## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 18, 15 pp.

### Spectral densities related to some fractional stochastic differential equations

Mirko D’Ovidio, Enzo Orsingher, and Ludmila Sakhno

#### Abstract

In this paper we consider fractional higher-order stochastic differential equations of the form \[ \left ( \mu + c_\alpha \frac{d^\alpha } {dt^\alpha } \right )^\beta X(t) = \mathcal{E} (t) , \quad \mu >0,\; \beta >0,\; \alpha \in (0,1) \cup \mathbb{N} \] where $\mathcal{E} (t)$ is a Gaussian white noise. We obtain explicitly the covariance functions and the spectral densities of the stochastic processes satisfying the above equations.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 18, 15 pp.

**Dates**

Received: 7 July 2015

Accepted: 17 February 2016

First available in Project Euclid: 25 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1456412898

**Digital Object Identifier**

doi:10.1214/16-ECP4411

**Mathematical Reviews number (MathSciNet)**

MR3485387

**Zentralblatt MATH identifier**

1343.60071

**Subjects**

Primary: 60K99: None of the above, but in this section 60G60: Random fields

**Keywords**

higher-order heat equations Weyl fractional derivatives airy functions spectral densities

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

D’Ovidio, Mirko; Orsingher, Enzo; Sakhno, Ludmila. Spectral densities related to some fractional stochastic differential equations. Electron. Commun. Probab. 21 (2016), paper no. 18, 15 pp. doi:10.1214/16-ECP4411. https://projecteuclid.org/euclid.ecp/1456412898