Parabolic Anderson model on Heisenberg groups: The Itô setting

TY - JOUR

T1 - Parabolic Anderson model on Heisenberg groups

T2 - The Itô setting

AU - Baudoin, Fabrice

AU - Ouyang, Cheng

AU - Tindel, Samy

AU - Wang, Jing

N1 - Publisher Copyright: © 2023 Elsevier Inc.

PY - 2023/7/1

Y1 - 2023/7/1

N2 - In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as [Formula presented], where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as [Formula presented]; (iii) Give some basic moment estimates for the solution u(t,x).

AB - In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as [Formula presented], where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as [Formula presented]; (iii) Give some basic moment estimates for the solution u(t,x).

KW - Gaussian noise

KW - Heisenberg group

KW - Parabolic Anderson model

KW - Solution to the stochastic heat equation

UR - http://www.scopus.com/inward/record.url?scp=85151855052&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2023.109920

DO - 10.1016/j.jfa.2023.109920

M3 - Journal article

AN - SCOPUS:85151855052

SN - 0022-1236

VL - 285

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

M1 - 109920

ER -