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

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.


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
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