Coupled Stochastic Squared Bessel Flows

• Coupled Stochastic Squared Bessel Flows are families of processes defined by coupled SDEs with shared or correlated noise, capturing coalescence and bifurcation phenomena.
• They underpin applications in random matrix theory, stochastic geometry, and interval partition evolutions with explicit transition kernels and joint distributions.
• The flows exhibit rich integrable and Markovian properties that enable advanced analysis of eigenvalue dynamics, local times, and regenerative structures in diffusions.

Coupled Stochastic Squared Bessel Flows are families of stochastic processes governed by coupled stochastic differential equations (SDEs) of squared Bessel (BESQ) type, with the coupling arising via shared or correlated driving noise. Such flows play a foundational role in random matrix theory, stochastic geometry of interval partitions, multi-particle diffusion systems, and the fine analysis of local times and bifurcation structures in diffusions with singularities or interfaces. The interplay between the parameter (index) of the BESQ process, coupling mechanisms (such as shared Brownian motion or space-time white noise), and boundary behaviors leads to a rich array of phenomena, including coalescing and bifurcating flows, jump-Markov processes of meeting points, determinantal integrable structures, and connections to point processes and stable Lévy processes.

1. Stochastic Differential Equations and Coupling Mechanisms

A single squared Bessel process of index $\delta\ge0$, starting at $x\ge0$, satisfies

$$dX_t = \delta\,dt + 2\sqrt{X_t}\,dW_t, \quad X_0 = x,$$

where $W_t$ is a standard Brownian motion. To construct a coupled pair of BESQ processes, $X_t$ and $Y_t$, with controlled correlation $\rho\in[-1,1]$ between their Brownian drivers, the system is given by

$$\begin{aligned} dX_t &= \delta\,dt + 2\sqrt{X_t}\,dW_t,\ dY_t &= \delta\,dt + 2\sqrt{Y_t}\,d\widetilde{W}_t,\ d\langle W, \widetilde{W}\rangle_t &= \rho\,dt. \end{aligned}$$

When both processes are driven by the same Brownian motion ($\rho=1$), the coupling is synchronous and yields contraction in $L^2$: $\mathbb{E}[(X_t(x) - Y_t(y))^2] \le (x - y)^2,$ exhibiting $L^2$–contractivity in Wasserstein-2 distance (Wiśniewolski, 2014). Alternative couplings, such as those constructed via space-time white noise in the context of BESQ flows, enable sophisticated coalescence and bifurcation properties, as well as coupling between flows of distinct indices (Aïdékon et al., 7 Nov 2024, Aïdékon et al., 5 Dec 2025).

2. Stochastic Flow Properties, Semigroup and Markovian Structure

Squared Bessel processes can be viewed as random flows $\Phi_t(x, \omega)$ that map initial data to solutions at time $t$ under a given noise realization $\omega$, possessing a cocycle or semigroup property: $$X_{s+t}(x, \omega) = \Phi_t\bigl(X_s(x, \omega), \theta_s\omega\bigr),$$ where $\theta_s$ is the path-shift operator. The family $\{\Phi_t\}$ thus forms a random dynamical system or stochastic flow (Wiśniewolski, 2014). For coupled flows, this underpins the construction of multi-point processes, reflection couplings (meeting and sticking of paths), and maximal couplings, where two flows coalesce and subsequently evolve identically after a meeting time via a strong Markov restart (Wiśniewolski, 2014, Aïdékon et al., 5 Dec 2025).

Coalescing BESQ flows driven by space-time white noise, in particular, guarantee that when two flow lines meet, they remain merged beyond the point of coalescence, leading to Markovian and regenerative structures in the set of meeting times or levels (Aïdékon et al., 7 Nov 2024, Aïdékon et al., 5 Dec 2025).

3. Explicit Laws: Transition Kernels, Joint Distributions, and Meeting Processes

The BESQ transition density with index $\delta$ is

$$q_t^\delta(x, y) = \frac{1}{2t}\left(\frac{y}{x}\right)^{\alpha/2} \exp\left(-\frac{x+y}{2t}\right) I_\alpha\left(\frac{\sqrt{xy}}{t}\right), \quad \alpha = \frac{\delta}{2} - 1,$$

with $I_\alpha$ the modified Bessel function (Wiśniewolski, 2014, Wu, 2021).

For coupled BESQ flows, the Laplace transform of the joint law for $(X_t, Y_t)$ with correlation $\rho$ is

$$\mathbb{E}\!\left[e^{-\lambda X_t(x) - \mu Y_t(y)}\right]=\exp\left(-x\,u_t(\lambda)-y\,u_t(\mu)- 2\rho\int_0^t\sqrt{u_{t-s}(\lambda) u_{t-s}(\mu)}\,ds\right)$$

where $u_t(\lambda)=\lambda/(1+2\lambda t)$ (Wiśniewolski, 2014).

Meeting levels between two coupled BESQ flow lines of different indices, when driven by the same white noise, are governed by pure-jump Feller Markov processes with explicit Beta-distributed jump measures, constructed via explicit formulas for the one-dimensional transition law and jump distributions. This structure provides a full Markovian description of bifurcations, coalescences, and the evolution of local-time flow for skew Brownian motion (Aïdékon et al., 7 Nov 2024).

4. Multi-Particle and Matrix-Valued Generalizations

Multi-dimensional coupled BESQ systems arise as non-colliding particle systems, the dynamics of eigenvalues in the Wishart/Laguerre ensemble, or the so-called Bessel/Dyson line ensembles. The SDEs for $N$ ordered particles are

$$dY_i(t) = 2\sqrt{Y_i(t)}\,dB_i(t) + \delta\,dt + 4\sum_{j\neq i}\frac{Y_i(t)}{Y_i(t)-Y_j(t)}\,dt, \quad Y_1(t) < \cdots < Y_N(t),$$

with reflecting boundary at $0$ (Wu, 2021, Graczyk et al., 2017, Małecki et al., 2019). Such systems possess unique strong non-colliding solutions under explicit parameter constraints and are contractive under monotone couplings (Graczyk et al., 2017). In the infinite-particle limit or under hard-edge scaling, these become line ensembles governed by determinantal point processes with explicit extended Bessel kernels (Wu, 2021, Benigni et al., 2021).

5. Applications: Random Partitions, Stochastic Geometry, and Interval Evolutions

Recent developments have harnessed coupled stochastic BESQ flows to construct random partitions of $\mathbb{R}_+\times \mathbb{R}$, arising from the bifurcation structure of two flows whose parameters differ by $\delta\in(0,2)$. The partition cells correspond to BESQ$^{-\delta}$ excursions, and are naturally embedded as marks in the jumps of a spectrally positive $(1+\delta/2)$-stable Lévy process; these "spindle" excursions have explicit lifetime laws, yielding Poissonian intensity for the marked point process of cell excursions (Aïdékon et al., 5 Dec 2025).

Moreover, the induced interval partition evolutions realized by slicing the plane at horizontal levels correspond to continuum interval partition diffusions (SSIP$(\alpha)$), with Poisson–Dirichlet $(\alpha,\alpha)$ or $(\alpha,0)$ distributions at fixed levels depending on region. This connects to the stochastic geometry of continuum random trees, stable shredded disks, and regenerative partition structures (Aïdékon et al., 5 Dec 2025).

6. Integrable and Determinantal Structures

BESQ flows admit rich integrable structures: along time-like (fixed index) or space-like (fixed time) paths, the law of the Bessel field is determinantal with explicit correlation kernels derived from hard-edge limits of Laguerre ensembles (Benigni et al., 2021). The Gibbs property in both time and index direction emerges as a non-colliding squared Bessel Gibbsian structure and as uniform measure on interlacing configurations, respectively (Benigni et al., 2021, Wu, 2021).

These structures are crucial for universality results in random matrix theory, especially characterizing the limiting local statistics at spectral hard edges or in multi-level stochastic processes (Małecki et al., 2019).

7. Advanced Boundary Phenomena and Extensions

Coupled BESQ flows with varying index encode non-trivial boundary behaviors, especially near zero:

• For $\delta<2$ processes are absorbed or reflect instantaneously at 0; for $\delta\ge2$ they stay strictly positive except possibly at the origin.
• Couplings between flows of dimension $\delta$ and $\delta'$ driven by the same white noise guarantee synchronization and yield explicit descriptions of the meeting processes, bifurcation times, and the full regenerative structure of certain exceptional sets (Aïdékon et al., 7 Nov 2024, Aïdékon et al., 5 Dec 2025).

Key extensions include time-inversion identities, generalized Lamperti correspondences between geometric Brownian motion and BESQ processes, and constructions of reflection or maximal couplings via time-shifted restarts post-coalescence (Wiśniewolski, 2014).

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Selected References

• “Inside the nature of squared Bessel process” (Wiśniewolski, 2014)
• “Meeting of squared Bessel flow lines and application to the skew Brownian motion” (Aïdékon et al., 7 Nov 2024)
• “Stochastic Flows and Marked Stable Processes” (Aïdékon et al., 5 Dec 2025)
• “The Bessel Line Ensemble” (Wu, 2021)
• “Determinantal structures for Bessel fields” (Benigni et al., 2021)
• “On squared Bessel particle systems” (Graczyk et al., 2017)
• “Universality classes for general random matrix flows” (Małecki et al., 2019)