### Abstract

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains O ⊂ Rd with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces Hγ,q _{p, θ} (O; T). The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted L_{p}(O)-Sobolev spaces into the scale of Besov spacesBα _{τ, τ}(O) 1/τ=α/d+1/p, α >0. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

Original language | English |
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Journal | Electronic Journal of Probability |

Volume | 18 |

DOIs | |

Publication status | Published - 2013 Sep 13 |

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### Keywords

- Adaptive numerical method
- Besov space
- Embedding theorem
- Hölder regularity in time
- L(L)-theory
- Lipschitz domain
- Nonlinear approximation
- Quasi-banach space
- Square root of Laplacian operator
- Stochastic partial differential equation
- Wavelet
- Weighted sobolev space

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

_{q}(L

_{p})-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains.

*Electronic Journal of Probability*,

*18*. https://doi.org/10.1214/EJP.v18-2478