Two-dimensional Brownian motion with dependent components: turning angle analysis

TL;DR

We address two-dimensional Brownian motion with correlated components, introducing a model with correlation $\rho$ and diffusivities $D_1,D_2$, and show that turning-angle statistics reveal dependencies beyond the mean-squared displacement. We derive exact increment autocovariances, cross-covariances, and a closed-form angle distribution $p_\alpha(\theta)$; we then obtain the turning-angle density $p_{\hat{\phi}}(\theta)$ via convolution, with a simple uniform case $p_{\hat{\phi}}(\theta)=1/\pi$ when $D_1=D_2$ and $\rho=0$. The paper validates the theory on real data—DJIA and S&P500—and on polystyrene-bead trajectories, and extends the framework to time-varying correlation $\rho(t)$, enabling detection of non-stationary coupling via sliding-window fits. These results provide a coordinate-invariant, probabilistic diagnostic for inter-component dependence in diffusion-like processes and offer practical tools for monitoring changing correlation in financial and physical systems.

Abstract

Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, as well as financial data. Most works dealing with multidimensional Brownian motion consider the different dimensions as independent components. In this article, we investigate a model of correlated Brownian motion in $\mathbb{R}^2$, where the individual components are not necessarily independent. We explore various statistical properties of the process under consideration, going beyond the conventional analysis of the second moment. Our particular focus lies on investigating the distribution of turning angles. This distribution reveals particularly interesting characteristics for processes with dependent components that are relevant to applications in diverse physical systems. Theoretical considerations are supported by numerical simulations and analysis of two real-world datasets: the financial data of the Dow Jones Industrial Average and the Standard and Poor's 500, and trajectories of polystyrene beads in water. Finally, we show that the model can be readily extended to trajectories with correlations that change over time.

Figures (11)

• Figure 1: Representative numerical simulations of two-dimensional Brownian motion trajectories with different correlations. In all panels $D_1 = D_2 = 0.5$, time step $1$ and trajectory length $2^{11}$. (a) Trajectories with $\rho=0$, (b) $\rho=-0.3$ and (c) $\rho=0.6$. Each trajectory is shown with a different color (yellow, blue, or green). For clarity of presentation, trajectories were shifted by 50 (yellow) and 100 units (green) to the right. The underlying randomness is the same for all panels with paths with the same color corresponding to equivalent trajectories.
• Figure 2: Contour plots of two-dimensional joint PDFs of the increments of BM with spatial correlations. The histograms on the top and right of the joint PDFs correspond to the increments of each individual component. (a) $\rho=0$, (b) to $\rho=-0.3$ and (c) to $\rho=0.6$.
• Figure 3: Ensemble mean square displacement of correlated BM. The solid blue line corresponds to the case with independent components ($\rho=0$), dashed orange to the negatively correlated case ($\rho=-0.3$), and the dash-dotted green line to the positively correlated case ($\rho=0.6$). For clarity, the MSDs corresponding to $\rho=-0.3$ and $\rho=0.6$ are shifted by 200 and 400, respectively.
• Figure 4: Schematic representation of the polar angles $\alpha_t^\Delta$ (marked with dark green lines) and turning angles $\phi_t$ (marked with dark red lines).
• Figure 5: Distribution of turning angles $\phi_t$ for the process with $D_1 = 2$, $D_2 = 1$, and $\rho = 0.7$. The solid blue histogram corresponds to the angles calculated using eq. (\ref{['eq:angle_def']}) from simulated data, and the dashed yellow line corresponds to $p_{\hat{\phi}}$ from eq. (\ref{['eq:angle_mod_final']}).