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Superdiffusive limits for Bessel-driven stochastic kinetics

Miha Brešar, Conrado da Costa, Aleksandar Mijatović, Andrew Wade

TL;DR

The paper addresses how a one-dimensional Itô process with a stochastic drift driven by an exogenous Bessel-type noise exhibits anomalous, superdiffusive diffusion. It develops a robust framework based on a decomposition into a drift-driven additive functional and a martingale, proving both a.s. growth-rate exponents and distributional limits for the scaled process; the key exponent is $\frac{1+\gamma+\alpha}{2}$, and the limiting law involves a functional of a squared-Bessel process. The authors establish a general theorem (applicable when an additive functional $A_t$ satisfies a set of asymptotic conditions) and then specialize to squared-Bessel noise $Y\sim {\mathrm BESQ}^{\delta}(y)$ to obtain precise weak limits. They further provide detailed asymptotics for squared-Bessel functionals, including tail bounds and excursion analyses, and discuss extensions to broader exogenous processes with similar asymptotic structure. Overall, the work clarifies how external stochastic drift drives anomalous diffusion in stochastic kinetic systems and delineates when and how such phenomena emerge in continuous-time dynamics.

Abstract

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous Bessel noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove a weak convergence result on the corresponding scale. We show how our result extends to admit, as exogenous noise processes, not only Bessel processes but more general processes satisfying certain asymptotic conditions.

Superdiffusive limits for Bessel-driven stochastic kinetics

TL;DR

The paper addresses how a one-dimensional Itô process with a stochastic drift driven by an exogenous Bessel-type noise exhibits anomalous, superdiffusive diffusion. It develops a robust framework based on a decomposition into a drift-driven additive functional and a martingale, proving both a.s. growth-rate exponents and distributional limits for the scaled process; the key exponent is , and the limiting law involves a functional of a squared-Bessel process. The authors establish a general theorem (applicable when an additive functional satisfies a set of asymptotic conditions) and then specialize to squared-Bessel noise to obtain precise weak limits. They further provide detailed asymptotics for squared-Bessel functionals, including tail bounds and excursion analyses, and discuss extensions to broader exogenous processes with similar asymptotic structure. Overall, the work clarifies how external stochastic drift drives anomalous diffusion in stochastic kinetic systems and delineates when and how such phenomena emerge in continuous-time dynamics.

Abstract

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous Bessel noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove a weak convergence result on the corresponding scale. We show how our result extends to admit, as exogenous noise processes, not only Bessel processes but more general processes satisfying certain asymptotic conditions.
Paper Structure (15 sections, 9 theorems, 53 equations, 1 figure)

This paper contains 15 sections, 9 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Motivation and discussion
  3. Motivation and literature
  4. An example motivated by a self-interacting random walk.
  5. Stochastic kinetic dynamics.
  6. Other stochastic drift models.
  7. Long-time behaviour of stochastic interest-rate models.
  8. Time-inhomogeneous one-dimensional diffusions.
  9. Overview and discussion of the proofs
  10. Strong law with stochastic drift
  11. Asymptotics of certain squared-Bessel functionals
  12. Existence, uniqueness and positivity for SDE
  13. Existence and pathwise uniqueness.
  14. Positivity.
  15. Bessel-type representation \ref{['eq:X_dynamics']} of $X=\sqrt{S}$.

Key Result

Theorem 1.1

Let $f:\mathbb{R}_+\times\mathbb{R}_+\to\mathbb{R}_+$ satisfy ass:parameters and assume $\delta \in (0,\infty)$. Suppose the adapted process $(S,Y,B)$ consists of an $\mathbb{R}$-valued Brownian motion $B$, a squared-Bessel process $Y$ with law ${\mathrm{BESQ}}^{\delta}(y)$ (defined via SDE eq:sq_be started at a deterministic level $S_0\in\mathbb{R}_+$. Then the $\mathbb{R}_+$-valued process $X=(X

Figures (1)

  • Figure 1: Simulated realizations (Euler scheme, step size of $1/10$) of a trajectory $t\mapsto X_t$ of $X$ (with $\gamma =0$, $\alpha =1/2$, i.e. $f(t,y)=\sqrt{y}$, and $\delta=1$) and the graph $t \mapsto t^{p}$ for exponent $p = (1+\alpha+\gamma)/2 = 3/4$ on the time interval $[0,T]$ with $T = 10^5$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.3
  • Proposition 3.1
  • proof
  • Remark 3.2: Squared-Bessel process
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['thm:general']}
  • ...and 9 more