Self-Similar DSY Cascades

• Self-Similar DSY Cascades are probabilistic branching processes on infinite trees with random branching times and intensities that exhibit scaling invariance.
• The framework uses multiplicative mechanisms and Markovian dependencies to generate power-law tails, multifractal statistics, and phase transitions between explosive and non-explosive regimes.
• Applications span turbulence modeling, rainfall prediction, and PDE representations by providing robust criteria for explosive behavior and measure concentration.

A Self-Similar Doubly Stochastic Yule Cascade (DSY) is a probabilistic branching process constructed on a (typically infinite binary) tree, where both the branching timing and the transition intensities are random and subject to self-similar scaling. DSY cascades generalize the classical Yule process through a second layer of stochasticity: the splitting rates at each node are random variables correlated along a branch, often governed by a Markovian or multiplicative mechanism. Self-similarity manifests when the law of the cascade measure exhibits scaling invariance, supporting features such as power-law tails, phase transitions between explosive and non-explosive regimes, and multifractal statistics. This framework forms the mathematical underpinning for modern models in turbulence, stochastic @@@@1@@@@ representations (notably generalized Navier–Stokes equations), phylogenetic tree structures, and sparse multifractal analysis.

1. Mathematical Formulation of DSY Cascades

DSY cascades are defined on an infinite tree $\mathbb{T}$ (often binary), where every vertex $v$ is assigned a holding time $T_v$ (usually exponentially distributed) and an associated random intensity $\lambda_v$. The holding time to split at vertex $v$ is then $\lambda_v^{-1}T_v$ (Dascaliuc et al., 2021). The sequence $(\lambda_{v})$ is itself random, with dependencies potentially described by a Markov chain or a multiplicative process. For self-similar DSY cascades, the intensity evolves along a branch $s$ by

$X_{s|n} = X_{\theta} R_1 R_2 \cdots R_n,$

where $(R_j)_{j \ge 1}$ are i.i.d. random variables and $X_{\theta}$ is the root state. Typically, the intensity function $\lambda(\cdot)$ satisfies a scaling identity: $\lambda(xy) = \lambda(x)\lambda(y).$ The total (branching) time up to the $n$th generation along a branch $v$ is given by

$$S_n = \sum_{j=0}^{n} \frac{T_{v|j}}{\lambda(X_{v|j})},$$

with the explosion time defined as

$\zeta = \lim_{n \to \infty} \min_{|v| = n} S_n.$

A DSY cascade is called non-explosive if $\mathbb{P}(\zeta = \infty) = 1$ (Dascaliuc et al., 2021, Dascaliuc et al., 2021), i.e., infinite genealogy cannot be realized in finite time; otherwise, it is explosive.

2. Self-Similarity and Scaling Laws

Self-similarity in DSY cascades refers to invariance under scaling the process in time, intensity, or measure. In continuous settings, a measure $\mu(t)$ satisfies the stochastic scaling law (Muzy et al., 2016, Muzy, 2018): $$\mu(s t) \stackrel{\text{law}}{=} s \, e^{\Omega_s} \, \delta_{\Gamma_s} \, \mu(t),$$ where $\Omega_s, \Gamma_s$ are random variables independent of $\mu(t)$, representing scaling factors for the measure and support (e.g., via a Bernoulli mask or cutout) (Muzy et al., 2016). In the tree context, the structure is self-similar under a pruning or rescaling operation, evidenced by invariance in branch statistics and Tokunaga coefficients (Kovchegov et al., 2016). Hierarchical branching processes derived from DSY formulations exhibit scaling exponents for moments or structure functions, e.g.,

$$M_q(t) = \mathbb{E}[\mu(t)^q] \sim t^{\zeta_q}, \quad \text{with} \quad \zeta_q = q + 1 - D - \psi(q),$$

where $D$ is a fractal dimension and $\psi(q)$ the cumulant generating function of the underlying infinitely divisible process.

3. Explosion, Phase Transition, and Criteria

The central dichotomy in DSY cascades is between explosive and non-explosive behavior. Explosion corresponds to finite cumulative branching time along some path, a phenomenon linked with nonuniqueness, potential blowup in associated PDEs, or measure concentration (Dascaliuc et al., 2021, Dascaliuc et al., 2021, Dascaliuc et al., 13 Sep 2025). Criteria are provided through spectral radius bounds in operator formulations: $$T_a f(x) = \frac{\lambda(x)}{a + \lambda(x)} \int f(y) p(x, dy),$$ where $p(x, dy)$ is the transition kernel and $a > 0$. Non-explosion is obtained if $\rho(T_a) < 1/2$ for some $a$ (binary tree); more generally, the critical values of mean offspring or moment conditions on multiplicative ratios $R_j$ control explosion. For self-similar cascades, the moment condition $\mathbb{E}[R_1^b] < 1/2$ (for suitable $b > 0$) yields non-explosive behavior, while $\mathbb{E}[R_{\max}^{-2\gamma}] < 1$ guarantees explosion (for dissipation exponent $\gamma$ in fractional NS cascades) (Dascaliuc et al., 13 Sep 2025). The spatial dimension $d$ and dissipation parameter $\gamma$ partition the regime into non-explosive (deeper supercritical) and explosive (critical) regions.

4. Multifractal, Cantor-Supported, and Doubly Stochastic Structures

DSY cascades can be constructed to support measures on random fractal sets of prescribed Hausdorff dimension (Muzy et al., 2016). By combining an infinitely divisible process (e.g., log-normal, log-Poisson) with a Poisson cutout (Bernoulli mask), the measure is restricted to a random Cantor set, with the parameter $D$ setting the fractal dimension. The probability that a small interval of size $\varepsilon$ covers the support scales as $P(\varepsilon) \sim \varepsilon^{1-D}$, and the distribution of void (gap) sizes behaves as $p(\tau) \sim \tau^{-D}$. In hierarchical tree cascades, similar multifractal scaling is realized via dyadic branching, multiplicative weights, and correlated splitting times. The multifractal spectrum $F(h)$ is linked via Legendre transform to structure function exponents $\zeta_q$ (Muzy, 2018). These constructions highlight the doubly stochastic nature—one layer controlling multiplicative weighting, another modulating support presence.

5. Applications in PDEs, Rainfall, and Network Models

DSY cascades serve as stochastic representations for solutions to nonlinear PDEs, notably fractional Navier–Stokes equations (FNSE, (Dascaliuc et al., 13 Sep 2025)). Solution processes are constructed as expectations over tree-indexed DSY cascades; explosion corresponds to finite-time blow-up or nonuniqueness in the associated equations. In $d = 2$, a closed-form for the FNSE solution process is obtained, featuring explicit geometric factors (products of signs and sine functions over the binary tree), and demonstrating loss of integrability for large initial data—though radial symmetry can ensure continuation beyond the breakdown.

Rainfall modeling leverages DSY analogs through multifractal Cantor-supported cascades that reproduce intermittent precipitation characteristics, dry period duration distributions, and seasonal fractal dimension variability (Muzy et al., 2016). In evolutionary biology, Yule-process-based DSY cascades and urn models explain power-law distributed genus-size frequencies and phylogenetic tree statistics (Lambert, 22 Sep 2024), with extensions to birth–death and preferential attachment. In turbulence modeling, continuous (wavelet-based) cascades present a DSY-compatible framework characterized by stochastic self-similarity, multifractal scaling laws, and leverage effect modeling (Muzy, 2018).

6. Connections, Generalizations, and Historical Perspective

Self-similar DSY cascades generalize numerous classical, deterministic, and discrete-time branching models. Aldous–Shields and Athreya’s percolation cell aging models, Galton–Watson trees, Tokunaga processes, and Simon’s preferential attachment schemes become special or limiting cases of the DSY framework (Kovchegov et al., 2016, Lambert, 22 Sep 2024). Historical analysis clarifies that Yule’s original work anticipated the branching and heavy-tailed statistics foundational to DSY; subsequent interpretations mistakenly attributed tree structures or network models solely to Yule. The doubly stochastic architecture—stochasticity in both arrival of new clusters and internal structure—proves crucial for explaining empirical heavy tails and emergence of scale-free features.

7. Summary Table: DSY Cascade Properties and Regimes

Feature	DSY Cascade Regime	Mathematical Characterization
Branching timing	Doubly randomized	$\lambda(X_v)^{-1} T_v$, $X_v$ Markov/mult.
Self-similarity	Scaling law	$$\mu(st) \stackrel{\text{law}}{=} s \cdot e^{\Omega_s} \delta_{\Gamma_s} \mu(t)$$
Explosion criterion	Explosive / non-explosive	Moment / spectral bounds ($\rho(T_a) < 1/2$, $\mathbb{E}[R^b] < 1/2$)
Multifractal spectrum	Power-law scaling	$\zeta_q = q + 1 - D - \psi(q)$
Cantor/fractal support	Sparsity control	Poisson cutout, box-counting dim. $D$
PDE representation	Solution process, FNSE	Expectation over DSY, stochastic recursion
Application domains	Turbulence, rainfall, trees	Network models, branching processes

Self-Similar DSY cascades provide a unified probabilistic and analytic structure for modeling hierarchical, branching, and self-similar phenomena observed in physical, biological, and mathematical systems. Their capacity to encode both multifractal measures (on fractal supports) and phase transitions between fading and explosive regimes underpins their significance in contemporary research.