The Stochastic Heat Equation with Fractional-Colored Noise: Existence of the Solution

The Stochastic Heat Equation with a Fractional-colored Noise: Existence of the Solution

Abstract. In this article we consider the stochastic heat equation ut −∆u = Ḃ in (0, T )× R, with vanishing initial conditions, driven by a Gaussian noise Ḃ which is fractional in time, with Hurst index H ∈ (1/2, 1), and colored in space, with spatial covariance given by a function f . Our main result gives the necessary and sufficient condition on H for the existence of the process solution. W...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R , whose solution is interpreted in the mild sense. The noise Ẇ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the ...

The Stochastic Wave Equation with Fractional Noise: a random field approach

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index H > 1/2. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in [10], when the noise is white in time. Under this condition, we show that the solution is L(Ω)-continuous. Similar results are o...

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

We consider the stochastic heat equation with a multiplicative colored noise term on Rd for d ≥ 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems...