\mathcal{L}_t$3

• A learnable (e.g., neural network) generator $\mathcal{L}_t$4 is trained to minimize a Bregman divergence, typically between conditional fields

$\mathcal{L}_t$5

where $\mathcal{L}_t$6 parametrize the generator action.

This encompasses and extends score matching for SDE-based diffusion models, flow matching for ODEs, and can include discrete and jump process models. Furthermore, superposition and hybridization of generators is enabled by the linearity of the Kolmogorov equation in the generator.

In graph generation (Stephenson et al., 3 Feb 2026), the generator is matched in Frobenius norm to the action of the Laplacian heat kernel generator, and a neural surrogate is trained to approximate this infinitesimal action along a noised diffusion path.

4. Analytical, Numerical, and Approximation Methodologies

A central application of infinitesimal generator matching is in the numerical approximation of nonlinear, convex, or sublinear Markov semigroups and their value functions. Stability and convergence can be established under generator convergence in appropriate topologies (Blessing et al., 2023):

• For convex monotone semigroups $\mathcal{L}_t$7 on function spaces, the "Γ-generator" or infinitesimal generator $\mathcal{L}_t$8 is defined via the mixed topology.
• Discrete Euler, Yosida regularizations, and Markov-chain approximations are matched to limiting generators if

$\mathcal{L}_t$9

on a core, ensuring convergence of semigroups and value functions.

In molecular dynamics, discretization of continuous diffusion generators to coarse-grained state spaces uses square-root approximations, matching the infinitesimal generator to a low-dimensional rate matrix by Voronoi projection, with entries given by geometric means of equilibrium weights (Donati et al., 2017).

In optimal control, kernel-based approaches estimate the infinitesimal generator from data and embed it into a reproducing kernel Hilbert space, matching sample-based empirical regression to the action of the true or controlled generator (Bevanda et al., 2024).

5. Infinite-Dimensional and Singular Settings

Generator matching is essential in the study of singular stochastic PDEs, such as the stochastic Burgers equation (Gubinelli et al., 2018). Due to singular drift terms, standard smooth test functions fail to lie in the domain of the generator. The resolution is to construct a "paracontrolled" domain: test functions are taken as Fock-space analogues of controlled distributions,

$f$0

where $f$1 is the regular (linear) part and $f$2 the singular (Burgers) part. On this domain, the infinitesimal generator can be matched to the semigroup, and a full martingale well-posedness theory is established.

6. Matching Infinitesimal Generators with Nonlinear and Unbounded Evolution

For general evolution operators (possibly unbounded), explicit matching between generator $f$3 and evolution operator $f$4 can be given via a generalized operator logarithm. Under suitable resolvent and spectral conditions, $f$5 can be represented as

$f$6

where $f$7 are Dunford–Riesz integrals over resolvent-based monomials of $f$8 (Iwata, 2021). This analytic log-calculus enables explicit identification in nonlinear, infinite-dimensional, and integrable contexts such as the Cole–Hopf transform and KdV hierarchy.

7. Applications and Theoretical Implications

Infinitesimal generator matching plays a foundational role in:

• Nonlinear PDE theory: enabling representation and analysis of fully nonlinear Hamilton–Jacobi–Bellman equations in both analytic and approximation frameworks (Kühn, 2019).
• High-dimensional generative modeling: unifying flow and diffusion model training, enabling Markov superpositions and multimodal generator constructions via direct infinitesimal matching (Holderrieth et al., 2024).
• Numerical stability and convergence: providing the key analytic link between discretized schemes and limiting models for nonlinear, convex, or uncertain evolution, sidestepping viscosity solutions by direct generator matching (Blessing et al., 2023).
• Operator-theoretic control: learning, regularizing, and implementing optimal feedback policies by regressing infinitesimal generator action in reproducing kernel Hilbert spaces (Bevanda et al., 2024).
• Infinite-dimensional and singular stochastic processes: constructing viable generator domains and solution theories in the presence of singular nonlinearities (Gubinelli et al., 2018).

The main theoretical implication is that infinitesimal generator matching provides the minimal analytic and computational interface between Markovian dynamics and their finite-time, statistical, or synthesized behaviors, with a unifying structure across disciplines and model classes.