## Electronic Journal of Probability

### Hölder continuity of the solutions to a class of SPDE’s arising from branching particle systems in a random environment

#### 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 $\epsilon >0$.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 105, 52 pp.

Dates
Accepted: 8 September 2019
First available in Project Euclid: 1 October 2019

https://projecteuclid.org/euclid.ejp/1569895474

Digital Object Identifier
doi:10.1214/19-EJP357

#### Citation

Hu, Yaozhong; Nualart, David; Xia, Panqiu. Hölder continuity of the solutions to a class of SPDE’s arising from branching particle systems in a random environment. Electron. J. Probab. 24 (2019), paper no. 105, 52 pp. doi:10.1214/19-EJP357. https://projecteuclid.org/euclid.ejp/1569895474

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