اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBogoliubov type recursions for renormalisation in regularity structures
43
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Hairers regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimers formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs as well as the proposal for resonance based numerical schemes for certain partial differential equations in numerical analysis introduced in a recent work by the first author.
قيم البحث
اقرأ أيضاً
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was contin
ued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen, Frohlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non-)construction of the focusing Hartree Gibbs measures in the three-dimensional setting. More precisely, we study a focusing $Phi^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $beta$. We first construct the focusing Hartree $Phi^4_3$-measure for $beta > 2$, while we prove its non-normalizability for $beta < 2$. Furthermore, we establish the following phase transition at the critical value $beta = 2$: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree $Phi^4_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the focusing Hartree $Phi^4_3$-measure for $beta > 2$ (and $beta = 2$ in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree $Phi^4_3$-measure. We also construct the defocusing Hartree $Phi^4_3$-measure for $beta > 0$.
The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time.
There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $dgeq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full t
emperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Phi^4_3$ model. As a corollary, we prove that the Brownian bri
dge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the p
article after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field $U$. If the velocity field is additionally assumed to be incompressible, then $Uequiv 1$ almost surely and we obtain a local central limit theorem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
