Behavior of the solution of a random semilinear heat equation. (English) Zbl 1152.60077

The authors consider the heat equation on \(\mathbb{R}\times \mathbb{R}\) with a quadratic dissipative term and a stationary random source \(\lambda(t)\) at the origin \[ u_{t}+u_{xx}-u^{2}+\lambda(t)\delta_{0}(x)=0. \] The solution of this equation could describe, under suitable rescaling, the density of a system of annihilating Brownian particles. There is a unique positive bounded solution on \(\mathbb{R}\times \mathbb{R}\times\Omega\) and it can be written as \[ u(t,x,\omega)=6(x+a(t,x,\omega))^{-2}. \] The authors prove that if \(\lambda\) is ergodic, a law of large numbers holds : \(\lim_{x\to\infty} a(0,x)={\overline a}\) a.s.; if in addition \(\lambda\) satisfies some mixing conditions a central limit theorem holds : as \(x\to\infty\), \(x(a(0,x)-{\overline a})\) has a limiting normal distribution. As a consequence, this means that, for every \(t\), as \(x\to\infty\), \[ u(t,x)\sim 6x^{-2}-12{\overline a}x^{-3}+12G(t)x^{-4}, \] with \(G(t)\) a Gaussian random variable.


60K37 Processes in random environments
35K55 Nonlinear parabolic equations
60G10 Stationary stochastic processes
60J60 Diffusion processes
35R60 PDEs with randomness, stochastic partial differential equations
37A25 Ergodicity, mixing, rates of mixing
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[1] Bricmont, Random walks in asymmetric random environments, Comm Math Phys 142 (2) pp 345– (1991) · Zbl 0734.60112
[2] Hairer, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann of Math (2) 164 (3) pp 993– (2006) · Zbl 1130.37038
[3] Hall, Martingale limit theory and its application (1980) · Zbl 0462.60045
[4] Lieberman, Second order parabolic differential equations (1996)
[5] Papanicolaou, The mathematics and physics of disordered media (Minneapolis, Minn., 1983) 1035 pp 391– (1983)
[6] Papanicolaou, Statistics and probability: essays in honor of C. R. Rao pp 547– (1982)
[7] Sznitman, Propagation of chaos for a system of annihilating Brownian spheres, Comm Pure Appl Math 40 (6) pp 663– (1987) · Zbl 0669.60094
[8] Sznitman, An invariance principle for isotropic diffusions in random environment, Invent Math 164 (3) pp 455– (2006) · Zbl 1105.60079
[9] Varadhan, Probability theory 7 (2001)
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