Intermittency for the wave and heat equations with fractional noise in time. (English) Zbl 1343.60081

Balan, Raluca M.; Conus, Daniel

Ann. Probab. 44, No. 2, 1488-1534 (2016).

The authors consider stochastic wave (SWE) and heat (SHE) equations \[ u_{tt}=\Delta u+u\dot W,\,u(0)=u_0,\,u_t(0)=v_0,\text{ resp. }u_t=\frac 12\Delta u+u\dot W,\,u(0)=u_0 \] on \(\mathbb R^d\), where the Gaussian noise \(\dot W\) is homogeneous in space (with spatial covariance kernel \(f\)) and behaves in time like a fractional Brownian motion with Hurst index \(H>1/2\), and the initial conditions \(u_0\), \(v_0\) are non-negative constants. The dimension \(d\) is restricted to \(d\leq 3\) in the case of the SWE. The spatial covariance kernel \(f\) of \(\dot W\) may be a bounded function, the Riesz kernel, a covariance of a fractional Brownian sheet or a Dirac measure (if \(d=1\)). The authors prove the existence of mild solutions (via the Malliavin calculus), uniqueness of solutions (except for the case of the SWE in \(\mathbb R^3\)), upper estimates for \(\mathbb E\) \(|u(t,x)|^p\) for \(p\geq 2\), lower estimates for \(\mathbb E\) \(|u(t,x)|^2\) and a Feyman-Kac-type representation for the second moment of solutions to the SWE.

Reviewer: Martin Ondreját (Praha)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H07	Stochastic calculus of variations and the Malliavin calculus
60G22	Fractional processes, including fractional Brownian motion

Keywords:

stochastic heat equation; stochastic wave equation; fractional Brownian motion; Malliavin calculus; intermittency

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