Filtered Collapses in Lévy Processes

• Filtered collapses are interactions between reflected Lévy processes and stochastic multiplicative resets that modulate heavy-tail behavior and yield stationary distributions.
• The mechanism introduces autoregressive recursions and Lindley-type equations, generalizing classical queueing models while preserving the heavy-tail exponent from jump distributions.
• This framework applies transform-based techniques to model state-dependent resets in systems like network buffers, risk theory, and energy storage for practical analytical solutions.

A filtered collapse is the interaction of a reflected Lévy process with stochastic multiplicative resets, or "collapses," occurring at Poisson epochs, resulting in innovative stationary distributions and new transform-based functional equations. The terminology is rooted in Boxma–Kella–Perry's construction of a strong Markov process on the nonnegative real line, where at stochastic reset times the process jumps instantaneously downward to a fraction of its pre-jump value. Filtered collapses provide both a methodological tool for modulating heavy-tail properties and a mathematically tractable model for systems with state-dependent random resets, with deep connections to autoregressive recursions and queuing theory (Boxma et al., 16 Jan 2025).

1. Formal Model: Lévy Reflection with Filtered Collapses

Let $X = \{X_t, t \geq 0\}$ be a real-valued Lévy process with

$$\psi(\theta) = \log\mathbb{E}[e^{-\theta X_1}] = c\,\theta + \tfrac12 \sigma^2\theta^2 + \int_{\mathbb{R}\setminus\{0\}}(e^{-\theta x} - 1 + \theta x \mathbf{1}_{|x| < 1})\,\nu(dx)$$

where $c, \sigma^2$ are drift and diffusion terms and $\nu$ is the jump measure.

Let $W_t = X_t + L_t$ be the Skorokhod reflection at zero, where $L_t = -\min_{0\leq s\leq t}(X_s \wedge 0)$ is the regulator. Collapses occur at independent Poisson epochs $\{ S_n \}$ of rate $\lambda$, at which point the process instantaneously jumps from $Z_{S_n-}$ to $Z_{S_n} = U_n Z_{S_n-}$, with i.i.d. $U_n \sim F \subset [0, 1]$.

Between collapses, the process evolves as the reflected Lévy process, restarting from the most recent post-collapse state.

The infinitesimal generator $A$ of this strong Markov process $\{Z_t\}$ is

$$A g(x) = c g'(x) + \frac12 \sigma^2 g''(x) + \int_{\mathbb{R}\setminus\{0\}}[g(x+y) - g(x) - y g'(x)\mathbf{1}_{|y|<1}]\,\nu(dy) + \lambda(\mathbb{E}[g(U x)] - g(x))$$

where $g$ is any smooth test function vanishing outside $[0,\infty)$.

2. Stationary Distribution and Lindley-Type Recursion

Assuming $\mathbb{P}(U<1) > 0$, the collapse mechanism induces ergodicity: all initial conditions eventually converge to a stationary law. The pre-collapse values $S_n = Z_{S_n-}$ satisfy the autoregressive recursion

$S_n = V_n + (U_n S_{n-1} - Y_n)^+, \quad n \geq 1$

where

• $V_n = W_{T_n}$ is the reflected Lévy process over an exponentially distributed holding time,
• $Y_n = L_{T_n}$ is the accumulated regulator,
• the increments are i.i.d., and independent of $U_n$.

This recursion generalizes the classical Lindley equation: the stationary law $Z^*$ is the limiting law of $S_n$.

3. Functional Equations for the Stationary Law

Let $h(x)$ be the stationary density and $H(x) = \mathbb{P}(Z^* \leq x)$. The stationary forward equation on $(0, \infty)$ is

$$0 = -\frac{d}{dx}\big[ c H'(x) \big] + \frac12 \sigma^2 H'''(x) + \int_0^\infty \left[ H(x+y) - H(x) - y H'(x) \mathbf{1}_{y<1} \right]\nu(dy) + \lambda \left[ \int_x^\infty \frac{1}{y}F(dx/y)\,H'(y)dy - H'(x) \right]$$

For the Laplace–Stieltjes transform $\phi(s) = \mathbb{E}[e^{-sZ^*}]$, define the "busy-period" transform for the reflected process

$$w(s) = \mathbb{E}[e^{-s W_T}], \quad T \overset{d}{=} \mathrm{Exp}(\lambda)$$

with

$w(s) = \frac{\lambda}{\lambda + \psi(s)}$

The key functional equation for the stationary transform is

$$\phi(s) = w(s) \frac{\mathbb{E}[\phi(s U)] - \phi(s)}{1 - w(s)}$$

or equivalently,

$$(1-w(s))\phi(s) = w(s)\,\mathbb{E}[\phi(sU)] - w(s)\phi(s)$$

The equation links the transform at argument $s$ to its value at contracted argument $s U$.

4. Explicit Solutions in Canonical Input Cases

4.1 Spectrally Positive Lévy Process

Suppose $\nu((-\infty,0]) = 0$, i.e., only upward jumps, not a subordinator. The Laplace exponent has right-inverse $\Phi(\lambda)$ defined by $\psi(\Phi(\lambda)) = \lambda$. Then

$w(s) = \frac{\Phi(\lambda)}{\Phi(\lambda) + s}$

The transform equation simplifies to

$$\phi(s) = \frac{\Phi(\lambda) + s}{s}\left( \mathbb{E}[\phi(sU)] - \phi(s) \right)$$

yielding a linear functional equation for $\phi$, solvable via Mellin or Laplace inversion for laws such as Uniform, Beta, or finite-atomic $U$.

4.2 Brownian and Compound Poisson Inputs

For Brownian motion with drift, $X_t = \mu t + \sigma B_t$, the law of $Z^*$ can be given via transforms: $$\phi(s) = \frac{(s-r_1)(s-r_2)}{s(r_1-r_2)} \left( 1 - D_1 \int_{r_2}^s (u - r_2)^{D_2 - 1} (u - r_1)^{-D_2 - 1} du \right)$$ where $r_1, r_2$ and $D_1, D_2$ are explicit functions of the model parameters.

For an M/M/1-type process, $X$ is a compound Poisson with rate $\nu$, jump size Exp($\alpha$), minus deterministic drift $\delta$. The corresponding Laplace exponent and transform equations are again solvable in closed form, yielding explicit expressions for the stationary transform in terms of incomplete Beta functions and poles.

5. Heavy-Tail and Filtering Effects

A principal feature of filtered collapse is its regulation of high excursions: the collapse mechanism "folds back" the heavy tail, yet allows the stationary process to inherit the same power-law index as the input jump distribution, rather than experiencing the typical one-degree shift known from classical M/G/1 queues. In particular, if the jump-size law has regular variation of index $-\alpha \in (-2, -1)$, then

$$\mathbb{P}(Z^* > x) \sim C \,\mathbb{P}(B > x), \quad x \to \infty$$

The introduction of multiplicative resets therefore guarantees the existence of all lower moments while preserving the principal heavy-tail exponent, a property of interest in queueing, risk theory, and stochastic networks.

6. Interpretation and Applications

Filtered collapses provide a unifying framework for models of state-dependent stochastic resets, energy storage with random discharges, and reflected random walks with scaling resets. The transform equations admit numerical or analytic solution for a wide range of Lévy inputs and collapse laws. Notably, the mechanism captures the self-stabilizing effect of random collapses while maintaining analytically tractable steady-state laws.

By incorporating the busy-period transform $w(s)$ and the self-similar map $s \mapsto sU$, filtered collapses allow practitioners to model and analyze systems with mixed random accumulation and multiplicative depletion. Applications encompass storage models, dam processes, network buffer regulation, and any regime where reflected growth and stochastic self-renewal coexist (Boxma et al., 16 Jan 2025).