Abstract:
Random-jump models of transport in disordered system are studied.
They are described by the master equation
$\dot P=-{\mathcal A}\xi P$, where $-{\mathcal A}$ is the generator of the spatially and
temporary uniform random walks process on a regular lattice, $\xi$ is the diagonal
operator, $\xi _{xy}=\xi _x \delta _{xy}$, where $\{\xi _x\}$ are independent
positive random variables with the same distribution. The case is elaborated
when ${\mathcal A}_{xy}={\mathcal A}_{yx}={\mathcal A}_{x-y,0}$, the transition rates are determined by multipole-type interactions and $\{\xi _x\}$ have several first negative moments
(the random-jump-rate model – with unbounded jumps). Methods of asymptotical
expansion of the propagator for small Laplace parameter values and for long times are developed. It is made also by means of functional integral representation. The influence of disorder and of interaction power on the long-time asymptotics is considered. A method of investigation for system with a forced drift along a certain direction is suggested. Some methods of reciprocal transformation of asymptotically–exactly solvable problems are
discussed and linkages with other known models are demonstrated. The $l_1$-norm
of resolvent is obtained for any Markov process with a countable set of states
and $l_1$-bounded generator.
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