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We study a class of nonlocal elliptic equations (− Δ) α / 2 ρ (x) − b (x) ⋅ ∇ ρ (x) = f (x) on a bounded domain Ω ⊂ R d . For f ∈ L q (Ω) ( q > d / α), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of L ∞ solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of strong solutions for stochastic differential equations in which the drift coefficient is square integrable in time variable and Hölder continuous in space variable. Moreover, we prove that the unique strong solution has a continuous modification, which is β -Hölder continuous in space variable for every β ∈ (0 , 1) , and as an L 2 (Ω × (0 , T)) valued function, it is differentiable as well. [ABSTRACT FROM AUTHOR]

We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations. [ABSTRACT FROM AUTHOR]

This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of parabolic approximation motivated by the method of vanishing viscosity given by Feng and Nualart (J. Funct. Anal. 2008, 255, 313–373). Furthermore, we prove the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f. [ABSTRACT FROM AUTHOR]