Kunita Flows in Stochastic Geometry

Updated 15 December 2025

Kunita flows are stochastic diffeomorphisms defined by SDEs with spatially inhomogeneous drift and noise, establishing a framework for random dynamical systems.
They leverage geometric constructs such as Riemannian metrics and invariant measures to analyze stability, entropy, and long-term behavior in stochastic systems.
Extensions to include jump processes and applications in shape analysis highlight the versatility of Kunita flows in modeling fluid mechanics and complex biological systems.

A Kunita flow is a stochastic flow of diffeomorphisms generated by the solution of a Stratonovich (or equivalently, Itô) stochastic differential equation (SDE) with spatially inhomogeneous drift and “Eulerian” (position-dependent) noise fields. Kunita flows are foundational in the mathematical theory of stochastic differential geometry, random dynamical systems, and their application to physical models such as fluid mechanics and random shape evolutions. Distinguished by their structure-preserving properties, Kunita flows provide the analytic underpinning of modern approaches to stochastic transport, geometry, and variational random processes.

1. Definition and Stochastic Differential Equations

Let $D \subseteq \mathbb{R}^d$ or a manifold $M$ , and $(\Omega, \mathcal{F}, P)$ be a probability space carrying a $m$ -dimensional Wiener process $W_t = (W_t^1, ..., W_t^m)$ . A Kunita flow $\varphi_t: D \rightarrow D$ (or $\varphi_{s,t}:M\to M$ ) is the strong solution to the SDE in Stratonovich form: $d\varphi_t(x) = b_t(\varphi_t(x)) \, dt + \sum_{i=1}^m \sigma_{i,t}(\varphi_t(x)) \circ dW_t^i, \quad \varphi_0(x) = x$ where $b_t \in \mathcal{X}(D)$ is the drift vector field and $\{\sigma_{i,t}\} \subset \mathcal{X}(D)$ are time-dependent noise fields. In components: $d\varphi_t^j(x) = b_t^j(\varphi_t(x))dt + \sum_{i=1}^m \sigma_{i,t}^j(\varphi_t(x)) \circ dW_t^i$ The equivalent Itô representation corrects the drift by

$\hat{u}_t = b_t + \frac{1}{2}\sum_{i=1}^m (\nabla \cdot \sigma_{i,t})\sigma_{i,t},$

ensuring existence of a flow of $C^l$ -diffeomorphisms (for vector fields of suitable smoothness) that satisfies the semigroup (cocycle) property $\varphi_{s,t}\circ\varphi_{t,u} = \varphi_{s,u}$ and has adaptation/measurability structure compatible with filtration $(\mathcal{F}_t)$ (Sommer et al., 12 Dec 2025).

2. Geometric Structure and Invariant Measures

If the collection $\{\sigma_{i,t}(x)\}$ spans $\mathbb{R}^d$ at every $x$ (ellipticity), the flow induces a time-dependent (or static, if noise is constant) Riemannian cometric and metric: $g_t^*(x) = \sum_{i=1}^{m} \sigma_{i,t}(x)\otimes\sigma_{i,t}(x), \quad g_t^{jk}(x) = \sum_{i=1}^m \sigma_{i,t}^j(x)\sigma_{i,t}^k(x).$ This metric underpins the Laplace–Beltrami operator $\Delta_{g_t}$ and Levi-Civita connection $\nabla^{g_t}$ , fundamental for the probabilistic and variational analysis of the flow (Grong et al., 2022). The generator acting on functions is

$L_t f = \frac{1}{2} \Delta_{g_t} f + z_t\cdot \nabla f,$

where $z_t$ incorporates drift and Itô–Stratonovich corrections.

Under additional stationarity and integrability conditions, Kunita flows are random dynamical systems (RDS) for which invariant measures absolutely continuous with respect to Lebesgue measure exist, and Lyapunov exponents can be defined almost everywhere (Biskamp, 2012).

3. Variational Principles and Most Probable Flows

The Kunita SDE framework enables a geometric Onsager–Machlup variational principle for pathwise likelihood of observing a trajectory $\gamma$ : $S[\gamma] = \frac{1}{2}\int_0^T \|\dot{\gamma}_t - z_t(\gamma_t)\|^2_{g_t}dt + \int_0^T f_t(\gamma_t)dt,$ with the scalar function $f_t$ encoding divergence, scalar curvature, and metric time-derivative terms. Minimizers of $S$ are characterized as most probable paths (MPPs), satisfying Euler–Lagrange (geodesic-type) ODEs with Christoffel symbols and noise-induced correction terms: $(D^t/dt)(\dot{\gamma}_t) + (\nabla^t_{\dot{\gamma}_t}z_t) - (\dot{z}_t + \nabla^t_{z_t}z_t) + \text{adjoint/metric-time terms} = \nabla^t f_t$ (Grong et al., 2022). In the limit of vanishing noise, the action reduces to deterministic drift minimization, while for non-trivial noise, path geometry is influenced by the stochastic Riemannian metric—most probable paths deviate from deterministic flows due to noise-induced "curvature" effects.

4. Decomposition, Extensions, and Jumps

Kunita flows on manifolds can be extended to semimartingales with jumps, typically interpreted in the Marcus sense: at each jump time, the solution instantaneously flows along the vector field for the jump size. The Itô–Ventzel–Kunita formula generalizes to this context for SDEs and flows with jumps, enabling a chain rule for composition and facilitating decomposition of flows (Melo et al., 2015, Lima et al., 3 Jan 2025).

Given a manifold $M$ with complementary distributions $\Delta^H \oplus \Delta^V = TM$ (e.g., foliations), a flow $\varphi_t$ admits, up to a stopping time, a local decomposition: $\varphi_t = \xi_t \circ \psi_t, \quad \xi_t \in \mathrm{Diff}(\Delta^H, M),\; \psi_t \in \mathrm{Diff}(\Delta^V, M),$ with explicit SDEs for each component. The decomposition is determined by local invertibility/Jacobian nondegeneracy, topological invariants like the attainability index, and may fail if the flow reaches geometric obstructions (e.g., failure of transversality) (Lima et al., 3 Jan 2025, Melo et al., 2015).

5. Applications in Shape Analysis and Stochastic Dynamics

Kunita flows act naturally on shape spaces through the action of $\mathrm{Diff}(\mathbb{R}^d)$ or its subgroups on objects such as embedded curves, surfaces, or finite landmark sets. The induced stochastic process $s_t = \varphi_t \cdot s_0$ inherits representation-independence, structure preservation, and equivariance (symmetry) properties. The law is determined entirely by the two-point motion due to the diffeomorphic nature of the flow (Sommer et al., 12 Dec 2025).

For landmark-based shape analysis, finite-dimensional SDEs result from pulling back spatial noise fields under $\varphi_t$ . In evolutionary biology and similar fields, Kunita flows underpin stochastic models of shape change, allowing for inference via bridge sampling and MCMC on model parameters. These models fully preserve the geometry of the underlying shape space and are compatible with infinite-dimensional generalizations.

6. Random Dynamical Systems, Entropy, and Pesin’s Formula

Stochastic flows of Kunita type fit into the framework of random dynamical systems (RDS). Under smoothness, integrability, and absolute continuity of invariant measure, classic results on Lyapunov exponents and entropy extend: Pesin’s formula applies to Kunita flows,

$h_\mu = \sum_{i: \lambda_i > 0} \lambda_i^+,$

linking metric entropy $h_\mu$ to the sum of positive Lyapunov exponents. Key steps involve the construction of stable manifolds, absolute continuity of leafwise conditional measures, and entropy bounds via multiplicative ergodic techniques. Applications include stochastic Ornstein–Uhlenbeck flows and spatially periodic SDEs with smooth coefficients (Biskamp, 2012).

7. Chain Rules and Stochastic Geometry for Differential Forms

The Kunita–Itô–Wentzell formula provides the analytic framework for stochastic transport of general $k$ -forms: $d(\varphi_t^* K)(t, x) = \varphi_t^* G(t, x) dt + \sum_i \varphi_t^* H_i(t, x) \circ dW^i_t + \varphi_t^* (\mathcal{L}_b K)(t, x) dt + \sum_j \varphi_t^* (\mathcal{L}_{\xi_j} K)(t, x) \circ dB^j_t$ (Léon et al., 2019). This formula is crucial in stochastic fluid mechanics for the structure-preserving SALT (Stochastic Advection by Lie Transport) class of SPDEs: it ensures the evolution of advected quantities (volume, vorticity, circulation) remains compatible with the underlying diffeomorphic stochastic flow, preserving symplectic and variational structures in stochastic settings.