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Functional limit theorems for a time-changed multidimensional Wiener process

Yuliia Mishura, René L. Schilling

TL;DR

This work analyzes the asymptotic behavior of a normalized time-changed multidimensional Brownian motion where the time change is driven by the additive functional $S_B(t)=\int_{0}^{t} f(B_s)\,ds$ with $f=1/\lambda$. By developing a functional limit theorem for superpositions of stochastic processes and examining both radial and diagonal octant limits of the intensity $\lambda$, the authors show that the time-changed process $B_{\tau_{nt}}/\sqrt{n}$ converges to a process $W(\nu^{-1}(t))$, a multidimensional skew Brownian motion–type limit, with the time-change $\nu$ encoded by occupation times in the octants. The results cover the case where $\lambda$ is separated from zero and extend to converging radial/diagonal limits, providing a unified framework for the limiting behavior of time-changed Brownian motion in high dimensions. The methodology combines additive-functional analysis, occupation-time arguments, and a robust superposition convergence theorem, offering insight into diffusion limits under irregular and degenerate conductivities and their potential applications to parabolic problems with irregular diffusion coefficients.

Abstract

We study the asymptotic behaviour of a properly normalized time-changed multidimensional Wiener process; the time change is given by an additive functional of the Wiener process itself. At the level of generators, the time change means that we consider the Laplace operator -- which generates a multidimensional Wiener process -- and multiply it by a (possibly degenerate) state-space dependent intensity. We assume that the intensity admits limits at infinity in each octant of the state space, but the values of these limits may be different. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time-changed multidimensional Wiener process. Among the possible limits there is a multidimensional analogue of skew Brownian motion.

Functional limit theorems for a time-changed multidimensional Wiener process

TL;DR

This work analyzes the asymptotic behavior of a normalized time-changed multidimensional Brownian motion where the time change is driven by the additive functional with . By developing a functional limit theorem for superpositions of stochastic processes and examining both radial and diagonal octant limits of the intensity , the authors show that the time-changed process converges to a process , a multidimensional skew Brownian motion–type limit, with the time-change encoded by occupation times in the octants. The results cover the case where is separated from zero and extend to converging radial/diagonal limits, providing a unified framework for the limiting behavior of time-changed Brownian motion in high dimensions. The methodology combines additive-functional analysis, occupation-time arguments, and a robust superposition convergence theorem, offering insight into diffusion limits under irregular and degenerate conductivities and their potential applications to parabolic problems with irregular diffusion coefficients.

Abstract

We study the asymptotic behaviour of a properly normalized time-changed multidimensional Wiener process; the time change is given by an additive functional of the Wiener process itself. At the level of generators, the time change means that we consider the Laplace operator -- which generates a multidimensional Wiener process -- and multiply it by a (possibly degenerate) state-space dependent intensity. We assume that the intensity admits limits at infinity in each octant of the state space, but the values of these limits may be different. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time-changed multidimensional Wiener process. Among the possible limits there is a multidimensional analogue of skew Brownian motion.
Paper Structure (6 sections, 8 theorems, 69 equations)

This paper contains 6 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Asymptotic behaviour of integral functionals of multi-dimensional Brownian motion
  3. Functional limit theorems for a time-changed multidimensional Wiener process
  4. A Functional limit theorem for the superposition of stochastic processes
  5. A functional limit theorem if the intensity $\lambda$ is separated from zero
  6. Functional limit theorems for the case of converging intensity $\lambda$

Key Result

Proposition 1.1

Let $\left\{X_t,\, t\geq 0\right\}$ be a $d$-dimensional Feller process with natural filtration $\mathcal{F}^X$, $\mathcal{F}_t^X = \sigma(X_s, s\leq t)$, and generator $(A, \mathcal{D}(A))$. Assume that the function $\lambda\in C_b(\mathds{R}^d)$ is real-valued and strictly positive. Set and denote by $\tau_t(\omega) = \inf\{u>0: S_X(u,\omega)>t\}$$(\inf\emptyset:=+\infty)$ the generalized right

Theorems & Definitions (18)

  • Proposition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • ...and 8 more