# Optimal Hölder Continuity of SHE and SWE with Rough Fractional Noise

@article{Hong2016OptimalHC, title={Optimal H{\"o}lder Continuity of SHE and SWE with Rough Fractional Noise}, author={Jialin Hong and Zhihui Liu}, journal={arXiv: Probability}, year={2016} }

We mainly focus on the stochastic heat equation (SHE), i.e., L = ∂t − ∂xx, and the stochastic wave equation (SWE), i.e., L = ∂tt − ∂xx, with specified initial data. In the SHE case we impose u(0, x) = u0(x), while in the SWE case we further impose ut(0, x) = v0(x). There has been a widespread interest in Holder continuity result for random fields. This type of sample path regularity is a key property that is needed early

## One Citation

Approximating Stochastic Evolution Equations with Additive White and Rough Noises

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2017

Optimal error estimates are obtained for the Galerkin approximations of stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to 1/2.

## References

SHOWING 1-10 OF 26 REFERENCES

SPDEs with rough noise in space: Holder continuity of the solution

- Mathematics
- 2016

We consider the stochastic wave and heat equations with affinemultiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H 2 ( 1 , 1 ).…

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…

SPDEs with rough noise in space: H\"older continuity of the solution

- Mathematics
- 2016

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in…

Stochastic heat equation with rough dependence in space

- Mathematics
- 2015

This paper studies the nonlinear one-dimensional stochastic heat equation
driven by a Gaussian noise which is white in time and which
has the covariance of a fractional Brownian motion with Hurst…

On Hölder continuity of the solution of stochastic wave equations in dimension three

- Mathematics
- 2014

In this paper, we study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path…

Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three

- Mathematics
- 2009

The authors of this title study the sample path regularity of the solution of a stochastic wave equation in spatial dimension d 3. The driving noise is white in time and with a spatially homogeneous…

Holder Continuity for the Stochastic Heat Equation With Spatially Correlated Noise

- Mathematics
- 2002

We study the Holder continuity in time and space of the solution of a stochastic heat equation with spatial parameter of any dimension d and spatially correlated noise. The conditions for this…

Corrections to: Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E. 's

- Mathematics
- 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…

Hitting probabilities for systems of non-linear stochastic heat equations with additive noise

- Mathematics
- 2007

We consider a system of d coupled non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional additive space-time white noise. We establish upper and lower bounds on hitting…

Approximating Stochastic Evolution Equations with Additive White and Rough Noises

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2017

Optimal error estimates are obtained for the Galerkin approximations of stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to 1/2.