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Alain Mazzolo

Publications and source records attributed to Alain Mazzolo.

At least 19 recordsLinked to original sources
arXiv2026

Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences

In this article, we consider the Benes process with drift $\mu(x)=\alpha \tanh(\alpha x + \beta)$, with $\alpha > 0$, $\beta \in \mathbb{R}$, that is, the diffusion defined by the stochastic differential equation $dX(t)=\alpha \tanh(\alpha X(t)+\beta)\,dt + dW(t)$, with an absorbing barrier at $x=a$. After deriving the propagator and key associated quantities--the first-passage-time distribution and the survival probability--we then condition this process to have various prescribed first-passage-time distributions. When the conditioning is imposed at an infinite time horizon, this procedure reveals the existence of different processes that share the same first-passage-time distribution as the Benes process, a phenomenon recently observed in the case of Brownian motion with drift. When the conditioning is imposed at a finite time horizon, the procedure shows that the conditioned Benes process and the Brownian motion with drift under the same conditioning exhibit identical behaviors. This strengthens an elegant result of Benjamini and Lee stating that Brownian motion and the Benes process share the same Brownian bridge, and it also connects with more recent findings obtained by conditioning two independent identical Brownian motions with drift, or two independent Benes processes that annihilate upon meeting. Moreover, we show that several conditioned Benes drifts converge near the absorbing boundary to the drift of the taboo diffusion, i.e., the diffusion conditioned to never reach the absorbing boundary, which motivates a parallel analysis of the taboo process itself. Using Girsanov's theorem, we derive its propagator, first-passage-time distribution, and conditioned versions, thereby further clarifying the structural relationships between Benes, Brownian, and taboo dynamics.

math-ph↗
arXiv2024

Exact solutions for the probability density of various conditioned processes with an entrance boundary

The probability density is a fundamental quantity for characterizing diffusion processes. However, it is seldom known except in a few renowned cases, including Brownian motion and the Ornstein-Uhlenbeck process and their bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In this paper, we utilize Girsanov's theorem, along with a variation of the method of images, to derive the exact expression of the probability density for diffusions that have one entrance boundary. Our analysis encompasses numerous families of conditioned diffusions, including the Taboo process and Brownian motion conditioned on its growth behavior, as well as the drifted Brownian meander and generalized Brownian excursion.

math-ph↗
arXiv2024

Probability density functions for photon propagation in a binary (isotropic-Poisson) statistical mixture with unmatched positives/negatives refractive indexes

The exact homogenized probability density function, for a photon making a step of length $s$ has been analytically derived for a binary (isotropic-Poisson) statistical mixture with unmatched refractive indexes. The companions, exact, homogenized probability density function for a photon to change direction (``scatter'') with an angle $\vartheta$, and the homogenized albedo, have also been obtained analytically. These functions also hold even in the case of negative refractive indexes and allow one to reduce hundreds of MC simulations of photon propagation in complex binary (isotropic-Poisson) statistical mixtures, to only one MC simulation, for an equivalent homogeneous medium. Note, that this is not an approximate approach, but a mathematically equivalent and exact result. Additionally, some tutorial examples of homogenized MC simulations are also given.

physics.optics↗
arXiv2023

First-passage time of a Brownian motion: two unexpected journeys

The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $\mu$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $\gamma(t) = \frac{a}{\sqrt{2 \pi t^3} } e^{-\frac{(a-\mu t)^2}{2 t}}, t>0$, which is normalized only if $\mu \geq 0$. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when $\mu <0$. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented.

cond-mat.stat-mech↗
arXiv2023

Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture

Monte Carlo (MC) simulations allowing to describe photons propagation in statistical mixtures represent an interest that goes way beyond the domain of optics, and can cover, e.g., nuclear reactor physics, image analysis or life science just to name a few. MC simulations are considered a ``gold standard'' because they give exact solutions (in the statistical sense), however, in the case of statistical mixtures they are enormously time consuming and their implementation is often extremely complex. For this reason, the aim of the present contribution is to propose a new approach that should allow us in the future to simplify the MC approach. This is done through an explanatory example, i.e.; by deriving the `exact' analytical expression for the probability density function of photons' random steps (single step function, SSF) propagating in a medium represented as a binary (isotropic-Poisson) statistical mixture. The use of the SSF reduces the problem to an `equivalent' homogeneous medium behaving exactly as the original binary statistical mixture. This will reduce hundreds time-consuming MC simulations to only one equivalent simple MC simulation. To the best of our knowledge the analytically `exact' SSF for a binary (isotropic-Poisson) statistical mixture has never been derived before.

cond-mat.stat-mech↗
arXiv2023

On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

For continuous-time ergodic Markov processes, the Kemeny time $\tau_*$ is the characteristic time needed to converge towards the steady state $P_*(x)$ : in real-space, the Kemeny time $\tau_*$ corresponds to the average of the Mean-First-Passage-Time $\tau(x,x_0) $ over the final configuration $x$ drawn with the steady state $P_*(x)$, which turns out to be independent of the initial configuration $x_0$; in the spectral domain, the Kemeny time $\tau_*$ corresponds to the sum of the inverses of all the non-vanishing eigenvalues $\lambda_n \ne 0 $ of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition : we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever.

cond-mat.stat-mech↗
arXiv2022

Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator

The non-equilibrium Fokker-Planck dynamics in an arbitrary force field $\vec f(\vec r)$ in dimension $N$ is revisited via the correspondence with the non-hermitian quantum mechanics in a scalar potential $V(\vec r)$ and a vector potential $\vec A(\vec r)$. The relevant parameters of irreversibility are then the $\frac{N(N-1)}{2}$ magnetic matrix elements $B_{nm}(\vec x ) =-B_{mn} (\vec x ) = \partial_n A_m (\vec x ) - \partial_m A_n (\vec x )$, while it is enlightening to explore the corresponding gauge transformations of the vector potential $\vec A(\vec r) $. This quantum interpretation is even more fruitful to study the statistics of all the time-additive observables of the stochastic trajectories, since their generating functions correspond to the same quantum problem with additional scalar and/or vector potentials. Our main conclusion is that the analysis of their large deviations properties and the construction of the corresponding Doob conditioned processes can be drastically simplified via the choice of an appropriate gauge for each purpose. This general framework is then applied to the special time-additive observables of Ornstein-Uhlenbeck trajectories in dimension $N$, whose generating functions correspond to quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e. to quantum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we finally focus on the Brownian gyrator in dimension $N=2$ in order to compute the large deviations properties of the entropy production of its stochastic trajectories and to construct the corresponding conditioned processes having a given value of the entropy production per unit time.

cond-mat.stat-mech↗
arXiv2022

Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes

For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $ \Sigma(t)=A(t)+B(t)$. Their asymptotic statistics for large time $t \to + \infty$ involves two very different cases : (i) when the diffusion process $X(t)$ is transient, the two local times $[A(t);B(t)]$ remain finite random variables $[A^*(\infty),B^*(\infty)]$ and we analyze their limiting joint distribution ; (ii) when the diffusion process $X(t)$ is recurrent, we describe the large deviations properties of the two intensive local times $a = \frac{A(t)}{t}$ and $b = \frac{B(t)}{t}$ and of their intensive sum $\sigma = \frac{\Sigma(t)}{t}=a+b$. These properties are then used to construct various conditioned processes $[X^*(t),A^*(t),B^*(t)]$ satisfying certain constraints involving the two local times, thereby generalizing our previous work [arXiv:2205.15818] concerning the conditioning with respect to a single local time $A(t)$. In particular for the infinite time horizon $T \to +\infty$, we consider the conditioning towards the finite asymptotic values $[A^*(\infty),B^*(\infty)]$ or $\Sigma^*(\infty) $, as well as the conditioning towards the intensive values $[a^*,b^*] $ or $\sigma^*$, that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local times in the regime of large deviations. This general construction is then applied to the simplest case where the unconditioned diffusion is the Brownian motion of uniform drift $\mu$.

cond-mat.stat-mech↗
arXiv2022

Conditioning diffusion processes with respect to the local time at the origin

When the unconditioned process is a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, the local time $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ at the origin $x=0$ is one of the most important time-additive observable. We construct various conditioned processes $[X^*(t),A^*(t)]$ involving the local time $A^*(T)$ at the time horizon $T$. When the horizon $T$ is finite, we consider the conditioning towards the final position $X^*(T)$ and towards the final local time $A^*(T)$, as well as the conditioning towards the final local time $A^*(T)$ alone without any condition on the final position $X^*(T)$. In the limit of the infinite time horizon $T \to +\infty$, we consider the conditioning towards the finite asymptotic local time $A_{\infty}^*<+\infty$, as well as the conditioning towards the intensive local time $a^* $ corresponding to the extensive behavior $A_T \simeq T a^*$, that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local time in the regime of large deviations. This general construction is then applied to generate various constrained stochastic trajectories for three unconditioned diffusions with different recurrence/transience properties : (i) the simplest example of transient diffusion corresponds to the uniform strictly positive drift $\mu(x)=\mu>0$; (ii) the simplest example of diffusion converging towards an equilibrium is given by the drift $\mu(x)=- \mu \, {\rm sgn}( x)$ of parameter $\mu>0$; (iii) the simplest example of recurrent diffusion that does not converge towards an equilibrium is the Brownian motion without drift $\mu=0$.

cond-mat.stat-mech↗
arXiv2022

Conditioning diffusion processes with killing rates

When the unconditioned process is a diffusion submitted to a space-dependent killing rate $k(\vec x)$, various conditioning constraints can be imposed for a finite time horizon $T$. We first analyze the conditioned process when one imposes both the surviving distribution at time $T$ and the killing-distribution for the intermediate times $t \in [0,T]$. When the conditioning constraints are less-detailed than these full distributions, we construct the appropriate conditioned processes via the optimization of the dynamical large deviations at Level 2.5 in the presence of the conditioning constraints that one wishes to impose. Finally, we describe various conditioned processes for the infinite horizon $T \to +\infty$. This general construction is then applied to two illustrative examples in order to generate stochastic trajectories satisfying various types of conditioning constraints : the first example concerns the pure diffusion in dimension $d$ with the quadratic killing rate $k(\vec x)= \gamma \vec x^2$, while the second example is the Brownian motion with uniform drift submitted to the delta killing rate $k(x)=k \delta(x)$ localized at the origin $x=0$.

cond-mat.stat-mech↗
arXiv2022

Conditioning two diffusion processes with respect to their first-encounter properties

We consider two independent identical diffusion processes that annihilate upon meeting in order to study their conditioning with respect to their first-encounter properties. For the case of finite horizon $T<+\infty$, the maximum conditioning consists in imposing the probability $P^*(x,y,T ) $ that the two particles are surviving at positions $x$ and $y$ at time $T$, as well as the probability $\gamma^*(z,t) $ of annihilation at position $z$ at the intermediate times $t \in [0,T]$. The adaptation to various conditioning constraints that are less-detailed than these full distributions is analyzed via the optimization of the appropriate relative entropy with respect to the unconditioned processes. For the case of infinite horizon $T =+\infty$, the maximum conditioning consists in imposing the first-encounter probability $\gamma^*(z,t) $ at position $z$ at all finite times $t \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This general framework is then applied to the explicit cases where the unconditioned processes are respectively two Brownian motions, two Ornstein-Uhlenbeck processes, or two tanh-drift processes, in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, the link with the stochastic control theory is described via the optimization of the dynamical large deviations at Level 2.5 in the presence of the conditioning constraints that one wishes to impose.

cond-mat.stat-mech↗
arXiv2022

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

When the unconditioned process is a diffusion living on the half-line $x \in ]-\infty,a[$ in the presence of an absorbing boundary condition at position $x=a$, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning consists in imposing the probability $P^*(y,T ) $ to be surviving at time $T$ and at the position $y \in ]-\infty,a[$, as well as the probability $\gamma^*(T_a ) $ to have been absorbed at the previous time $T_a \in [0,T]$. When the time horizon is infinite $T=+\infty$, the conditioning consists in imposing the probability $\gamma^*(T_a ) $ to have been absorbed at the time $T_a \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This case of infinite horizon $T=+\infty$ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passage-time properties at position $a$. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift $\mu$ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

cond-mat.stat-mech↗
arXiv2020

A universal property of random trajectories in bounded domains

The celebrated invariance property states that particles entering a bounded domain, with isotropic and uniform incidence, spend on average $\langle \ell \rangle=4V/S$ length inside, no matter how they scatter. We show that this remarkable property is merely the infinite-length limit of an even broader law: for any curves randomly placed and oriented in space -- stochastic or deterministic, generated by ballistic or diffusive dynamics, with possible stopping or branching, in two or more dimensions -- $ \displaystyle \frac{1}{\langle \ell \rangle}= \frac{1}{\langle L\rangle}+ \frac{1}{\langle \sigma \rangle} $, with $\langle\ell\rangle$ its mean in-domain path, $\langle L\rangle$ its mean total length, and $\langle\sigma\rangle$ the mean chord of the domain, a known geometric quantity related to the volume-to-surface ratio. Derived solely from the kinematic formula of integral geometry, the result is independent of step-length statistics, memory, absorption, and branching, making it equally relevant to photons in turbid tissue, active bacteria in micro-channels, cosmic rays in molecular clouds, or neutron chains in nuclear reactors. Monte-Carlo simulations spanning straight needles, Y-shapes, and isotropic random walks in 2D and 3D confirm the universality and demonstrate how a local measurement of $\langle \ell \rangle$ yields $\langle L\rangle$ without ever tracking the full trajectory.

math-ph↗
arXiv2019

Strongly constrained stochastic processes: the multi-ends Brownian bridge

In a recent article, Krapivsky and Redner (J. Stat. Mech. 093208 (2018)) established that the distribution of the first hitting times for a diffusing particle subject to hitting an absorber is independent of the direction of the external flow field. In the present paper, we build upon this observation and investigate when the conditioning on the diffusion leads to a process that is totally independent of the flow field. For this purpose, we adopt the Langevin approach, or more formally the theory of conditioned stochastic differential equations. This technique allows us to derive a large variety of stochastic processes: in particular, we introduce a new kind of Brownian bridge ending at two different final points and calculate its fundamental probabilities. This method is also very well suited for generating statistically independent paths. Numerical simulations illustrate our findings.

cond-mat.stat-mech↗
arXiv2018

Sweetest taboo processes

Brownian dynamics play a key role in understanding the diffusive transport of micro particles in a bounded environment. In geometries containing confining walls, physical laws determine the behavior of the random trajectories at the boundaries. For impenetrable walls, imposing reflecting boundary conditions to the Brownian particles leads to dynamics described by reflecting stochastic differential equations. In practice, these stochastic differential equations as well as their refinements are quite challenging to handle, and more importantly, many physical processes are better modeled by processes conditioned to stay in a prescribed bounded region. In the mathematical literature, these processes are known as taboo processes, and despite their simplicity, at least compared to the reflecting stochastic differential equations approach, are surprisingly not much exploited in physics. This paper explores some aspect of taboo processes and other constrained processes in simple geometries: Interval in one dimension, circular annulus in two dimensions, hollow sphere in three dimensions, and more. In particular, for the two-dimensional taboo process in a circular annulus, the Gaussian behavior of the stochastic angle is established.

cond-mat.stat-mech↗
arXiv2017

Poisson-Box Sampling algorithms for three-dimensional Markov binary mixtures

Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories. Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media. CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions. In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in $d$-dimensional binary Markov mixtures. In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Larsen and Pomraning and extended by Brantley: for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients. PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.

cond-mat.stat-mech↗
arXiv2017

Constraint Ornstein-Uhlenbeck bridges

In this paper, we study the Ornstein-Uhlenbeck bridge process (i.e. the Ornstein-Uhlenbeck process conditioned to start and end at fixed points) constraints to have a fixed area under its path. We present both anticipative (in this case, we need the knowledge of the future of the path) and non-anticipative versions of the stochastic process. We obtain the anticipative description thanks to the theory of generalized Gaussian bridges while the non-anticipative representation comes from the theory of stochastic control. For this last representation, a stochastic differential equation is derived which leads to an effective Langevin equation. Finally, we extend our theoretical findings to linear bridge processes.

cond-mat.stat-mech↗
arXiv2017

Monte Carlo particle transport in random media: the effects of mixing statistics

Particle transport in random media obeying a given mixing statistics is key in several applications in nuclear reactor physics and more generally in diffusion phenomena emerging in optics and life sciences. Exact solutions for the ensemble-averaged physical observables are hardly available, and several approximate models have been thus developed, providing a compromise between the accurate treatment of the disorder-induced spatial correlations and the computational time. In order to validate these models, it is mandatory to resort to reference solutions in benchmark configurations, typically obtained by explicitly generating by Monte Carlo methods several realizations of random media, simulating particle transport in each realization, and finally taking the ensemble averages for the quantities of interest. In this context, intense research efforts have been devoted to Poisson (Markov) mixing statistics, where benchmark solutions have been derived for transport in one-dimensional geometries. In a recent work, we have generalized these solutions to two and three-dimensional configurations, and shown how dimension affects the simulation results. In this paper we will examine the impact of mixing statistics: to this aim, we will compare the reflection and transmission probabilities, as well as the particle flux, for three-dimensional random media obtained by resorting to Poisson, Voronoi and Box stochastic tessellations. For each tessellation, we will furthermore discuss the effects of varying the fragmentation of the stochastic geometry, the material compositions, and the cross sections of the transported particles.

cond-mat.stat-mech↗