×

Periodic and invariant measures for stochastic wave equations. (English) Zbl 1047.35106

Let \(B_k(t)\in \mathbb{R}^1\) be mutually independent standard Brownian motions. The author studies the semilinear wave equation with random noise \[ u_{tt}+2\alpha u_t-\Delta u+\beta u= f(t,x,u)+\sum_{k=1}^{\infty}g_k(t,x,u)\frac{dB_k(t)}{dt},\tag{1} \]
\[ u(0)=u_0,\quad u_t(0)=u_1\tag{2} \] and states conditions that guarantee the existence of a pathwise unique solution to (1), (2) in a certain functional class. In addition, he proves that if \(X(t)=(u, u_t)\) is a solution to (1), (2), the family of its distributions is tight, and there exist both a periodic measure and an invariant measure for \(X(t)\). Some results are obtained as well for the problem (1), (2) in a bounded domain \( G\) with smooth boundary \(\partial G\) under zero boundary conditions.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L15 Initial value problems for second-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: EuDML EMIS