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Register a new userContinuous stochastic processes with non-local memory
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We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into expression for the higher-order transition probability function and stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with prescribed pair correlation function. As illustration, some examples of numerical simulation of the processes with non-local memory are presented.
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We study the random processes with non-local memory and obtain new solutions of the Mori-Zwanzig equation describing non-markovian systems. We analyze the system dynamics depending on the amplitudes $ u$ and $mu_0$ of the local and non-local memory and pay attention to the line in the ($ u$, $mu_0$)-plane separating the regions with asymptotically stationary and non-stationary behavior. We obtain general equations for such boundaries and consider them for three examples of the non-local memory functions. We show that there exist two types of the boundaries with fundamentally different system dynamics. On the boundaries of the first type, the diffusion with memory takes place, whereas on borderlines of the second type, the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and non-stationarity, as well as for diffusive and noise-induced-resonance borderlines between them.
We present a treatment of non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and for upto frst order expansion of stretched exponential function for very small $tau rightarrow 0$. A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.
We investigate the standard deviation $delta v(tsamp)$ of the variance $v[xbf]$ of time series $xbf$ measured over a finite sampling time $tsamp$ focusing on non-ergodic systems where independent configurations $c$ get trapped in meta-basins of a generalized phase space. It is thus relevant in which order averages over the configurations $c$ and over time series $k$ of a configuration $c$ are performed. Three variances of $v[xbf_{ck}]$ must be distinguished: the total variance $dvtot = dvint + dvext$ and its contributions $dvint$, the typical internal variance within the meta-basins, and $dvext$, characterizing the dispersion between the different basins. We discuss simplifications for physical systems where the stochastic variable $x(t)$ is due to a density field averaged over a large system volume $V$. The relations are illustrated for the shear-stress fluctuations in quenched elastic networks and low-temperature glasses formed by polydisperse particles and free-standing polymer films. The different statistics of $svint$ and $svext$ are manifested by their different system-size dependence
We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.
The time of the first occurrence of a threshold crossing event in a stochastic process, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional sensor monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the $textit{first detection time}$, which denotes the time when the sensor detects the threshold crossing event for the first time. In this work, a birth-death process monitored by a random intermittent sensor is studied, for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to, and is obtainable from, the first passage time distribution. Our analytical results display an excellent agreement with simulations. Further, this framework is demonstrated in several applications -- the SIS compartmental and logistic models, and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.
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