arXiv subjects

Marco Bianucci

Publications and source records attributed to Marco Bianucci.

5 recordsLinked to original sources

arXiv2026

Exact and limit results for the CTRW in presence of drift and position dependent noise intensity

Continuous-time random walks (CTRWs) with drift and position-dependent jumps provide a general framework for describing a wide range of natural and engineered systems. We analyze the stochastic differential equation associated with this class of models, in which the driving noise consists of spike (shot) events, and we derive two exact analytical results. First, we obtain a closed-form expression for the $n$-time correlation functions of The noise, expressed as a sum over all $2^{n-1}$ ordered partitions of the observation times (Proposition 2). Second, using the $G$-cumulant formalism, we derive an \emph{exact} non-local master equation (ME) for the probability density function of the CTRW variable, valid without invoking diffusive limits, fractional scaling assumptions, or closure hypotheses (Proposition 3). In interaction representation, this ME retains the same structural form as that of the standard CTRW without drift or position-dependent jumps. Our main result is the emergence of a \emph{universal local master equation}: at long times, the exact non-local ME is universally and accurately approximated by a time-local ME whose only coefficient is the instantaneous renewal rate $R(t)$. From this equation, exact in the well known Poissonian case, both local and global properties of the PDF can be readily inferred. For example, the temporal behavior of the PDF is directly controlled by that of the rate function $R(t)$: if the waiting-time distribution decays as a power law with exponent $\mu>2$, then $R(t)\to const$ and the system converges to the Poissonian equilibrium. By contrast, for $\mu<2$, the rate decays in time and the effective diffusion induced by the noise slowly weakens, without leading to a stationary state. Numerical experiments confirm its remarkable accuracy even far beyond regimes where a naive time-scale separation would justify it.

cond-mat.stat-mech↗

arXiv2025

Universal behaviors of the multi-time correlation functions of random processes with renewal: the step noise case (the random velocity of a L\'evy walk)

Stochastic processes with renewal properties are powerful tools for modeling systems where memory effects and long-time correlations play a significant role. In this work, we study a broad class of renewal processes where a variable's value changes according to a prescribed Probability Density Function (PDF), $p(\xi)$, after random waiting times $\theta$. This model is relevant across many fields, including classical chaos, nonlinear hydrodynamics, quantum dots, cold atom dynamics, biological motion, foraging, and finance. We derive a general analytical expression for the $n$-time correlation function by averaging over process realizations. Our analysis identifies the conditions for stationarity, aging, and long-range correlations based on the waiting time and jump distributions. Among the many consequences of our analysis, two new key results emerge. First, for Poissonian waiting times, the correlation function quickly approaches that of telegraphic noise. Second, for power-law waiting times with $\mu>2$, , \emph{any $n$-time correlation function asymptotically reduces to the two-time correlation evaluated at the earliest and latest time points}. This second result reveals a universal long-time behavior where the system's full statistical structure becomes effectively two-time reducible. Furthermore, if the jump PDF $p(\xi)$ has fat tails, this convergence becomes independent of the waiting time PDF and is significantly accelerated, requiring only modest increases in either the number of realizations or the trajectory lengths. Building upon earlier work that established the universality of the two-point correlation function (i.e., a unique formal expression depending solely on the variance of $\xi$ and on the waiting-time PDF), the present study extends that universality to the full statistical description of a broad class of renewal-type stochastic processes.

cond-mat.stat-mech↗

arXiv2024

Colored Stochastic Multiplicative Processes with Additive Noise Unveil a Third-Order PDE, Defying Conventional FPE and Fick-Law Paradigms

Research on stochastic differential equations (SDE) involving both additive and multiplicative noise has been extensive. In situations where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast factor, including phenomena like thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. This work focuses on a significant class of such systems, particularly those characterized by linear drift and multiplicative noise, extensively explored in the literature. Conventionally, multiplicative stochastic processes are also treated as white noise in existing studies. However, when considering colored multiplicative noise, the emphasis has been on characterizing the far tails of the probability density function (PDF), regardless of the spectral properties of the noise. In the absence of additive noise and with a general colored multiplicative SDE, standard perturbation approaches lead to a second-order PDE known as the Fokker-Planck Equation (FPE), consistent with Fick's law. This investigation unveils a notable departure from this standard behavior when introducing additive white noise. At the leading order of the stochastic process strength, perturbation approaches yield a \textit{third-order PDE}, irrespective of the white noise intensity. The breakdown of the FPE further signifies the breakdown of Fick's law. Additionally, we derive the explicit solution for the equilibrium PDF corresponding to this third-order PDE Master Equation. Through numerical simulations, we demonstrate significant deviations from outcomes derived using the FPE obtained through the application of Fick's law.

math.ST↗

arXiv2020

Optimal FPE for non-linear 1d-SDE. I: Additive Gaussian colored noise

Many complex phenomena occurring in physics,chemistry, biology, finance, etc. can be reduced, by some projection process, to a 1-d stochastic Differential Equation (SDE) for the variable of interest. Typically, this SDE is both non-linear and non-markovian, so a Fokker Planck equation (FPE), for the probability density function (PDF), is generally not obtainable. However, a FPE is desirable because it is the main tool to obtain relevant analytical statistical information such as stationary PDF and First Passage Time. This problem has been addressed by many authors in the past, but due to an incorrect use of the interaction picture (the standard tool to obtain a reduced FPE) previous theoretical results were incorrect, as confirmed by direct numerical simulation of the SDE. We will show, in general, how to address the problem and we will derived the correct best FPE from a perturbation approach. The method followed and the results obtained have a general validity beyond the simple case of exponentially correlated Gaussian driving used here as an example; they can be applied even to non Gaussian drivings with a generic time correlation.

cond-mat.stat-mech↗

arXiv2019

About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach

More than fifty years ago, in a couple of seminal works Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept, implicitly introduced by Laplace in 1810. Kubo's idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo's idea, today not yet fully exploited. It results that for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo-Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it {\em does not} depend on the specific nature of the moments, but just on the definition of the generalized cumulants.

math-ph↗