Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRandom Time Change and Related Evolution Equations: Time Asymptotic Behavior
67
0
0.0
(
0
)
Added by
Jos\\'e Lu\\'is da Silva Dr.
Publication date
2019
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The class of subordinators for which asymptotic analysis may be realized is described.
rate research
Read More
We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in cite{A-E} in order to understand the structure of normal martingales in $RR^n$.The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in $CC^N$. We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.
We study continuous-time (variable speed) random walks in random environments on $mathbb{Z}^d$, $dge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space-time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjostrand representation of gradient models with certain non-strictly convex potentials.
It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wide interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of $mathbb{R}^d$-valued Markov processes with the components of an independent multivariate inverse subordinator. As a possible application, we present a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks.
The steady-state simplified Pn (SPn) approximations to the linear Boltzmann equation have been proven to be asymptotically higher-order corrections to the diffusion equation in certain physical systems. In this paper, we present an asymptotic analysis for the time-dependent simplified Pn equations up to n = 3. Additionally, SPn equations of arbitrary order are derived in an ad hoc way. The resulting SPn equations are hyperbolic and differ from those investigated in a previous work by some of the authors. In two space dimensions, numerical calculations for the Pn and SPn equations are performed. We simulate neutron distributions of a moving rod and present results for a benchmark problem, known as the checkerboard problem. The SPn equations are demonstrated to yield significantly more accurate results than diffusion approximations. In addition, for sufficiently low values of n, they are shown to be more efficient than Pn models of comparable cost.
We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rateis given by the directional free energy.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
