We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $\kappa$, in dimension $d$, to accuracy $\varepsilon$ in R\'enyi divergence, in $\widetilde O(\kappa^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(\kappa d^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.
We establish the sharp logarithmic order $(\log N)^{-1/2}$ for the expected $p$-Wasserstein distance, induced by the supremum norm, between the empirical law of $N$ independent copies of a continuous It\^o process and their common path law. We only assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>p\geq1$. Under this assumption, we use an adaptive random time interval partition argument, which leads to a $(\log n)^{-1/2}$ functional quantization rate. A general transfer principle then converts the quantization estimate into a mean estimate and nonasymptotic deviation bounds for equal-weight empirical laws. Applications include empirical path-law estimates for path-dependent SDEs and a path-space propagation-of-chaos estimate for path-dependent McKean--Vlasov interacting particle systems.
We derive convergence rates for the approximation of the Fr\'echet distribution $\mathcal{F}(\alpha)$ with parameter $\alpha>0$ by sequences of renormalized maxima in the extreme value theorem. Our proofs rely on the application of the infinitesimal generator approach to Stein's method to max-stable distributions, using the family of Markov semi-groups recently introduced in \cite{CostacequePhD, Costaceque24}. We develop two different approaches to compute rates of convergence; the first one relies on the second-order regular variation assumption, while the second one requires the existence of a density function for the base distribution. In particular, with the first approach, our bounds are expressed using the Kolmogorov distance, and the Wasserstein distance when $\alpha>1$. The second approach allows also rates for a smooth H{\"o}lder distance when $\alpha \in (0,1)$. In both cases, we also obtain convergence rates for moments when they exist.
The Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions and relies on a bound on the average KL divergence along Hamiltonian dynamics.
Siddharth Mitra, Vishwak Srinivasan, Xiuyuan Wang et al.· arXiv.org· 1 citation
We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a"shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.
In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The It\^o--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.
We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\geq 2$. We show the Hausdorff dimension of the set of points in $\mathbb{R}^{d}$ whose best approximation constant lies in $[\varepsilon,\varepsilon(1+\delta\varepsilon^{d})]$, for some constant $\delta>0$, is positive, and for a slightly larger set approaches full dimension as $\varepsilon \to 0$. We also show that the box dimension of the $d$-dimensional Lagrange spectrum is bounded from below by $1-\frac{1}{d+1}$. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. As an additional result we use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.
Unknown authors· 0 citations
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