Andersen's parabolic problem on a regular tree
We consider the Cauchy parabolic problem with random potential on a regular tree of a \(T^ \nu\)-connected graph without any cycles, each of whose points has precisely \(\nu\) neighboring points in the succeeding layer of the tree and precisely one neighboring point in the preceding layer: \[ {\partial c(t,x)\over \partial t} = \kappa \Delta c(t,x) + \xi(x) c(t,x), \qquad c(0,x) \equiv 1.\tag{1} \] \((t,x)\) are defined so that \(t \geq 0\), \(t \in \mathbb{R}^ 1\), while \(x \in T^ \nu\) is a point of the tree and \(\kappa\) is a numerical coefficient that characterizes the diffusion. The symbol \(\Delta\) denotes a discrete Laplace operator that acts on real-valued functions \(f : T^ \nu\to \mathbb{R}^ 1\) in the following way: \[ \Delta f(x) := \sum_{| x - x'| = 1} (f(x') - f(x)); \quad x,x' \in T^ \nu. \] For a random potential \(\xi(x)\), it is assuemd that \(\Sigma = \{\xi(x), x\in T^ \nu\}\) is a family of independent identically distributed random variables that are independent of \(t\) such that \[ \ln P \{| \xi(x)| > | y|\} = -| y|^ p, \qquad p > 1\tag{2} \] (equality is understood asymptotically as \(| y| \to \infty\)).
Cauchy parabolic problem with random potential
scientific article; zbMATH DE number 771980
scientific article