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Recursive formulae in regularity structures
Springer Science and Business Media LLC - - 2018
Yvain Bruned
Path properties of the solution to the stochastic heat equation with Lévy noise
Springer Science and Business Media LLC - Tập 7 - Trang 123-168 - 2018
Carsten Chong, Robert C. Dalang, Thomas Humeau
We consider sample path properties of the solution to the stochastic heat equation, in $${\mathbb {R}}^d$$ or bounded domains of $${\mathbb {R}}^d$$ ...... hiện toàn bộ
Fluctuations of parabolic equations with large random potentials
Springer Science and Business Media LLC - Tập 3 - Trang 1-51 - 2014
Yu Gu, Guillaume Bal
In this paper, we present a fluctuation analysis of a type of parabolic equations with large, highly oscillatory, random potentials around the homogenization limit. With a Feynman–Kac representation, the Kipnis–Varadhan’s method, and a quantitative martingale central limit theorem, we derive the asymptotic distribution of the rescaled error between heterogeneous and homogenized solutions under dif...... hiện toàn bộ
Correction to: Convergent numerical approximation of the stochastic total variation flow
Springer Science and Business Media LLC - Tập 11 - Trang 1732-1739 - 2022
Ĺubomír Baňas, Michael Röckner, André Wilke
We correct two errors in our paper [4]. First error concerns the definition of the SVI solution, where a boundary term which arises due to the Dirichlet boundary condition, was not included. The second error concerns the discrete estimate [4,  Lemma 4.4], which involves the discrete Laplace operator. We provide an alternative proof of the estimate in spatial dimension ...... hiện toàn bộ
A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation
Springer Science and Business Media LLC - Tập 11 - Trang 1379-1415 - 2022
Amirali Hannani, Stefano Olla
We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperat...... hiện toàn bộ
Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances
Springer Science and Business Media LLC - Tập 7 - Trang 240-296 - 2018
Sebastian Andres, Stefan Neukamm
We study the random conductance model on the lattice $${\mathbb {Z}}^d$$ , i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the condu...... hiện toàn bộ
Strong Feller semigroups and Markov processes: a counterexample
Springer Science and Business Media LLC - - Trang 1-16 - 2023
Lucian Beznea, Iulian Cîmpean, Michael Röckner
The aim of this note is to show, by providing an elementary way to construct counterexamples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counterexample is the Brownian semigroup on ...... hiện toàn bộ
Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line
Springer Science and Business Media LLC - Tập 8 - Trang 509-543 - 2019
Timur Yastrzhembskiy
We prove a Stroock–Varadhan’s type support theorem for a stochastic partial differential equation on the real line with a noise term driven by a cylindrical Wiener process on $$L_2 ({\mathbb {R}})$$ . The main ingredients of the proof are V. Mackevičius’s approach to support theorem for diffusion processes and N.V. Krylov’s $$L_p$$ -theory of SPDEs.
Statistical null-controllability of stochastic nonlinear parabolic equations
Springer Science and Business Media LLC - Tập 10 - Trang 190-222 - 2021
Víctor Hernández-Santamaría, Kévin Le Balc’h, Liliana Peralta
In this paper, we consider forward stochastic nonlinear parabolic equations with a control localized in the drift term. Under suitable assumptions, we prove the small-time global null-controllability with a truncated nonlinearity. We also prove the “statistical” local null-controllability of the true system. The proof relies on a precise estimation of the cost of null-controllability of the stocha...... hiện toàn bộ
2D-defocusing nonlinear Schrödinger equation with random data on irrational tori
Springer Science and Business Media LLC - - 2020
Chenjie Fan, Yumeng Ou, Gigliola Staffilani, Hong Wang
We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.
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