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This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. 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. [ABSTRACT FROM AUTHOR]

This paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non-homogeneous boundary condition. We first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general L 1 -data. Our results allow us to give a positive answer to an open problem posed by Bauzet, Vallet and Wittbold in [4] . [ABSTRACT FROM AUTHOR]

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen–Cahn equation does not have finite time singularities and the unique solution exists globally. [ABSTRACT FROM AUTHOR]

Abstract: The paper is concerned with the problem of non-existence of global solutions for a class of stochastic reaction–diffusion equations of Itô type. Under some sufficient conditions on the initial state, the nonlinear term and the multiplicative noise, it is proven that, in a bounded domain , there exist positive solutions whose mean -norm will blow up in finite time for , while, if , the previous result holds in any compact subset of . Two examples are given to illustrate some application of the theorems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô''s- and Tanaka''s-type formulae associated to . [Copyright &y& Elsevier]