Brownian Potential Research Articles

Scaling limit theorem for transient random walk in random environment

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.

A Diffusion Process with a Random Potential Consisting of Two Contracted Self-Similar Processes

We study a limiting behavior of a one-dimensional diffusion process with a random potential. The potential consists of two independent contracted self-similar processes with different indices for the right and the left hand sides of the origin. Brox (1986) and Schumacher (1985) studied a diffusion process with a Brownian potential, and showed, roughly speaking, after a long time with high probability the process is at the bottom of a valley. Their result was extended to a diffusion process in an asymptotically self-similar random environment by Kawazu, Tamura and Tanaka (1989). Our model is a variant of their models. But we show, roughly speaking, after a long time it is possible that our process is not at the bottom of a valley. We also study asymptotic behaviors of the minimum process and the maximum process of our process.

Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model.

The Sinai model of a tracer diffusing in a quenched Brownian potential is a much-studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian bridge on a finite interval (0,L) and then periodically repeated over the whole real line and study the power spectrum S(f) of the diffusive process x(t) in such a potential. We show that for most of realizations of x(t) in a given realization of the potential, the low-frequency behavior is S(f)∼A/f^{2}, i.e., the same as for standard Brownian motion, and the amplitude A is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of A of arbitrary, negative, or positive order k and demonstrate that they exhibit a multifractal dependence on k and a rather unusual dependence on the temperature and on the periodicity L, which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of A has a log-normal left tail and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process x(t).

Renewal structure and local time for diffusions in random environment

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0 0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Levy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

A Diffusion Process with a Brownian Potential Including a Zero Potential Part

A one-dimensional diffusion process with a Brownian potential including a zero potential part is studied. The maximum process and the minimum process of the diffusion process are also investigated.

Localization and number of visited valleys for a transient diffusion in random environment

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0 < \kappa < 1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$.We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on adecomposition of the trajectory of $W_{\kappa}$ in the neighborhood of$h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.;

Theoretical study and pathways for nanoparticle capture during solidification of metal melt

Nanocomposites can provide exciting physical, chemical, and mechanical properties for numerous applications. The solidification processing method has great potential for economical fabrication of bulk nanocomposites, especially for those with crystalline materials as the matrix, such as metal matrix nanocomposites (MMNCs). However, it is extremely difficult to effectively capture nanoparticles (less than 100 nm) into the solidification fronts during solidification. It is thus very important to initiate a theoretical study to examine the physics that governs the interactions between nanoparticles and the solidification front, and to provide enabling pathways for effective nanoparticle capture during solidification. The aim of this paper is to establish a theoretical framework for the fundamental understanding of nanoparticle capture during solidification of metal melt in order to obtain bulk MMNCs. A thermodynamically favorable condition is set as the starting point for further theoretical analysis of the three-party model system, namely a nanoparticle–metal-melt–solidification front. Three key interaction potentials, the interfacial energy at short range (0.2–0.4 nm), the van der Waals potential (especially at a longer range beyond 0.4 nm and up to ∼10 nm) and the Brownian potential, were studied. Three possible pathways for nanoparticle capture were thus devised: viscous capture, Brownian capture and spontaneous capture. Spontaneous capture is proposed as the most favorable for nanoparticle capture during solidification of metal melt. The theoretical model of nanoparticle capture from this study will serve as a powerful tool for future experimental studies to realize exciting functionalities offered by bulk MMNCs.

Intermittency and ageing for the symbiotic branching model

Dans le cadre du modèle de branchement symbiotique introduit dans [Stochastic Process. Appl. 114 (2004) 127–160], nous montrons que le vieillissement et l’intermittence présentent différents comportements suivant les cas où la corrélation est négative, positive ou nulle. Notre approche permet de prouver et d’affiner de manière élémentaire des résultats classiques concernant les seconds moments du modèle parabolique d’Anderson avec potentiel Brownien. Nous raffinons aussi quelques résultats récents de vieillissement pour des diffusions interactives à noyaux généraux à portée infinie.

Estimates on the Speedup and Slowdown for a Diffusion in a Drifted Brownian Potential

We study a model of diffusion in a brownian potential. This model was firstly introduced by T. Brox (1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma, introduced at first by K. Kawazu and H. Tanaka (1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (2008). Our results are in agreement with their results in the discrete case.

Rates of convergence of a transient diffusion in a spectrally negative Lévy potential

We consider a diffusion process $X$ in a random L\'{e}vy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation \begin{eqnarray*}\cases{dX_t=d\beta_t-{1/2}\mathbb{V}'(X_t) dt,\cr X_0=0,}\end{eqnarray*} ($\beta$ B. M. independent of $\mathbb{V}$). We study the rate of convergence when the diffusion is transient under the assumption that the L\'{e}vy process $\mathbb{V}$ does not possess positive jumps. We generalize the previous results of Hu--Shi--Yor for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there exists $0<\kappa<1$ such that $\mathbf{E}[e^{\kappa\mathbb{V}_1}]=1$, then $X_t/t^{\kappa}$ converges to some nondegenerate distribution. These results are in a way analogous to those obtained by Kesten--Kozlov--Spitzer for the transient random walk in a random environment.

Annealed tail estimates for a Brownian motion in a drifted Brownian potential

We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [J. Math. Soc. Japan 49 (1997) 189–211] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [Trans. Amer. Math. Soc. 351 (1999) 3915–3934] using stochastic calculus. We studied large deviations [Ann. Probab. 29 (2001) 1173–1204], showing among other results that the rate function in the annealed setting, that is, after averaging over the potential, has a flat piece in the ballistic regime. In this paper we focus on this subexponential regime, proving that the probability of deviating below the almost sure speed has a polynomial rate of decay, and computing the exponent in this power law. This provides the continuous-time analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment [Comm. Math. Phys. 181 (1996) 667–683]. Our method takes a completely different route, making use of Lamperti’s representation together with an iteration scheme.

Limit theorems for a diffusion process with a one-sided Brownian potential

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t &gt; 0, the sample space describing the random potential is divided into two parts, Ãt and B̃t, both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ãt or on B̃t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).

Limit theorems for a diffusion process with a one-sided Brownian potential

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t &amp;gt; 0, the sample space describing the random potential is divided into two parts, à t and B̃ t , both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on à t or on B̃ t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).

Moderate deviations for diffusions with Brownian potentials

We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani’s lemma and Lamperti’s representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571–609] who studied the corresponding problem for Sinai’s random walk in random environment.

Shape of a Barkhausen pulse

The average shape of the pulse in Barkhausen noise has been recently proposed as a tool to compare models and experiments. We compute theoretically the pulse shape of Barkhausen noise in a model describing the motion of a domain wall in an effective Brownian potential. In this framework, the pulse shape is related to the properties of the excursion of a random process in a $c\log(x)-kx$ potential. We record the Barkhausen noise in polycristalline ${FeSi}$ materials, and compare the pulse shape with the one predicted by the domain wall model.

Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model

The real-space renormalization group (RSRG) method introduced previously for the Brownian landscape is generalized to obtain the joint probability distribution of the subset of the important extrema at large scales of other one-dimensional landscapes. For a large class of models we give exact solutions obtained either by the use of constrained path-integrals in the continuum limit, or by solving the RSRG equations via an Ansatz which leads to the Liouville equation. We apply in particular our results to the toy model energy landscape, which consists in a quadratic potential plus a Brownian potential, which describes, among others, the energy of a single domain wall in a 1D random field Ising model (RFIM) in the presence of a field gradient. The measure of the renormalized landscape is obtained explicitly in terms of Airy functions, and allows to study in detail the Boltzmann equilibrium of a particle at low temperature as well as its non-equilibrium dynamics. For the equilibrium, we give results for the statistics of the absolute minimum which dominates at zero temperature, and for the configurations with nearly degenerate minima which govern the thermal fluctuations at very low-temperature. For the dynamics, we compute the distribution over samples of the equilibration time, or equivalently the distribution of the largest barrier in the system. We also study the properties of the rare configurations presenting an anomalously large equilibration time which govern the long-time dynamics. We compute the disorder averaged diffusion front, which interpolates between the Kesten distribution of the Sinai model at short rescaled time and the reaching of equilibrium at long rescaled time. Finally, the method allows to describe the full coarsening (i.e., many domain walls) of the 1D RFIM in a field gradient as well as its equilibrium.

Localization of thermal packets and metastable states in the Sinai model.

We consider the Sinai model describing a particle diffusing in a one-dimensional random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: all thermal trajectories starting from the same initial condition in the same sample remain within a finite distance of each other even in the limit of infinite time. More precisely, he has proved that the disorder average P(t)(y) of the distribution of the relative distance y=x(t)-m(t) with respect to the (disorder-dependent) most probable position m(t), converges in the limit t--> infinity, towards a distribution P(G)(y) defined as a functional of two independent Bessel processes. In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint distribution P( infinity )(y,u) of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute the localization parameters Y(k), representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place in the infinite-time limit, and the correlation function C(l) representing the disorder-averaged probability density that two particles of the thermal packet are at a distance l from each other. We, moreover, prove that our results for Y(k) and C(l) exactly coincide with the thermodynamic limit L--> infinity of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. So even if the Sinai dynamics on the infinite line is always out-of-equilibrium since it consists in jumps in deeper and deeper wells, the particles of the same thermal packet can nevertheless be considered asymptotically as if they were at thermal equilibrium in a Brownian potential. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.

Large Deviations for a Brownian Motion in a Drifted Brownian Potential

We derive a large deviation principle both quenched and annealed for a one-dimensional diffusion process in a drifted Brownian environment providing the continuous time analogue of what Comets, Gantert and Zeitouni recently establish for the random walk in random environment. A key-ingredient, Kotani’s lemma, allows us to compute the corresponding rate functions. The results are more explicit than in the discrete-time setting.

A Diffusion Process with a One-Sided Brownian Potential

A Diffusion Process with a One-Sided Brownian Potential

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.