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Meeting of squared Bessel flow lines and application to the skew Brownian motion

Elie Aïdékon, Chengshi Wang, Yaolin Yu

Abstract

We study the meeting level between squared Bessel (BESQ) flow lines of different dimensions, and show that it gives rise to a jump Markov process. We apply these results to the skew Brownian flow introduced by Burdzy and Chen \cite{burdzy2001local} and Burdzy and Kaspi \cite{burdzy2004lenses}. It allows us to extend the results of \cite{burdzy2001local} and of Gloter and Martinez \cite{gloter2013distance} describing the local time flow of skew Brownian motions. Finally, we compute the Hausdorff dimension of exceptional times revealed by Burdzy and Kaspi \cite{burdzy2004lenses} when skew Brownian flow lines bifurcate.

Meeting of squared Bessel flow lines and application to the skew Brownian motion

Abstract

We study the meeting level between squared Bessel (BESQ) flow lines of different dimensions, and show that it gives rise to a jump Markov process. We apply these results to the skew Brownian flow introduced by Burdzy and Chen \cite{burdzy2001local} and Burdzy and Kaspi \cite{burdzy2004lenses}. It allows us to extend the results of \cite{burdzy2001local} and of Gloter and Martinez \cite{gloter2013distance} describing the local time flow of skew Brownian motions. Finally, we compute the Hausdorff dimension of exceptional times revealed by Burdzy and Kaspi \cite{burdzy2004lenses} when skew Brownian flow lines bifurcate.

Paper Structure

This paper contains 22 sections, 45 theorems, 74 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Structure of the paper
  4. Acknowledgements
  5. BESQ flows
  6. BESQ processes
  7. Martingale measures
  8. Basic facts about BESQ flows
  9. Bifurcation events
  10. BESQ flows with varying parameters
  11. Decomposition of BESQ flows along a flow line
  12. Meeting of flow lines
  13. Meeting of a forward flow line from the left
  14. Meeting of a forward flow line from the right
  15. Meeting of a forward and a dual line
  16. ...and 7 more sections

Key Result

Theorem 1.1

The process $(U(r)-r,\, r\ge 0)$ is a homogeneous Feller process starting from $0$. For any $r>0$, $\frac{U(r)}{r}$ has distribution ${\mathcal{B}}(\frac{2-\widehat{\delta}+\delta}{2},\frac{\widehat{\delta}}{2})$. Conditionally on $U(0)=z>0$, the process $U$ stays constant then jumps at time $\mathr

Figures (7)

  • Figure 1: The black line represents $Y$/$Y^*$ in picture (a)/(b), both starting from $(0,0)$. The blue lines in both pictures represent the ${\rm BESQ}^\delta$ flow $\mathcal{S}$. In picture (b), the black line goes down and the blue line goes up.
  • Figure 2: Schematic representation of the bifurcation point. The black lines represent the flow $\mathcal{S}^1$ and the blue lines represent the flow $\mathcal{S}^2$. Here, $\mathcal{S}^1_{r,x}(a) > \mathcal{S}^2_{r,x}(a-)$ for some $x>r$, where $\mathcal{S}^1$ and $\mathcal{S}^2$ are two $\rm BESQ$ flows of dimensions $\delta_1<\delta_2$.
  • Figure 3: The black line represents a ${\rm BESQ}(\overline{\delta_0}\,|_{r_1}\, \overline{\delta_1})$ flow line $Y=(Y_x,\, x\ge r_0)$. The blue lines represent the ${\rm BESQ}(\overline{\delta_2})$ flow $\mathcal{S}$, both driven by $\mathcal{W}$. The shaded gray area represents $\mathcal{W}_Y^-$, while the white area indicates $\mathcal{W}_Y^+$.
  • Figure 5: The blue lines represent the ${\rm BESQ}^\delta$ flow lines starting from different points $(0, r_i)$, $i=1,2$. The black line represents $Y^0$, the ${\rm BESQ}(\delta \,|_z\, \widehat{\delta})$ flow line starting from $(0, 0)$. The levels $U(r_i)$, $i=1,2$, are the meeting levels as defined in \ref{['def:U']}.
  • Figure 6: The blue lines represent the ${\rm BESQ}(\delta'\,|_0\,\delta)$ flow lines starting from different points $(0, -r_i)$, $i=1,2$. The black line represents $Y^0$, the ${\rm BESQ}(\delta \,|_z\, \widehat{\delta})$ flow line starting from $(0, 0)$. The levels $U(-r_i)$, $i=1,2$, are the meeting levels as defined in \ref{['def:U-']}.
  • ...and 2 more figures

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 80 more