Translator Disclaimer
2019 Hölder continuity of the solutions to a class of SPDE’s arising from branching particle systems in a random environment
Yaozhong Hu, David Nualart, Panqiu Xia
Electron. J. Probab. 24: 1-52 (2019). DOI: 10.1214/19-EJP357

Abstract

We consider a $d$-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure $X_{t}^{n}$ converges weakly in the Skorohod space $D([0,T];M_{F}(\mathbb{R} ^{d}))$ and the limit has a density $u_{t}(x)$, where $M_{F}(\mathbb{R} ^{d})$ is the space of finite measures on $\mathbb{R} ^{d}$. We also derive a stochastic partial differential equation $u_{t}(x)$ satisfies. By using the techniques of Malliavin calculus, we prove that $u_{t}(x)$ is jointly Hölder continuous in time with exponent $\frac{1} {2}-\epsilon $ and in space with exponent $1-\epsilon $ for any $\epsilon >0$.

Citation

Download Citation

Yaozhong Hu. David Nualart. Panqiu Xia. "Hölder continuity of the solutions to a class of SPDE’s arising from branching particle systems in a random environment." Electron. J. Probab. 24 1 - 52, 2019. https://doi.org/10.1214/19-EJP357

Information

Received: 18 October 2018; Accepted: 8 September 2019; Published: 2019
First available in Project Euclid: 1 October 2019

zbMATH: 07142899
Digital Object Identifier: 10.1214/19-EJP357

Subjects:
Primary: 60H07, 60H15, 60J68

JOURNAL ARTICLE
52 PAGES


SHARE
Vol.24 • 2019
Back to Top