Extremal behavior of the heat random field (Q881408)

The authors consider the Cauchy problem for the \(n\)-dimensional heat equation \({\partial u\over\partial t}=\mu\Delta u\), \(t>0\), \(x\in \mathbb R^{n}\), \(\mu>0\), subject to the random initial condition \(u(0,x)=\eta(x)\), \(x\in \mathbb R^{n}\), where \(u=u(t,x)\), \(t>0\), \(x\in \mathbb R^{n}\); \(\Delta\) is Laplacian and \(\eta(x)\), \(x\in \mathbb R^{n}\), is a random field. The exponential type inequality for the distribution of the supremum of the solution of the considered Cauchy problem is obtained.