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 study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $\rho$ and a target measure $\mu$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $\mu$, we establish the existence of global weak so...
Antonin Chodron de Courcel, Matthew Rosenzweig· 0 citations
We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $\tau$ on $[0,1]$. We show that the rescaled process $a^H(B_{\tau+\,\cdot\,/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero,...
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous L\'evy--It\^o diffusions driven by a Brownian motion and an independent non-truncated pure-jump L\'evy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space togeth...
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 num...
Fan Chen, Sinho Chewi, Alexander Rakhlin et al.· 1 citation
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up...
Joseph Najnudel, Thammadol Tansrivorarat· 0 citations
This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel'fand triple built from a L\'evy measure $\lambda_{\beta}(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $\lambda_{\beta}(u...
Yun-Ching Chang· arXiv.org· 0 citations
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