Varadhan, S. R. S.; Zygouras, Nikos Behavior of the solution of a random semilinear heat equation. (English) Zbl 1152.60077 Commun. Pure Appl. Math. 61, No. 9, 1298-1329 (2008). 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. Reviewer: Mihai Gradinaru (Rennes) Cited in 1 ReviewCited in 2 Documents MSC: 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 Keywords:annihilating Brownian particles; random semilinear heat equation; space-time Bessel process; Bessel bridge; Feynman-Kac representation; stationary process; ergodic theorem × Cite Format Result Cite Review PDF Full Text: DOI References: [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) · doi:10.1142/3302 [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) · doi:10.1090/cln/007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.