{"id":1032,"date":"2026-06-04T08:44:13","date_gmt":"2026-06-04T08:44:13","guid":{"rendered":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?p=1032"},"modified":"2026-06-04T13:30:25","modified_gmt":"2026-06-04T13:30:25","slug":"on-couplings-for-kinetic-langevin-diffusions","status":"publish","type":"post","link":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/","title":{"rendered":"On Couplings for Kinetic Langevin Diffusions"},"content":{"rendered":"<p><em>Joint work with Sonja Cox and Roy Schieven (University of Amsterdam). Paper: <a href=\"https:\/\/arxiv.org\/abs\/2605.31088\">arXiv:2605.31088<\/a>.<\/em><\/p>\n<p>Two copies of the same diffusion, started apart. How can you get them to coalesce? For kinetic Langevin dynamics \u2014\u00a0the SDE behind the diffusive-to-ballistic speedup in modern sampling \u2014 the surprising answer is that the optimal coupling, with respect to the natural joint filtration, is provably non-Markovian: it strictly beats every Markovian strategy in TV. We call this the <em>non-Markovian advantage<\/em>.<\/p>\n<p>This work was announced at the <a href=\"https:\/\/math.ethz.ch\/fim\/activities\/conferences\/scalable-mcmc-sampling.html\">FIM workshop on Scalable MCMC Sampling at ETH Zurich<\/a>, a terrific event organized by Yuansi Chen, Francesco Pedrotti, and Peter Whalley. The program drew researchers from probability, statistics, and machine learning, and the conversations across those communities made the week particularly rewarding. \u00a0I also got a chance to walk around Lake Zurich and hike up Uetliberg.<\/p>\n<h2>1. The setting: sampling via kinetic Langevin<\/h2>\n<p>Fix a target distribution \u03bc_target(dx) \u221d exp(-U(x)) dx on R^d, with U in C^1. Two natural diffusions leave \u03bc_target invariant, possibly after marginalization.<\/p>\n<p><strong>Overdamped Langevin.<\/strong><\/p>\n<p>dX_t = -\u2207U(X_t) dt + \u221a2 dW_t.<\/p>\n<p><strong>Kinetic Langevin.<\/strong> Lift to R^(2d) by introducing a velocity coordinate,<\/p>\n<p>dX_t = V_t dt,<br \/>\ndV_t = -\u2207U(X_t) dt &#8211; \u03b3 V_t dt + \u221a(2\u03b3) dW_t,<\/p>\n<p>which preserves the Boltzmann-Gibbs measure \u03bc = \u03bc_target \u2297 N(0, I_d).<\/p>\n<p>Two structural facts are worth pausing on. First, noise enters only through the velocity coordinate; it reaches the position indirectly, via the kinematic constraint dX = V dt. This makes kinetic Langevin a <em>hypoelliptic<\/em> SDE: the noise covariance is degenerate, yet the law of (X_t, V_t) admits a smooth density in all 2d coordinates for t &gt; 0. Second, kinetic Langevin can be viewed as a stochastic relaxation of Newton&#8217;s equations of motion in a potential U, augmented by friction \u03b3 V_t dt and noise \u221a(2\u03b3) dW_t in fluctuation-dissipation balance.<\/p>\n<h2>2. Why hypoellipticity is worth the trouble<\/h2>\n<p>The degeneracy is exactly what makes the dynamics powerful. Under a Poincar\u00e9 inequality with constant m:<\/p>\n<ul>\n<li>Overdamped Langevin (Bakry, Gentil &amp; Ledoux, 2014, Thm. 4.2.5) relaxes on a <em>diffusive<\/em> timescale T_relax \u224d m^(-1).<\/li>\n<li>Critically-tuned kinetic Langevin (Cao, Lu &amp; Wang, 2023; Eberle &amp; L\u00f6rler, 2024) relaxes on a <em>ballistic<\/em> timescale T_relax \u224d m^(-1\/2).<\/li>\n<\/ul>\n<p>A square-root speedup in ill-conditioned problems: the probabilistic analog of Nesterov acceleration in optimization. This is the motivation for studying kinetic Langevin and, more practically, its splitting discretizations.<\/p>\n<h2>3. From SDE to MCMC: the OBABO splitting<\/h2>\n<p>For sampling, we discretize. The kinetic Langevin generator admits a clean decomposition into three solvable flows:<\/p>\n<ul>\n<li><strong>O_h<\/strong>, the Ornstein\u2013Uhlenbeck step, exact in distribution;<\/li>\n<li><strong>\u03b8^(A)_h<\/strong>, the kinematic flow (X, V) \u21a6 (X + hV, V);<\/li>\n<li><strong>\u03b8^(B)_h<\/strong>, the velocity kick by -h\u2207U(X).<\/li>\n<\/ul>\n<p>The <strong>OBABO splitting<\/strong> (Bussi &amp; Parrinello, 2007; Leimkuhler &amp; Matthews, 2013) assembles them symmetrically,<\/p>\n<p>\u03a6^(OBABO)_h = O_(h\/2) \u2218 \u03b8^(B)_(h\/2) \u2218 \u03b8^(A)_h \u2218 \u03b8^(B)_(h\/2) \u2218 O_(h\/2).<\/p>\n<p>The composition is second-order weakly accurate. Each step is a deterministic map of two Gaussian increments \u03be_k = (\u03be_k^(1), \u03be_k^(2)) ~ N(0, I_(2d)),<\/p>\n<p>Z^h_(k+1) = \u03a8^h_(Z^h_k)(\u03be_(k+1)), Z^h_k = (X^h_k, V^h_k).<\/p>\n<p>The central question is when the corresponding Markov chain (Z_h^k) realizes the acceleration. Writing \u03ba = L\/m for the condition number of the target (m\u00b7I \u2aaf \u2207\u00b2U \u2aaf L\u00b7I), and t_mix(\u03bd, \u03b5) = inf{n : d_TV(\u03bd P^n, \u03bc) \u2264 \u03b5} for the TV mixing time from initial law \u03bd, the paper is motivated by:<\/p>\n<blockquote><p><strong>Question.<\/strong> <em>For which initial laws \u03bd, if any, does OBABO attain accelerated mixing t_mix(\u03bd, \u03b5) \u224d \u221a\u03ba \u00b7 log(1\/\u03b5)?<\/em><\/p><\/blockquote>\n<h2>4. Couplings and total variation<\/h2>\n<p>Two pieces of standard probabilistic machinery underlie our approach.<\/p>\n<p><strong>Coupling characterization of TV.<\/strong> The identity<\/p>\n<p>d_TV(\u03bd_1, \u03bd_2) = inf P(X \u2260 X\u0303),<\/p>\n<p>where the infimum ranges over couplings (X, X\u0303) with X ~ \u03bd_1 and X\u0303 ~ \u03bd_2, reduces upper bounds on TV to the construction of a coupling and a bound on the disagreement probability.<\/p>\n<p><strong>Wasserstein-to-TV regularization.<\/strong> If, for every pair of point masses,<\/p>\n<p>d_TV(\u03b4_z P^n, \u03b4_z\u0303 P^n) \u2264 C_n \u00b7 |z &#8211; z\u0303|,<\/p>\n<p>then by joint convexity of TV along a W_1-optimal coupling,<\/p>\n<p>d_TV(\u03bd P^n, \u03bd\u0303 P^n) \u2264 C_n \u00b7 W_1(\u03bd, \u03bd\u0303).<\/p>\n<p>Combined with Wasserstein contraction in the chain, this transfers Wasserstein control to TV control, which is what one ultimately wants for mixing.<\/p>\n<p>The problem now reads: build a coupling of two OBABO chains started at z and z\u0303 that brings their laws together in TV. Hypoellipticity makes this hard. The very degeneracy that yields the acceleration frustrates attempts to force two copies to meet.<\/p>\n<h2>5. A brief history of hypoelliptic couplings<\/h2>\n<p>The hypoelliptic coupling program has a long arc.<\/p>\n<ul>\n<li><strong>Stochastic oscillator<\/strong> (Ben Arous, Cranston &amp; Kendall, 1995). The first hypoelliptic coupling: a co-adapted switching between synchronous and antithetic Brownian drivers makes two copies of the stochastic oscillator coincide in finite time almost surely.<\/li>\n<li><strong>Kolmogorov diffusion<\/strong> (W_t, \u222b_0^t W_s ds) (Banerjee &amp; Kendall, 2016). The first impossibility theorem: no Markovian coupling matches the asymptotic TV decay rate.<\/li>\n<li><strong>Wasserstein contraction for kinetic Langevin<\/strong> (Eberle, Guillin &amp; Zimmer, 2019). A hybrid coupling \u2014 synchronous on a contractive hyperplane in phase space, reflection transverse to it \u2014 gives quantitative Wasserstein convergence without requiring global convexity of U. Discrete-time analogues followed in work of Cheng et al. (2018, 2020), Dalalyan &amp; Riou-Durand (2020), Leimkuhler-Paulin-Whalley (2024), and Schuh &amp; Whalley (2025).<\/li>\n<li><strong>TV mixing via Wasserstein-to-TV<\/strong> (Roberts\u2013Rosenthal, 2002; Madras\u2013Sezer, 2010; Monmarch\u00e9, 2021; Gouraud et al., 2025; Chak\u2013Monmarch\u00e9, 2025). Hypoelliptic smoothing transfers Wasserstein bounds to TV bounds. Chak &amp; Monmarch\u00e9&#8217;s recent work constructs an explicit coalescence map \u03a8^n_(z, z\u0303) from an ansatz, with the resulting Wasserstein-to-TV bound closed by Lemma 15 of B.-R. &amp; Eberle (2023) \u2013 our starting point below.<\/li>\n<\/ul>\n<h2>6. Preliminaries<\/h2>\n<h3>6.1 A general lemma: TV bound between a reference and a perturbed Gaussian<\/h3>\n<p>Closing a Wasserstein-to-TV bound for OBABO comes down to controlling the total variation distance d_TV between two laws on noise space \u2014 the reference Gaussian Law(\u03be) and its pushforward Law(\u03a8\u207f_(z, z\u0303)(\u03be)) under the coalescence map. The following sharp lemma of B.-R. &amp; Eberle (2023) provides the required bound.<\/p>\n<p><strong>Lemma<\/strong> (<a href=\"https:\/\/arxiv.org\/abs\/2105.00887\">B.-R. &amp; Eberle, 2023<\/a>; Lem. 15). <em>Let \u03be ~ N(0, I_d) and \u03a6 : \u211d^d \u2192 \u211d^d be a C^1 diffeomorphism. Then<\/em><\/p>\n<p><em>d_TV( Law(\u03be), Law(\u03a6(\u03be)) ) \u2264 \u221a(2 KL),<\/em><\/p>\n<p><em>where<\/em><\/p>\n<p><em>KL = E[ (1\/2) |\u03a6(\u03be) \u2212 \u03be|^2 + tr(D\u03a6(\u03be) \u2212 I) \u2212 log|det D\u03a6(\u03be)| ].<\/em><\/p>\n<p><em>Proof sketch.<\/em> Pinsker&#8217;s inequality gives d_TV \u2264 \u221a(KL\/2). A change of variables on the pushforward density yields<\/p>\n<p>KL = E[ (1\/2) |\u03a6(\u03be) \u2212 \u03be|^2 + (\u03a6(\u03be) \u2212 \u03be) \u00b7 \u03be \u2212 log|det D\u03a6(\u03be)| ].<\/p>\n<p>Gaussian integration by parts (Stein&#8217;s identity) handles the cross term:<\/p>\n<p>E[ (\u03a6(\u03be) \u2212 \u03be) \u00b7 \u03be ] = E tr( D\u03a6(\u03be) \u2212 I ),<\/p>\n<p>since for any C^1 vector field F : \u211d^d \u2192 \u211d^d, E[ F(\u03be) \u00b7 \u03be ] = E[ div F(\u03be) ]. Substituting recovers the displayed KL. \u25a1<\/p>\n<p><strong>Remark (OBABO).<\/strong> For the coalescence map \u03a8\u207f_(z, z\u0303) considered next, the i-th coordinate \u03be\u0303_i depends only on \u03be_1, \u2026, \u03be_(i-1), so the Jacobian D\u03a8\u207f_(z, z\u0303) is block lower-triangular with identity diagonal. Hence<\/p>\n<p>tr( D\u03a8\u207f_(z, z\u0303) \u2212 I ) \u2261 0, det D\u03a8\u207f_(z, z\u0303) \u2261 1,<\/p>\n<p>and the lemma collapses to a pure second-moment bound,<\/p>\n<p>d_TV( Law(\u03be), Law(\u03a8\u207f_(z, z\u0303)(\u03be)) ) \u2264 E[ |\u03a8\u207f_(z, z\u0303)(\u03be) \u2212 \u03be|^2 ]^(1\/2).<\/p>\n<p>Choosing the interior gap trajectory y_1, \u2026, y_(n-1) to minimize this second moment is a classical minimum-energy LQ control problem, presented later in \u00a78 \u2014 and the resulting bound is one of our main theorems in \u00a77.1.<\/p>\n<h3>6.2 The coalescence map and its Malliavin derivative<\/h3>\n<p>The three results that follow share a common object \u2014 the coalescence map \u2014 and a common framing in terms of differentiation on noise space. We fix both here.<\/p>\n<p>Write \u03a8\u207f_z : \u211d^(2dn) \u2192 \u211d^(2d) for the <em>chain map<\/em> of OBABO: given a noise sequence \u03be = (\u03be_1, \u2026, \u03be_n), the value \u03a8\u207f_z(\u03be) is the n-step state started at z and driven by \u03be. The <em>coalescence map<\/em> \u03a8\u207f_(z, z\u0303) : \u211d^(2dn) \u2192 \u211d^(2dn) is the corresponding noise transport: given \u03be driving the chain started at z, the image \u03be\u0303 := \u03a8\u207f_(z, z\u0303)(\u03be) is the noise that, applied to the chain started at z\u0303, makes the two chains meet at time n,<\/p>\n<p>\u03a8\u207f_(z\u0303)( \u03a8\u207f_(z, z\u0303)(\u03be) ) = \u03a8\u207f_z(\u03be).<\/p>\n<p>The figure below records the construction via the <em>gap trajectory<\/em> y_k := z\u0303_k \u2212 z_k. The endpoints are fixed by y_0 = z\u0303 \u2212 z and y_n = 0; the interior values y_1, \u2026, y_(n-1) are free, and chosen later by LQ optimization.<\/p>\n<figure id=\"attachment_1033\" aria-describedby=\"caption-attachment-1033\" style=\"width: 648px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1033\" src=\"http:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-300x151.jpg\" alt=\"\" width=\"648\" height=\"326\" srcset=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-300x151.jpg 300w, https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-1024x514.jpg 1024w, https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-768x386.jpg 768w, https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-1536x772.jpg 1536w, https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map.jpg 1909w\" sizes=\"(max-width: 648px) 100vw, 648px\" \/><figcaption id=\"caption-attachment-1033\" class=\"wp-caption-text\"><em>The coalescence map. Solid blue: the z-chain. Dashed orange: the z\u0303-chain, driven by the transported noise \u03be\u0303 = \u03a8\u207f_(z, z\u0303)(\u03be). The chains meet exactly at the terminal horizon and generically never before; the gap trajectory y_k = z\u0303_k \u2212 z_k is chosen by LQ optimization.<\/em><\/figcaption><\/figure>\n<p>The natural framework for the analysis is Malliavin calculus. The chain map \u03a8\u207f_z is a smooth functional of the Gaussian noise on \u211d^(2dn), and the Jacobian D\u03a8\u207f_(z, z\u0303) \u2014 its Malliavin derivative with respect to \u03be \u2014 encodes how the coalescing noise depends on the driving noise. Two structural facts about OBABO make this Jacobian especially well-behaved. First, the i-th component \u03be\u0303_i depends only on \u03be_1, \u2026, \u03be_(i-1), so D\u03a8\u207f_(z, z\u0303) is block lower-triangular. Second, the diagonal blocks are the identity. Consequently det D\u03a8\u207f_(z, z\u0303) \u2261 1 and tr(D\u03a8\u207f_(z, z\u0303) \u2212 I) \u2261 0, and the KL bound of Lemma 15 in B.-R. &amp; Eberle (2023) collapses to a second-moment cost on the gap trajectory. Optimizing that cost gives the explicit non-Markovian coupling.<\/p>\n<h2>7. Three results<\/h2>\n<h3>7.1 A quantitative TV bound for OBABO<\/h3>\n<p><strong>Theorem<\/strong> (<a href=\"https:\/\/arxiv.org\/abs\/2605.31088\">B.-R.\u2013Cox\u2013Schieven, 2026<\/a>; Thm. 3.2). <em>For \u03b3, h &gt; 0, n a positive integer, and U in C^2(R^d) with \u2207U being L-Lipschitz, for all z, z\u0303 in R^(2d),<\/em><\/p>\n<p><em>d_TV(\u03c0_n(\u03b4_z), \u03c0_n(\u03b4_z\u0303))<\/em><br \/>\n<em> \u2264 \u03b3^(-1\/2) \u00b7 [ 5\u00b7(hn)^(-3\/2)<\/em> <em>+ (12 + 5\u03b3)\u00b7(hn)^(-1\/2)<\/em><br \/>\n<em> + (1 + hn\/(1 + \u03b3hn)) \u00b7 L \u00b7 (\u03b3h)^(1\/2) \u00b7 (1 &#8211; exp(-\u03b3h))^(-1\/2) \u00b7 (hn)^(1\/2) ]<\/em><br \/>\n<em> \u00b7 |z\u0303 &#8211; z|.<\/em><\/p>\n<p>A few features of the bound are worth noting:<\/p>\n<ul>\n<li>It holds for all h &gt; 0 and all positive integers n \u2014 no hn \u2264 1 restriction and no h \u2264 h_0 smallness condition.<\/li>\n<li>It only requires \u2207U Lipschitz; the Hessian-Lipschitz assumption of Chak &amp; Monmarch\u00e9 (2025) is not needed.<\/li>\n<li>It yields a Wasserstein-to-TV regularization for arbitrary initial laws \u03bd, \u03bd\u0303 via the convexity argument above.<\/li>\n<li>It is realized by an <em>explicit non-Markovian coupling<\/em>.<\/li>\n<\/ul>\n<h3>7.2 An impossibility theorem<\/h3>\n<p>The natural place to test optimality is a quadratic potential. Take U(x) = \u03b1|x|^2, \u03b1 \u2265 0, in the overdamped regime \u03b3^2 &gt; 4\u03b1. The drift matrix has eigenvalues<\/p>\n<p>\u03bb_\u00b1 = -\u03b3\/2 \u00b1 (1\/2)\u00b7\u221a(\u03b3^2 &#8211; 4\u03b1), \u03bb_- &lt; \u03bb_+ \u2264 0.<\/p>\n<p>If the initial gap \u0394z = (\u0394x, \u0394v) lies in the \u03bb_- eigenspace \u2014 that is, \u03bb_- \u00b7 \u0394x = \u0394v \u2014 then<\/p>\n<p>d_TV(Law(Z_t), Law(Z\u0303_t)) \u2272 exp(\u03bb_- t) \u00b7 |\u0394z|.<\/p>\n<p>The question is whether a Markovian coupling can match this rate. The answer is no.<\/p>\n<p><strong>Theorem<\/strong> (<a href=\"https:\/\/arxiv.org\/abs\/2605.31088\">B.-R.\u2013Cox\u2013Schieven, 2026<\/a>; continuous time Thm. 4.3). <em>Under \u03b3^2 &gt; 4\u03b1 \u2265 0: if \u03bb_- \u00b7 \u0394x = \u0394v and \u0394x \u2260 0, then d_TV \u2264 C \u00b7 exp(\u03bb_- t) \u00b7 |\u0394z| (upper). For every Markovian coupling \u03bc and every \u0394z,<\/em><\/p>\n<p><em>\u03bc(Z_t \u2260 Z\u0303_t) \u2265 c_\u03bc \u00b7 min( t^(-1\/2), exp(\u03bb_+ t) ) for t \u2265 t_\u03bc.<\/em><\/p>\n<p>A discrete-time version yields, for every h &gt; 0 and every Markovian \u03bc_h,<\/p>\n<p>\u03bc_h(Z^h_k \u2260 Z\u0303^h_k) \u2265 c \u00b7 min( c_(\u03bc_h) \u00b7 (hk+1)^(-1\/2), c_(\u03bc_h) \u00b7 exp(\u03bb_+ hk), h^(-1) \u00b7 exp(\u03bb_- hk) ).<\/p>\n<p>Since \u03bb_- &lt; \u03bb_+, the Markovian lower bound is strictly slower than the upper bound for non-Markovian couplings. The result extends Banerjee-Kendall (2016) from the Kolmogorov diffusion to kinetic Langevin, ruling out an entire class of strategies.<\/p>\n<h3>7.3 An exact meeting probability for iterated one-shot<\/h3>\n<p>The canonical Markovian candidate is the <em>iterated one-shot coupling<\/em>: at each step, maximize the meeting probability via a reflection coupling. It satisfies the <em>now-equals-forever<\/em> property (Z^h_s = Z\u0303^h_s implies Z^h_t = Z\u0303^h_t for t \u2265 s, almost surely) and is asymptotically optimal for overdamped Euler-Maruyama (Durmus &amp; Moulines, 2019). For kinetic Langevin, the impossibility result says it must be suboptimal \u2014 but it is natural to ask by how much.<\/p>\n<p><strong>Theorem<\/strong> (<a href=\"https:\/\/arxiv.org\/abs\/2605.31088\">B.-R.\u2013Cox\u2013Schieven, 2026<\/a>; Thm. 5.1). <em>Let (Z_k, Z\u0303_k) be the iterated one-shot coupling of the linear chain Z_(k+1) = A_(k+1) Z_k + B_(k+1) \u03be_(k+1) with A_k, B_k non-singular. Then<\/em><\/p>\n<p><em>P(Z_n \u2260 Z\u0303_n) = 2 \u00b7 \u03a6( 1 \/ (2 \u00b7 \u0398_n^(1\/2)) ) &#8211; 1,<\/em><\/p>\n<p><em>where \u0398_n = \u03a3_(k=1)^n [ 1 \/ |B_k^(-1) \u03a0_k \u0394z|^2 ] and \u03a0_k = A_k \u00b7 A_(k-1) \u00b7 \u22ef \u00b7 A_1.<\/em><\/p>\n<p>This sharpens Durmus\u2013Moulines (2019, Thm. 19) from inequality to equality in the linear-drift case. It is used as a lower bound on P(Z_n \u2260 Z\u0303_n), not a TV upper bound. For free kinetic Langevin (\u03b1 = 0) with initial gap \u0394z = (\u0394x, -\u03b3 \u00b7 \u0394x), the probability of not meeting degrades like h^(-1) \u00b7 exp(\u03bb_- hk) as the step size shrinks \u2014 <em>exactly saturating<\/em> the corresponding term in the impossibility lower bound.<\/p>\n<h2>8. The non-Markovian construction<\/h2>\n<p>The coupling that realizes the upper bound is non-Markovian.<\/p>\n<ol>\n<li>Sample \u03be ~ N(0, I_(2dn)) \u2014 the noise driving the first chain over the entire horizon [0, n].<\/li>\n<li>Compute the proposal \u03be\u0303* := \u03a8^n_(z, z\u0303)(\u03be) \u2014 the noise the second chain <em>would<\/em> need to coalesce with the first at time n.<\/li>\n<li>Maximally couple \u03be and \u03be\u0303*. On the acceptance event, set \u03be\u0303 = \u03be\u0303*: the chains meet at time n. On rejection, draw \u03be\u0303 independently: the chains evolve independently.<\/li>\n<\/ol>\n<p>By construction,<\/p>\n<p>P(Z^h_n \u2260 Z\u0303^h_n) = d_TV( Law(\u03be), Law(\u03a8^n_(z, z\u0303)(\u03be)) ).<\/p>\n<p>The coupling is non-Markovian because \u03be\u0303 depends on the entire \u03be at once. Intermediate chains generally disagree; the chains meet only at the terminal horizon, and generically never before.<\/p>\n<p><strong>Choosing \u03a8 optimally.<\/strong> The trajectory \u03a8^n_(z, z\u0303) is chosen to minimize the TV cost. In the force-free case (\u2207U \u2261 0), Lemma 15 of B.-R. &amp; Eberle (2023) reduces the TV bound to a controlled L^2 cost on the gap trajectory y_k := z\u0303_k &#8211; z_k,<\/p>\n<p>d_TV \u2264 (1\/2) \u00b7 ( \u03a3_(k=1)^n |E_k|^2 )^(1\/2), E_(k+1) = -L_h^(-1) \u00b7 (y_(k+1) &#8211; A_h \u00b7 y_k),<\/p>\n<p>with boundary conditions y_0 = z\u0303 &#8211; z and y_n = 0. This is a classical <em>minimum-energy linear\u2013quadratic control problem<\/em>, solvable in closed form via the controllability Gramian \u03a3_(h,n):<\/p>\n<p>\u03a3_(k=1)^n |E_k|^2 = | \u03a3_(h,n)^(-1\/2) \u00b7 A_h^n \u00b7 \u0394z |^2.<\/p>\n<p>For a general potential, the same trajectory is used and \u2207U is handled as a Lipschitz perturbation \u2014 the design principle of Eberle\u2013Guillin\u2013Zimmer.<\/p>\n<p>One technical point makes the OBABO bound clean. The Jacobian D\u03a8^n_(z, z\u0303) is block lower-triangular with identity diagonal, because the i-th coalescence noise depends only on \u03be_1, \u2026, \u03be_(i-1). Hence det D\u03a8^n_(z, z\u0303) \u2261 1 and tr( D\u03a8^n_(z, z\u0303) &#8211; I ) \u2261 0, so the KL bound from Lemma 15 reduces to the second-moment term<\/p>\n<p>d_TV \u2264 E[ |\u03a8^n_(z, z\u0303)(\u03be) &#8211; \u03be|^2 ]^(1\/2).<\/p>\n<h2>9. The non-Markovian advantage<\/h2>\n<p>Taken together, the three theorems give a complete picture. The natural strategy for coupling kinetic Langevin in total variation \u2014 the iterated reflection coupling that works so cleanly for overdamped dynamics \u2014 provably cannot capture the sharp asymptotic rate. The coupling that does is global: it solves a classical minimum-energy control problem on the gap trajectory, and forces the chains to meet only at the terminal horizon. The very hypoellipticity that delivers the ballistic speedup is what makes the Markov property an obstruction.<\/p>\n<p>Establishing the corresponding accelerated TV mixing bound t_mix(\u03bd, \u03b5) \u224d \u03ba^(1\/2) log(1\/\u03b5) reduces, via the Wasserstein-to-TV regularization of \u00a76.1, to proving W_1 contraction for OBABO at rate \u03ba^(-1\/2). This remains open.<\/p>\n<p>Related directions include KL and R\u00e9nyi divergence bounds (B.-R., Mitra &amp; Wibisono, 2026), other splittings (Schuh &amp; Whalley, 2025), and Metropolis-adjusted variants (B.-R. &amp; Oberd\u00f6rster, 2024, EJP).<\/p>\n<h2>Acknowledgements<\/h2>\n<p>Many thanks to Pierre Monmarch\u00e9, Andreas Eberle, and Stefan Oberd\u00f6rster for fruitful discussions.<\/p>\n<h2>References<\/h2>\n<ul>\n<li>Bakry, D., Gentil, I., &amp; Ledoux, M. (2014). <em>Analysis and Geometry of Markov Diffusion Operators<\/em>. Grundlehren der mathematischen Wissenschaften, vol. 348. Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-3-319-00227-9\">DOI<\/a><\/li>\n<li>Banerjee, S., &amp; Kendall, W. S. (2016). Coupling the Kolmogorov diffusion: maximality and efficiency considerations. <em>Advances in Applied Probability<\/em>, 48(A), 15\u201335. <a href=\"https:\/\/doi.org\/10.1017\/apr.2016.40\">DOI<\/a><\/li>\n<li>Ben Arous, G., Cranston, M., &amp; Kendall, W. S. (1995). Coupling constructions for hypoelliptic diffusions: two examples. In <em>Stochastic Analysis (Ithaca, NY, 1993)<\/em>, Proc. Sympos. Pure Math., vol. 57, 193\u2013212. American Mathematical Society. <a href=\"https:\/\/doi.org\/10.1090\/pspum\/057\/1335472\">DOI<\/a><\/li>\n<li>Bou-Rabee, N., Cox, S., &amp; Schieven, R. (2026). On couplings for kinetic Langevin diffusions. <em>arXiv preprint<\/em>. <a href=\"https:\/\/arxiv.org\/abs\/2605.31088\">arXiv:2605.31088<\/a><\/li>\n<li>Bou-Rabee, N., &amp; Eberle, A. (2023). Mixing time guarantees for unadjusted Hamiltonian Monte Carlo. <em>Bernoulli<\/em>, 29(1), 75\u2013104. <a href=\"https:\/\/doi.org\/10.3150\/21-bej1450\">DOI<\/a><\/li>\n<li>Bou-Rabee, N., Mitra, S., &amp; Wibisono, A. (2026). Tail-sensitive KL and R\u00e9nyi convergence of unadjusted Hamiltonian Monte Carlo via one-shot couplings. <em>arXiv preprint<\/em>. <a href=\"https:\/\/arxiv.org\/abs\/2601.09019\">arXiv:2601.09019<\/a><\/li>\n<li>Bou-Rabee, N., &amp; Oberd\u00f6rster, S. (2024). Mixing of Metropolis-adjusted Markov chains via couplings: the high acceptance regime. <em>Electronic Journal of Probability<\/em>, 29, Paper No. 89. <a href=\"https:\/\/doi.org\/10.1214\/24-ejp1150\">DOI<\/a><\/li>\n<li>Bussi, G., &amp; Parrinello, M. (2007). Accurate sampling using Langevin dynamics. <em>Physical Review E<\/em>, 75(5), 056707. <a href=\"https:\/\/doi.org\/10.1103\/PhysRevE.75.056707\">DOI<\/a><\/li>\n<li>Cao, Y., Lu, J., &amp; Wang, L. (2023). On explicit L\u00b2-convergence rate estimate for underdamped Langevin dynamics. <em>Archive for Rational Mechanics and Analysis<\/em>, 247, Paper No. 90. <a href=\"https:\/\/doi.org\/10.1007\/s00205-023-01922-4\">DOI<\/a><\/li>\n<li>Chak, M., &amp; Monmarch\u00e9, P. (2025). Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case. <em>IMA Journal of Numerical Analysis<\/em>, draf045. <a href=\"https:\/\/doi.org\/10.1093\/imanum\/draf045\">DOI<\/a><\/li>\n<li>Cheng, X., Chatterji, N. S., Bartlett, P. L., &amp; Jordan, M. I. (2018). Underdamped Langevin MCMC: A non-asymptotic analysis. In <em>Conference on Learning Theory (COLT)<\/em>, 300\u2013323. PMLR.<\/li>\n<li>Cheng, X., Chatterji, N. S., Abbasi-Yadkori, Y., Bartlett, P. L., &amp; Jordan, M. I. (2020). Sharp convergence rates for Langevin dynamics in the nonconvex setting. <em>arXiv preprint<\/em>. <a href=\"https:\/\/arxiv.org\/abs\/1805.01648\">arXiv:1805.01648<\/a><\/li>\n<li>Dalalyan, A. S., &amp; Riou-Durand, L. (2020). On sampling from a log-concave density using kinetic Langevin diffusions. <em>Bernoulli<\/em>, 26(3), 1956\u20131988. <a href=\"https:\/\/doi.org\/10.3150\/19-BEJ1178\">DOI<\/a><\/li>\n<li>Durmus, A., &amp; Moulines, \u00c9. (2019). High-dimensional Bayesian inference via the unadjusted Langevin algorithm. <em>Bernoulli<\/em>, 25(4A), 2854\u20132882. <a href=\"https:\/\/doi.org\/10.3150\/18-BEJ1073\">DOI<\/a><\/li>\n<li>Eberle, A., Guillin, A., &amp; Zimmer, R. (2019). Couplings and quantitative contraction rates for Langevin dynamics. <em>Annals of Probability<\/em>, 47(4), 1982\u20132010. <a href=\"https:\/\/doi.org\/10.1214\/18-AOP1299\">DOI<\/a><\/li>\n<li>Eberle, A., &amp; L\u00f6rler, F. (2024). Non-reversible lifts of reversible diffusion processes and relaxation times. <em>Probability Theory and Related Fields<\/em>. <a href=\"https:\/\/doi.org\/10.1007\/s00440-024-01308-x\">DOI<\/a><\/li>\n<li>Gouraud, N., Le Bris, P., Majka, A., &amp; Monmarch\u00e9, P. (2025). HMC and underdamped Langevin united in the unadjusted convex smooth case. <em>SIAM\/ASA Journal on Uncertainty Quantification<\/em>, 13(1), 278\u2013303. <a href=\"https:\/\/doi.org\/10.1137\/23M1608963\">DOI<\/a><\/li>\n<li>Leimkuhler, B., &amp; Matthews, C. (2013). Rational construction of stochastic numerical methods for molecular sampling. <em>Applied Mathematics Research Express. AMRX<\/em>, 2013(1), 34\u201356. <a href=\"https:\/\/doi.org\/10.1093\/amrx\/abs010\">DOI<\/a><\/li>\n<li>Leimkuhler, B. J., Paulin, D., &amp; Whalley, P. A. (2024). Contraction and convergence rates for discretized kinetic Langevin dynamics. <em>SIAM Journal on Numerical Analysis<\/em>, 62(3), 1226\u20131258. <a href=\"https:\/\/doi.org\/10.1137\/23M1556289\">DOI<\/a><\/li>\n<li>Madras, N., &amp; Sezer, D. (2010). Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances. <em>Bernoulli<\/em>, 16(3), 882\u2013908. <a href=\"http:\/\/www.jstor.org\/stable\/25735016\">JSTOR<\/a><\/li>\n<li>Monmarch\u00e9, P. (2021). High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion. <em>Electronic Journal of Statistics<\/em>, 15(2), 4117\u20134166. <a href=\"https:\/\/doi.org\/10.1214\/21-ejs1888\">DOI<\/a><\/li>\n<li>Roberts, G. O., &amp; Rosenthal, J. S. (2002). One-shot coupling for certain stochastic recursive sequences. <em>Stochastic Processes and their Applications<\/em>, 99(2), 195\u2013208. <a href=\"https:\/\/doi.org\/10.1016\/S0304-4149(02)00096-0\">DOI<\/a><\/li>\n<li>Schuh, K., &amp; Whalley, P. A. (2025). Convergence of kinetic Langevin samplers for non-convex potentials. <em>arXiv preprint<\/em>. <a href=\"https:\/\/arxiv.org\/abs\/2405.09992\">arXiv:2405.09992<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Joint work with Sonja Cox and Roy Schieven (University of Amsterdam). Paper: arXiv:2605.31088. Two copies of the same diffusion, started apart. How can you get them to coalesce? For kinetic &hellip; <a href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/\" class=\"\">Read More<\/a><\/p>\n","protected":false},"author":2614,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10,8],"tags":[],"class_list":["post-1032","post","type-post","status-publish","format-standard","hentry","category-events","category-preprints"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>On Couplings for Kinetic Langevin Diffusions - Nawaf Bou-Rabee<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"On Couplings for Kinetic Langevin Diffusions - Nawaf Bou-Rabee\" \/>\n<meta property=\"og:description\" content=\"Joint work with Sonja Cox and Roy Schieven (University of Amsterdam). 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Solid blue: the z-chain. Dashed orange: the z\u0303-chain, driven by the transported noise \u03be\u0303 = \u03a8\u207f_(z, z\u0303)(\u03be). The chains meet exactly at the terminal horizon and generically never before; the gap trajectory y_k = z\u0303_k \u2212 z_k is chosen by LQ optimization.\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"On Couplings for Kinetic Langevin Diffusions\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#website\",\"url\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/\",\"name\":\"Nawaf Bou-Rabee\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#\/schema\/person\/f9a7363ddc14a33eef77b520728dbde0\",\"name\":\"Nawaf Bou-Rabee\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f3f33e8b5e6e2430997aea3dfec1a454?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f3f33e8b5e6e2430997aea3dfec1a454?s=96&d=mm&r=g\",\"caption\":\"Nawaf Bou-Rabee\"},\"description\":\"Math Professor at Rutgers\",\"sameAs\":[\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/\"],\"url\":\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/author\/nb361\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"On Couplings for Kinetic Langevin Diffusions - Nawaf Bou-Rabee","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/","og_locale":"en_US","og_type":"article","og_title":"On Couplings for Kinetic Langevin Diffusions - Nawaf Bou-Rabee","og_description":"Joint work with Sonja Cox and Roy Schieven (University of Amsterdam). Paper: arXiv:2605.31088. Two copies of the same diffusion, started apart. How can you get them to coalesce? For kinetic &hellip; Read More","og_url":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/","og_site_name":"Nawaf Bou-Rabee","article_published_time":"2026-06-04T08:44:13+00:00","article_modified_time":"2026-06-04T13:30:25+00:00","og_image":[{"url":"http:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-300x151.jpg"}],"author":"Nawaf Bou-Rabee","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Nawaf Bou-Rabee","Est. reading time":"14 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/","url":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/","name":"On Couplings for Kinetic Langevin Diffusions - Nawaf Bou-Rabee","isPartOf":{"@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#website"},"primaryImageOfPage":{"@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#primaryimage"},"image":{"@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#primaryimage"},"thumbnailUrl":"http:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map-300x151.jpg","datePublished":"2026-06-04T08:44:13+00:00","dateModified":"2026-06-04T13:30:25+00:00","author":{"@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#\/schema\/person\/f9a7363ddc14a33eef77b520728dbde0"},"breadcrumb":{"@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#primaryimage","url":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map.jpg","contentUrl":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/06\/coalescence-map.jpg","width":1909,"height":959,"caption":"The coalescence map. Solid blue: the z-chain. Dashed orange: the z\u0303-chain, driven by the transported noise \u03be\u0303 = \u03a8\u207f_(z, z\u0303)(\u03be). The chains meet exactly at the terminal horizon and generically never before; the gap trajectory y_k = z\u0303_k \u2212 z_k is chosen by LQ optimization."},{"@type":"BreadcrumbList","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/"},{"@type":"ListItem","position":2,"name":"On Couplings for Kinetic Langevin Diffusions"}]},{"@type":"WebSite","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#website","url":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/","name":"Nawaf Bou-Rabee","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#\/schema\/person\/f9a7363ddc14a33eef77b520728dbde0","name":"Nawaf Bou-Rabee","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f3f33e8b5e6e2430997aea3dfec1a454?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f3f33e8b5e6e2430997aea3dfec1a454?s=96&d=mm&r=g","caption":"Nawaf Bou-Rabee"},"description":"Math Professor at Rutgers","sameAs":["https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/"],"url":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/author\/nb361\/"}]}},"_links":{"self":[{"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/posts\/1032"}],"collection":[{"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/users\/2614"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/comments?post=1032"}],"version-history":[{"count":15,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/posts\/1032\/revisions"}],"predecessor-version":[{"id":1048,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/posts\/1032\/revisions\/1048"}],"wp:attachment":[{"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/media?parent=1032"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/categories?post=1032"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-json\/wp\/v2\/tags?post=1032"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}