Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals

Abstract

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on bounded intervals. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

Stochastic Burgers’ equation with fractional derivative driven by multiplicative noise

“…The main contribution of this paper is to establish the existence, uniqueness, and regularity properties of mild solution to time-space fractional SBE driven by multiplicative noise, which generalizes many previous works [6,11,12]. The rest of the paper is organized as follows.…”

“…Yang [11] proposed some estimates on the solution of space-fractional SBE and given the invariant measure. Lv and Duan [12] investigated the existence of martingale solutions and weak solutions for space-fractional SBE on a bounded domain. However, to the best of our knowledge, there are no existing works for the time-and space-fractional SBE, which is a fascinating and useful problem.…”

Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

“…For the Burgers equation with the fractional Laplacian operator driven by the normal noise, Lv and Duan [19] established the existence of the martingale solution and the weak solution. Brzezniak, Debbi and Goldys [2] further studied the ergodicity properties of the system.…”

“…A probability measure µ ∈ M(L 2 (D)) is called an invariant measure if P * t µ = µ for all t ≥ 0. Using the same arguments as Lv and Duan [19], and Brzezniak, Debbi and Goldys [2], it is easily to derive the well-posedness and the existence of invariant measure of SFBE (1) as follows.…”

Nonlocal elliptic equations involving measures

“…For fractional operator, we can take the classical fractional Sobolev space as its work space. But for nonlocal operator, we must take nonlocal Sobolev space as its work space, see Remark 2.1 in [12]. Here nonlocal Sobolev space is the weight fractional Sobolev space.…”