### Abstract

Language | English |
---|---|

Pages | 278-318 |

Number of pages | 41 |

Journal | Stochastic and Partial Differential Equations: Analysis and Computations |

Volume | 5 |

Issue number | 2 |

Early online date | 25 Nov 2016 |

DOIs | |

Publication status | Published - 30 Jun 2017 |

### Fingerprint

### Keywords

- stochastic partial differential equations
- evolutionary equations
- stochastic equations of mathematical physics
- weak solutions

### Cite this

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*Stochastic and Partial Differential Equations: Analysis and Computations*, vol. 5, no. 2, pp. 278-318. https://doi.org/10.1007/s40072-016-0088-8

**A solution theory for a general class of SPDEs.** / Süß, André; Waurick, Marcus.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A solution theory for a general class of SPDEs

AU - Süß, André

AU - Waurick, Marcus

PY - 2017/6/30

Y1 - 2017/6/30

N2 - In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of RdRd. The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green’s function is not required.

AB - In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of RdRd. The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green’s function is not required.

KW - stochastic partial differential equations

KW - evolutionary equations

KW - stochastic equations of mathematical physics

KW - weak solutions

UR - https://link.springer.com/journal/40072

U2 - 10.1007/s40072-016-0088-8

DO - 10.1007/s40072-016-0088-8

M3 - Article

VL - 5

SP - 278

EP - 318

JO - Stochastic and Partial Differential Equations: Analysis and Computations

T2 - Stochastic and Partial Differential Equations: Analysis and Computations

JF - Stochastic and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 2

ER -