ترغب بنشر مسار تعليمي؟ اضغط هنا

Convergence in Wasserstein Distance for Empirical Measures of Semilinear SPDEs

116   0   0.0 ( 0 )
 نشر من قبل Feng-Yu Wang
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English
 تأليف Feng-Yu Wang




اسأل ChatGPT حول البحث

The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.

قيم البحث

اقرأ أيضاً

81 - Feng-Yu Wang 2020
Let $X_t$ be the (reflecting) diffusion process generated by $L:=Delta+ abla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $partial M$, where $Vin C^1(M)$ such that $mu(d x):= e^{V(x)}d x$ is a probability measure. We es timate the convergence rate for the empirical measure $mu_t:=frac 1 t int_0^t delta_{X_sd s$ under the Wasserstein distance. As a typical example, when $M=mathbb R^d$ and $V(x)= c_1- c_2 |x|^p$ for some constants $c_1in mathbb R, c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either $d<frac{4(p-1)}p$ or $dge 4$ and $ptoinfty$.
228 - Feng-Yu Wang , Bingyao Wu 2021
Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $Vin C^2(M)$ such that $mu(d x):=e^{V(x)}d x$ is a probability measure, where $d x$ is the volume measure, and let $L=Delta+ abla V$. The exact convergence rate in Wasse rstein distance is derived for empirical measures of subordinations for the (reflecting) diffusion process generated by $L$.
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $infty-$Wasserstein distance betwe en the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slepv{c}ev to the case that the true distribution has an unbounded density.
81 - Panpan Ren , Feng-Yu Wang 2020
The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(mu_t,mu_infty)^2 +{rm Ent}(mu_t|mu_infty)le c {rm e}^{-lambda t} minbig{W_2(mu_0, mu_infty)^2,{rm Ent}(mu_0|mu_infty)big}, tge 1,$$ whe re $c,lambda>0$ are constants, $mu_t$ is the distribution of the solution at time $t$, $mu_infty$ is the unique invariant probability measure, ${rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.
Consider the empirical measure, $hat{mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $mathbb{P}$ on the unit interval. For fixed $mathbb{P}$ the Wasserstein distance between $hat{mathbb{P}}_N$ and $mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $mathbb{P}$ is a convex combination between the uniform distribution supported on the two points ${0,1}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic confidence regions for the underlying measure $mathbb{P}$. We also suggest extensions to higher dimensions with numerical evidence.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا