Coupling by Reflection (II)

A general coupling technique for characterizing a broad range of diffusions.

Limitations of Synchronous Coupling

Given a $\kappa$-strongly convex drift $U$, we can apply the synchronous coupling for the diffusion process

\[\begin{align} \mathrm{d}X_t = U(X_t)\mathrm{d}t+\mathrm{d}W_t\notag\\ \mathrm{d}Y_t = U(Y_t)\mathrm{d}t+\mathrm{d}W_t.\notag\\ \end{align}\]

Eliminating the Brownian motion, we obtain a contractivity property

\[\begin{align} \|X_t-Y_t\|\leq \|X_0-Y_0\|^2 \exp(-\kappa t).\notag \end{align}\]

However, we cannot easily obtain the desired contraction when $U$ is not strongly convex. To address this issue, one should consider a more general coupling method based on a specic metric instead of the standard Euclidean metric. The diffusions may not contract almost surely, but rather in the average sense.

Reflection Coupling

Define the coupling time $T_c=\inf \{ t\geq 0 | X_t =Y_t \}$. By definition, we know that $X_t=Y_t$ for $t\geq T_c$ (Chen & Li, 1989) (Eberle, 2011) (Eberle, 2016) (N. Bou-Rabee & Zimmer, 2020). When the drift term $U$ is zero, we observe that $\|X_t-Y_t\|$ remains fixed for any $t$ and synchronous coupling doesn’t induce any contraction.

Let’s explore an alternative coupling where the Brownian motion moves in the opposite direction. We anticipate with some probability the processes will merge [Why?].

\[\begin{align} \mathrm{d}X_t &= U(X_t)\mathrm{d}t+\mathrm{d}W_t\notag\\ \mathrm{d}Y_t &= U(Y_t)\mathrm{d}t+(\mathrm{I} - 2\cdot e_t e_t^{\intercal})\mathrm{d}W_t,\notag\\ \end{align}\]

where $e_t=\mathbb{I}_{[X_t\neq Y_t]}\cdot \frac{X_t-Y_t}{\|X_t-Y_t\|}$ and one can identify that $\widetilde W_t=\int_0^t \big[\mathrm{I} - 2\cdot e_s e_s^{\intercal} \big]\mathrm{d} s$ is also a Brownian motion. In addition, $e_t e_t^{\intercal}$ is the orthogonal projection onto the unit vector $e_t$ [Hint] and you can easily check that $e_t$ is the eigenvector of $\mathrm{I} - 2\cdot e_t e_t^{\intercal}$ with one eigenvalue $-1$..


We first show $\exp(c\cdot t)f(G_t)$ is a supermartingale, where $G_t=\|X_t-Y_t\|$.

Apply Ito’s lemma to $f(G_t)$, where $f$ is a concave function to induce a new distance metric $d_f(X, Y)=f(\|X-Y\|)$ (Eberle, 2011).

\[\begin{align} \mathrm{d} f(G_t)=2f'(R_t)\mathrm{d}W_t+\bigg\{f'(G_t)\cdot \bigg\langle U(X_t)-U(Y_t), \frac{X_t-Y_t}{\|X_t-Y_t\|}\bigg\rangle +2f''(G_t)\bigg\} \mathrm{d}t.\notag \end{align}\]

Assume $\langle U(X_t)-U(Y_t), X_t-Y_t\rangle \leq -\kappa(r) \frac{\|X_t-Y_t\|^2}{2}$, where $\kappa(r)$ is not necessarily positive

\[\begin{align} \bigg\langle U(X_t)-U(Y_t), \frac{X_t-Y_t}{\|X_t-Y_t\|}\bigg\rangle \leq -\frac{1}{2} \cdot G_t \cdot\kappa(G_t). \notag \end{align}\]

Further including the integration factor $\exp(c\cdot t)$, we have

\[\begin{align} \dfrac{\mathrm{d} \bigg[\exp(c\cdot t)f(G_t)\bigg]}{\exp(c\cdot t)}\leq 2f'(R_t) \mathrm{d}W_t + \bigg[-\frac{1}{2} G_t \cdot\kappa(G_t) f'(G_t)+2f''(G_t)+c \cdot f(G_t)\bigg]\mathrm{d}t. \notag \end{align}\]

In other words, it induces a supermartingale when we have

\[\begin{align} -\frac{1}{2} G_t \cdot\kappa(G_t) f'(G_t)+2f''(G_t)+c \cdot f(G_t)\leq 0.\notag \end{align}\]

It implies that a proper $f$ may help us obtain the desired result

\[\begin{align} \mathrm{E}[f(\|X_t-Y_t\|)] \leq f(\|X_0-Y_0\|)\cdot \exp(-c\cdot t).\notag \end{align}\]

How to build such an $f$

A simple case when $c = 0$

We propose to find a $f$ that satisfies

\[\begin{align} f''(G_t)\leq \frac{1}{4} G_t \cdot\kappa(G_t) f'(G_t).\notag \end{align}\]

The worst case is given by $f(R)=\int_0^{R} f’(s) \mathrm{d}s$, where $f’$ is solved by Growall inequality

\[\begin{align} f'(R)&=\exp\bigg\{\int_0^R\frac{1}{4} s \cdot\kappa(G_t) \mathrm{d}s\bigg\}.\notag \end{align}\]

Extention to $c>0$

We aim to obtain the following dimension-independent bound in $R, L\in [0, \infty)$ (Eberle, 2011).

The general idea is to permit strong convexity outside of a ball with a given radius, within which local non-convexity is allowed.

$-\mathbb{I}_{[\|X_t-Y_t\|< R]} L{\|X_t-Y_t\|^2}\leq \langle U(X_t)-U(Y_t), X_t-Y_t\rangle \leq \mathbb{I}_{[\|X_t-Y_t\|\geq R]} K{\|X_t-Y_t\|^2}.$

  1. Chen, M., & Li, S. (1989). Coupling Methods for Multidimensional Diffusion Processes. Annals of Probability.
  2. Eberle, A. (2011). Reflection Coupling and Wasserstein Contractivity without Convexity. Comptes Rendus Mathematique.
  3. Eberle, A. (2016). Reflection couplings and contraction rates for diffusions. Prob. Theo. Related. Fields.
  4. N. Bou-Rabee, A. E., & Zimmer, R. (2020). Coupling and convergence for Hamiltonian Monte Carlo. Annals of Applied Probability.