Stochastic Koopman Operator (SKO)

• Stochastic Koopman Operator (SKO) is an expectation-based operator that evolves observables in stochastic systems through conditional expectations, distinguishing state and function spaces.
• It utilizes the generator of a Markov semigroup and the Kolmogorov backward equation to capture spectral properties and non-multiplicative eigenfunction behaviors unlike deterministic operators.
• SKO is applied in rare-event simulation, stochastic control, filtering, and quantum dynamics, with numerical methods like EDMD offering robust data-driven approximations.

The Stochastic Koopman Operator (SKO) is the expectation-based Koopman operator associated with a stochastic process, random dynamical system, or Markov diffusion, and it advances observables by conditional expectation rather than by composition with a deterministic map. In discrete time, a standard form is
[
(Kf)(x)=\mathbb E[f(x_{n+1})\mid x_n=x],
]
while for a time-homogeneous SDE one writes
[
(\mathcal K^t f)(x)=\mathbb E[f(X_t)\mid X_0=x].
]
In this sense, the SKO is the Markov semigroup acting on observables, and its infinitesimal generator is the backward operator in the Kolmogorov backward equation [2101.07330, 2201.12062, 1408.4408].

1. Definition, state space, and observable space

A recurrent feature of the SKO literature is the distinction between the state space and the function space of observables. For deterministic discrete-time dynamics (x_{n+1}=Tx_n), the Koopman operator is the composition operator (Kf=f\circ T). In the stochastic setting, this composition law is replaced by conditional expectation, so the object of evolution is no longer a pointwise pullback of observables but an averaged future observable conditioned on the present state [2507.23498, 2506.21844].

For finite-state Markov chains, the operator becomes especially concrete. If (M={1,\dots,m}) and (p_{ij}=\mathbb P(x_{n+1}=j\mid x_n=i)), then
[
Kf(i)=\sum_{j=1}^m p_{ij}f(j),
]
so on (L^2(M,\mu)\cong \mathbb R^m) the SKO acts as multiplication by the transition matrix (P=[p_{ij}]{i,j=1}^m) [2507.23498]. For continuous-time diffusions, the same operator-theoretic idea appears as a one-parameter semigroup ({\mathcal K^t}{t\ge 0}), with
[
\mathcal K^t f(x)=\mathbb E[f(X_t)\mid X_0=x].
]

The distinction between states and observables is not merely formal. A recent discussion of partial observation in stochastic systems emphasizes that, once noise is present, one must not identify the action of the Koopman operator with state evolution itself: the SKO evolves functions and returns conditional expectations, and this distinction becomes essential under hidden variables and delay-coordinate reconstructions [2506.21844].

2. Semigroups, generators, and backward equations

For time-homogeneous SDEs
[
dX_t=A(X_t)\,dt+B(X_t)\,dW_t,
]
the SKO is the Markov semigroup of the diffusion, and its generator coincides with the infinitesimal generator of the SDE. In the notation used in the rare-event literature,
[
\mathcal A f=\langle A(x),\nabla f\rangle+\operatorname{Tr}[Q(x)\nabla^2 f],\qquad Q(x)=\tfrac12 B(x)B(x)^*,
]
and (\mathcal A) is exactly the generator of the SKO semigroup [2101.07330]. The same identification is used in quantum-mechanical applications, where the stochastic Koopman operator is written as
[
(\mathcal K^{t,\Delta_t}f)(x)=\mathbb E[f(X_{t+\Delta_t})\mid X_t=x],
]
with generator
[
\mathcal L_t f=b(x,t)\cdot \nabla f+\tfrac12 a(x,t):\nabla^2 f,
]
the backward Kolmogorov operator [2201.12062].

This generator viewpoint is central because the Kolmogorov backward equation
[
\frac{\partial \Phi}{\partial t}+\mathcal A\Phi=0
]
is precisely the PDE representation of the SKO semigroup. In rare-event simulation, one writes
[
\Phi(t,x)=\mathcal K^{T-t}f(x),
]
so approximating the SKO directly yields an approximation of the backward solution required for the Doob transform and importance sampling [2101.07330]. In stochastic control, related backward semigroups appear after logarithmic or desirability transforms of the stochastic Hamilton–Jacobi–Bellman equation; there the relevant operator is a backward Kolmogorov-type generator, sometimes augmented by a potential term, and only a specific observable sector is needed for control synthesis [2012.05514].

A controlled Itô setting makes the same structure explicit at the infinitesimal level. For
[
dx_t=f(x_t,u_t)dt+\sum_{1\le y\le r} g_y(x_t,u_t)\,dB_y,
]
the infinitesimal stochastic Koopman action on an observable (\varphi_j) splits into a drift term, an Itô correction, and diffusion-direction terms,
[
d\varphi_j(x_t)=\big(L_u^f\varphi_j(x_t)+L_u^b\varphi_j(x_t)\big)\,dt+\sum_{1\le y\le r}L_u^y\varphi_j(x_t)\,dB_y,
]
which is the starting point for bilinear lifted models and filtering constructions [2505.14369].

3. Spectral theory and the limits of deterministic intuition

In deterministic Koopman theory, many constructions rely on eigenpairs and on the multiplicative closure of eigenfunctions. The stochastic setting retains the spectral viewpoint but alters several of its structural consequences. A recent operator-theoretic analysis proves that for discrete-time deterministic Koopman operators acting by composition, the spectrum on (L^2) is multiplicatively closed, but for the expectation-based stochastic Koopman operator associated with discrete-time stochastic processes this lattice structure fails in general [2507.23498].

The obstruction is fundamental. Deterministic Koopman operators are multiplicative:
[
K(fg)=Kf\cdot Kg,
]
whereas for the SKO
[
K(fg)(x)=\mathbb E[f(x_{n+1})g(x_{n+1})\mid x_n=x]
]
is generally not equal to
[
Kf(x)\,Kg(x)=\mathbb E[f(x_{n+1})\mid x_n=x]\;\mathbb E[g(x_{n+1})\mid x_n=x].
]
Accordingly, product closure of eigenfunctions, power closure, and multiplicative spectral lattices cannot be transferred naively from deterministic theory to the SKO [2507.23498].

This difference has practical consequences for modal interpretation. In one line of work on stochastic oscillators, the eigenfunctions of the backward operator (L_{\bm X}^+) are used to define stochastic asymptotic phase and amplitude. If
[
L_{\bm X}^{+}\overline{Q_1}=\Lambda_1\overline{Q_1},\qquad \Lambda_1=\mu_1+i\omega_1,
]
then the stochastic phase is defined as
[
\Phi(\bm X)=\operatorname{Arg}\,\overline{Q_1}(\bm X),
]
while a dominant real eigenfunction (\overline{Q_2}) defines a stochastic amplitude coordinate. These quantities evolve exponentially in expectation and provide a unified phase–amplitude description for noisy limit cycles, noise-induced oscillations, and semiclassical quantum oscillators [2106.13633].

A different spectral emphasis appears in singular-value-based approaches. For stochastic dynamical systems, recent work formulates the Koopman operator as
[
(Kg)(x)=\mathbb E[g(X_{t+1})\mid X_t=x]
]
but targets its left and right singular functions rather than eigenfunctions. This is motivated by irreversible and non-normal stochastic dynamics, where singular subspaces remain informative even when eigenfunctions are harder to interpret or approximate directly [2507.07222].

4. Numerical approximation and data-driven learning

The modern computational literature on the SKO is anchored by EDMD. The Williams–Kevrekidis–Rowley construction takes snapshot pairs ((x_m,y_m)) and a dictionary ({\psi_1,\dots,\psi_K}), forms
[
\mathbf G=\frac1M\sum_{m=1}^M \mathbf\Psi(x_m)^*\mathbf\Psi(x_m),\qquad
\mathbf A=\frac1M\sum_{m=1}^M \mathbf\Psi(x_m)^*\mathbf\Psi(y_m),
]
and defines the finite approximation (\mathbf K=\mathbf G^+\mathbf A). When the data are generated by a Markov process rather than a deterministic map, the same algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which the paper identifies with the stochastic Koopman setting [1408.4408].

Noise, however, changes the behavior of DMD-type estimators in nontrivial ways. A robust approximation study shows that many standard DMD constructions become biased when observables are noisy or when time-delay coordinates are built from random trajectories. In particular, Hankel DMD is biased for random dynamics because delayed coordinates contain randomness from the dynamics itself. The proposed remedy is a cross-correlation-based “noise resistant / robust DMD” that replaces ordinary snapshot correlations by correlations with a suitable dual observable, and the same idea extends to a robust Hankel DMD capable of using a single observable measured over a single trajectory [2011.00078].

For continuous-time SDEs, generator-based methods are often preferred. In rare-event simulation, gEDMD is used to approximate eigenfunctions of the infinitesimal generator directly from basis functions (\psi_k) and their images (\mathcal A\psi_k), yielding a finite generator approximation
[
K=d\Psi_X\Psi_X^+.
]
This is then combined with regression of the target observable to approximate the backward solution and the Doob transform [2101.07330]. A related equation-based method bypasses trajectory data altogether for polynomializable SDEs: it computes selected Koopman matrix elements directly from the backward generator (\mathcal L^\dagger) using monomial expansions, a resolvent approximation of (e^{\mathcal L^\dagger T}), and extrapolation, and it is compared with EDMD on noisy van der Pol and Ornstein–Uhlenbeck examples [2111.07213].

Recent large-scale operator-learning work shifts from eigenfunctions to singular functions. For stochastic, time-homogeneous Markov processes, a low-rank approximation objective
[
L_{\mathrm{lora}}(f,g)=-2\,\operatorname{tr}(\langle f,g\rangle)+\operatorname{tr}!\bigl(\langle f,f\rangle_{\rho_0}\langle g,g\rangle_{\rho_1}\bigr)
]
is proposed as a stable alternative to VAMP-type objectives for learning dominant singular subspaces of the stochastic Koopman operator without backpropagating through inverse square roots, SVDs, or matrix inversions during training [2507.07222]. RKHS-based online learning forms another strand: it relates Koopman learning to conditional mean embeddings and develops an operator stochastic approximation algorithm with sparsification, but it is best understood as an online stochastic learning method for a projected Koopman operator rather than as a foundational paper on intrinsically stochastic dynamics [2501.16489].

5. Applications: rare events, control, filtering, and quantum systems

One prominent SKO application is rare-event simulation. The SKO semigroup is linked to the Kolmogorov backward equation, and its eigenfunctions are used to approximate
[
\tilde\Phi(t,x)=\sum_{i=1}^N f_i e^{\lambda_i(T-t)}\phi_i(x),
]
which then yields an approximate Doob transform
[
\tilde u(t,x)=B(x)^*\frac{\sum_{i=1}^N f_i e^{\lambda_i(T-t)}\nabla\phi_i(x)}{\sum_{i=1}^N f_i e^{\lambda_i(T-t)}\phi_i(x)}.
]
This construction turns SKO eigenfunctions into a practical mechanism for importance sampling in SDEs, including non-normal, non-gradient, oscillatory, and rank-deficient-noise systems [2101.07330].

In filtering and controlled stochastic systems, SKO ideas can be used to lift nonlinear Itô dynamics into bilinear stochastic models. For controlled nonlinear Itô SDEs, finite-dimensional Koopmanization yields
[
dz_t=\big(A^f(u_t)+A^b(u_t)\big)z_t\,dt+\sum_{1\le y\le r}D_y(u_t)z_t\,dB_y+\sum_{1\le y\le r}F_y(u_t)\,dB_y,
]
and, when the observation map is linear in the lifted observables, this leads to a generalized Riccati-type filtering equation in Koopman coordinates [2505.14369]. By contrast, several control papers are Koopman-based yet not SKO papers in the strict operator-theoretic sense. An EKF–Koopman optimal-control formulation replaces the original stochastic problem by a deterministic certainty-equivalent information-state surrogate and then applies a deterministic Koopman operator to that surrogate; it is therefore conceptually adjacent to SKO but not an expectation-based SKO construction [2402.18554]. Likewise, Koopman-based stochastic MPC for vehicle lateral control treats Koopman approximation error as a stochastic residual in an MPC problem, rather than defining a stochastic Koopman operator for the plant itself [2310.10214].

A further application area is quantum and semiclassical dynamics. By ground-state transformation, Schrödinger operators can be converted into backward generators of diffusion processes, so that quantum spectral problems become spectral problems for a stochastic Koopman generator. The same paper also uses Nelson’s stochastic mechanics and data-driven Koopman approximations to analyze coherent structures and to solve imaginary-time Schrödinger equations through stochastic-control formulations [2201.12062].

6. Scope, generalizations, and competing usages

A central misconception in the recent literature is that every model labeled “stochastic Koopman” refers to the same operator. In the standard operator-theoretic sense, the SKO is the conditional-expectation operator on observables of a stochastic process. Some works explicitly depart from that meaning. The “Stochastic Adversarial Koopman” model, for example, applies learned linear maps to the mean and log-standard deviation of a Gaussian latent code; it is a probabilistic deep Koopman embedding in latent space, not the classical expectation-based SKO on observables of a stochastic process [2109.05095]. Similar caution applies to Koopman-based stochastic control and online learning papers whose stochasticity lies mainly in residual modeling or in stochastic approximation of the learner rather than in the operator definition itself [2310.10214, 2501.16489].

The most explicit generalization of SKO in the current literature is the Distributional Koopman Operator (DKO). Here the state space (M) is replaced by the space of probability measures (\mathcal P(M)), the transfer operator (T_t) propagates distributions forward in time, and the Koopman operator on distributional observables is defined by
[
\mathcal D_t h=h(T_t\pi).
]
For linear observables
[
h(\pi)=\int_M \hat h(x)\,d\pi(x),
]
one has
[
\mathcal D_t h=\int_M \mathcal S_t\hat h\,d\pi(x),
\qquad
\mathcal D_t h=\mathcal S_t\hat h,
]
so the DKO contains the SKO on Dirac measures and extends it to nonlinear observables of distributions, such as variances and products of moments [2504.11643].

Several open problems remain explicit. A recent spectral paper leaves unresolved the question of which stochastic processes, if any, possess a Koopman spectrum with a deterministic-style multiplicative lattice structure [2507.23498]. Distributional Koopman theory identifies observable learning and statistical learning theory on (\mathcal P(M)) as future directions [2504.11643]. Partial-observation work indicates that delay embedding remains beneficial in stochastic systems and reports empirical power-law behavior of reconstruction accuracy with respect to additive-noise amplitude, but treats the relation between that exponent and hidden-variable effects as an open interpretive problem [2506.21844]. A plausible implication is that the future development of SKO theory will depend less on importing deterministic folklore and more on explicitly respecting the Markov-semigroup, function-space, and observation-level structure that stochastic dynamics impose.