# Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

@article{Herrell2018SharpSR, title={Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise}, author={Rande K. Herrell and Renming Song and Dongsheng Wu and Yimin Xiao}, journal={Stochastic Analysis and Applications}, year={2018}, volume={38}, pages={747 - 768} }

Abstract In this article, we study the following stochastic heat equation where is the generator of a Lévy process X in B is a fractional-colored Gaussian noise with Hurst index in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution in both time and space variables. Under mild conditions, we establish the exact uniform…

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## References

SHOWING 1-10 OF 61 REFERENCES

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

- Mathematics
- 2007

In this article we consider the stochastic heat equation ut −�u = u B in (0,T) ×R d , with vanishing initial conditions, driven by a Gaussian noise u B which is fractional in time, with Hurst index H…

Stochastic heat equation driven by fractional noise and local time

- Mathematics
- 2007

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with…

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

- Mathematics
- 2014

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of…

SPDEs with affine multiplicative fractional noise in space with index 14 < H < 1 2

- 2015

In this article, we consider the stochastic wave and heat equations on R with nonvanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional…

Feynman–Kac formula for heat equation driven by fractional white noise

- Mathematics
- 2009

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to…

Quadratic variations for the fractional-colored stochastic heat equation

- Mathematics
- 2014

Using multiple stochastic integrals and Malliavin calculus, we analyze the quadratic variations of a class of Gaussian processes that contains the linear stochastic heat equation on $\mathbf{R}^{d}$…

EXTENDING MARTINGALE MEASURE STOCHASTIC INTEGRAL WITH APPLICATIONS TO SPATIALLY HOMOGENEOUS

- 1999

We extend the definition of Walsh’s martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green’s function is not a function but a Schwartz…

Sample paths of the solution to the fractional-colored stochastic heat equation

- Mathematics
- 2015

Let {u(t,x),t ∈ [0,T],x ∈ ℝd} be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the…

Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2

- Mathematics
- 2012

In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst…

Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise

- Mathematics
- 2016

This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $$H\in (0,1/2)$$H∈(0,1/2). We establish the Feynman–Kac representation of the solution…