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On the chaotic character of the stochastic heat equation, before the onset of intermitttency. (English) Zbl 1286.60060
Let \(W = \{ W(t,x) \}\), \(t \geq 0\), \(x \in {\mathbb R}\), denote a real-valued Brownian sheet indexed by \((t,x)\) \(\in\) \({\mathbb R}_+ \times {\mathbb R}\), namely, \(W\) is a centered Gaussian process with covariance \[ \mathrm{Cov}( W(t,x), W(s,y)) = \min (t,s) \times \min ( | x |, | y | ) \times 1_{(0, \infty)} (x,y). \tag{1} \] This paper treats the nonlinear stochastic heat equation \[ \frac{\partial}{\partial t} u_t(x) = \frac{k}{2} \frac{\partial^2}{\partial x^2} u_t(x) + \sigma( u_t(x) ) \frac{\partial^2}{\partial t \partial x} W(t,x), \tag{2} \] where \(x \in {\mathbb R}\), \(t > 0\), \(\sigma : {\mathbb R} \to {\mathbb R}\) is a nonrandom and Lipschitz continuous function, \(k > 0\) is a fixed viscosity parameter and the initial function \(u_0 : {\mathbb R} \to {\mathbb R}\) is bounded, nonrandom and measurable. The mixed partial derivative \(\partial^2 W(t,x)/\partial t \partial x\) is the space-time white noise, and is defined as a generalized Gaussian random field. It is well known that the stochastic heat equation (2) has a weak solution \(\{ u_t(x) \}\), \(t > 0\), \(x \in {\mathbb R}\}\), that is jointly continuous, and also that it is unique up to evanescence. Moreover, the solution can be written in mild form as the a.s. solution to the following stochastic integral equation: \[ u_t(x) = ( p_t * u_0)(x) + \iint_{ (0,t) \times R} p_{t-s}(y-x) \sigma( u_s(y)) W(ds dy), \tag{3} \] where \(p_t(z)\) denotes the free-space heat kernel. The authors establish that the global behavior of the solution depends in a critical manner, on the structure of the initial function \(u_0\), at every fixed time \(t > 0\). This sensitivity to the initial data implies that the solution to the stochastic heat equation is chaotic at fixed times, before the onset of intermittency. Actually, under suitable conditions on \(u_0\) and \(\sigma\), \[ \sup_{ x \in {\mathbb R} } u_t(x) < \infty \quad \text{a.s.}, \tag{4} \] when \(u_0\) has compact support; while when \(u_0\) is bounded uniformly away from zero, \[ \limsup_{ | x | \to \infty} \frac{ u_t(x)}{ ( \log | x | )^{1/6} } > 0 \tag{5} \] holds with probability one. For other related works, see, e.g. [M. Foondun and D. Khoshnevisan, Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 4, 895–907 (2010; Zbl 1210.35305); C. Mueller, Stochastics Stochastics Rep. 37, No. 4, 225–245 (1991; Zbl 0749.60057)].

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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