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The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution
Probability · Submitted 3 Mar 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution.
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The abstract, sentence by sentence
In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, 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. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. | background 0.62 | 0.55 | 0.03 | asserted 0.90 |
| s1 | Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. | result 0.70 | 0.94 | 0.90 | asserted 1.00 |
| s2 | When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. | result 0.99 | 0.86 | 0.36 | asserted 1.00 |
| s3 | When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$. | result 1.00 | 0.87 | 0.05 | asserted 1.00 |
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The paper
Original abstract
In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, 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. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$.
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