Brownian Local Time Research Articles

Resolvent stochastic processes

A family r λ r_\lambda , λ ∈ C \lambda \in \mathbb {C} , of complex stochastic processes is introduced, which makes it possible to construct a probabilistic representation for the resolvent of the operator − 1 2 d 2 d x 2 -\frac {1}{2}\frac {d^2}{dx^2} . For λ = 0 \lambda =0 , the process r λ r_\lambda coincides with the Brownian local time process.

Limit theorems for general functionals of Brownian local times

Limit theorems for general functionals of Brownian local times

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time . If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density . The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location and the amount of time spent at the origin. The continuum limit involves rescaling by a factor , where is the lattice spacing. In the limit , the rescaled functional becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

On local times of Martin-Löf random Brownian motion

In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-Löf random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed.

Local and global comparisons of the Airy difference profile to Brownian local time

Local and global comparisons of the Airy difference profile to Brownian local time

Boundary homogenization for patchy surfaces trapping patchy particles.

Trapping diffusive particles at surfaces is a key step in many systems in chemical and biological physics. Trapping often occurs via reactive patches on the surface and/or the particle. The theory of boundary homogenization has been used in many prior works to estimate the effective trapping rate for such a system in the case that either (i) the surface is patchy and the particle is uniformly reactive or (ii) the particle is patchy and the surface is uniformly reactive. In this paper, we estimate the trapping rate for the case that the surface and the particle are both patchy. In particular, the particle diffuses translationally and rotationally and reacts with the surface when a patch on the particle contacts a patch on the surface. We first formulate a stochastic model and derive a five-dimensional partial differential equation describing the reaction time. We then use matched asymptotic analysis to derive the effective trapping rate, assuming that the patches are roughly evenly distributed and occupy a small fraction of the surface and the particle. This trapping rate involves the electrostatic capacitance of a four-dimensional duocylinder, which we compute using a kinetic Monte Carlo algorithm. We further use Brownian local time theory to derive a simple heuristic estimate of the trapping rate and show that it is remarkably close to the asymptotic estimate. Finally, we develop a kinetic Monte Carlo algorithm to simulate the full stochastic system and then use these simulations to confirm the accuracy of our trapping rate estimates and homogenization theory.

Renewal equationsfor single-particle diffusion through a semipermeable interface.

Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability κ_{0} can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflected BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by κ_{0}. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equationthat relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equationcan be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. In each case we show how the solution of the renewal equationsatisfies the classical diffusion equationwith a permeable boundary condition at the interface. That is, the probability flux across the interface is continuous and proportional to the difference in densities on either side of the interface. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy tailed.

A probabilistic model of diffusion through a semi-permeable barrier

Diffusion through semi-permeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogues such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability κ 0 is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or the right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by κ 0 . A new round is then immediately started in either direction with equal probability. In this article, we use a combination of renewal theory, Laplace transforms and Green’s function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modelling diffusion through a semi-permeable barrier. This includes modifications of the diffusion process away from the barrier (e.g. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.

Last passage isometries for the directed landscape

Consider the restriction of the directed landscape \({\mathcal {L}}(x, s; y, t)\) to a set of the form \(\{x_1, \dots , x_k\} \times \{s_0\} \times {\mathbb {R}}\times \{t_0\}\). We show that on any such set, the directed landscape is given by a last passage problem across k locally Brownian functions. The k functions in this last passage isometry are built from certain marginals of the extended directed landscape. As applications of this construction, we show that the Airy difference profile is locally absolutely continuous with respect to Brownian local time, that the KPZ fixed point started from two narrow wedges has a Brownian-Bessel decomposition around its cusp point, and that the directed landscape is a function of its geodesic shapes.

The fractional smoothness of integral functionals driven by Brownian motion

In this paper, we study the fractional smoothness of integral functionals driven by Brownian motion. By using Yamada identities, we also obtain the fractional smoothness of Hilbert transform and fractional derivative with respect to Brownian local time, which recovers the results in Eddahbi et al. (2000).

Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk

This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett. 108 (2012) 040601). In particular, the Bessel process of dimension 4=2+2 appears in the descriptions as a path decomposition of Brownian motion at a local minimum and the Ray–Knight description of Brownian local times near the minimum.

Hedging Strategies in Commodity Markets – Rolling Intrinsic and Delta Hedging for Virtual Power Plants

ABSTRACT Hedging on commodity markets is usually done by applying either the rolling intrinsic strategy or the canonical delta hedge strategy. In this paper we introduce, compare and discuss both hedging strategies in the context of virtual power plants (VPP). We formulate the precise relationship of the two strategies mathematically. Our main result is that they are not only very similar regarding hedge construction but also that both strategies are equal in expectation. The proof involves some stochastic calculus and the Brownian local time. We illustrate our findings with simulated data as well as in prototypical market scenarios. These studies show that the rolling intrinsic hedge comes with a riskier profile than the delta hedge.

Continuity in law of some additive functionals of bifractional Brownian motion

ABSTRACTLet be a bifractional Brownain motion with indices and . We prove the continuity in law, in some anisotropic Besov spaces, with respect to H and K. Our result generalizes those obtained by Jolis and Viles [Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theor. Probab. 20(2) (2007), pp. 133–152] of the fractional Brownian motion local time and gives a new result for the generalized fractional derivatives with kernel depending on slowly varying function of the local time of . Notice that their result was generalized by Wu and Xiao [Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844] for wide class of anisotropic gaussian random fields satisfying some condition (A) which is not satisfied by . To prove our result, we use the decomposition in law of given by Lei and Nualart [A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624]. Our result is also new in the space of continuous functions.

The Local Time Method for Targeting and Selection

We propose a framework for targeting and selection (T&S), a new problem class in simulation optimization where the objective is to select a simulation alternative whose mean performance matches a prespecified target as closely as possible. T&S resembles the more well-known problem of ranking and selection but presents unexpected challenges: for example, a one-step look-ahead method may produce statistically inconsistent estimates of the values, even under very standard normality assumptions. We create a new and fundamentally different approach, based on a Brownian local time model, that exhibits characteristics of two widely studied methodologies, namely expected value of information and optimal computing budget allocation. We characterize the asymptotic sampling rates of this method and relate them to the convergence rates of metrics of interest. The local time method outperforms benchmarks in experiments, including problems where the modeling assumptions of T&S are violated. The online appendix is available at https://doi.org/10.1287/opre.2018.1731 .

On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema

This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.

Squared Bessel processes of positive and negative dimension embedded in Brownian local times

The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell $ of a reflecting Brownian motion $|B|$ by a multiple $\mu $ of its local time process $\ell $ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.

A limit theorem for moments in space of the increments of Brownian local time

We prove a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. [Ann. Prob. 38 (2010) 396–438] and Rosen [Stoch. Dyn. 11 (2011) 5–48], which were later reproven by Hu and Nualart [Electron. Commun. Probab. 15 (2010) 396–410] and Rosen [In Séminaire de Probabilités XLIII (2011) 95–104 Springer] are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins’ semimartingale decomposition, the Kailath–Segall identity and an asymptotic Ray–Knight theorem by Pitman and Yor.

Strong approximation of stochastic processes at random times and application to their exact simulation

We study the convergence rates of strong approximations of stochastic processes (possibly non semi-martingales) at random times (possibly non stopping times). Examples include Brownian local times at random points, Fractional Brownian motions or diffusion processes at Brownian time. These strong approximation results allow to design an exact simulation scheme.

The fractional derivative for fractional Brownian local time with Hurst index large than 1 / 2

Let \(B^H\) be a fractional Brownian motion with Hurst index \(\frac{1}{2}<H<1\) and the weighted local time \({\mathscr {L}}^H(x,t)\). In this paper, we consider the process $$\begin{aligned} {\mathcal S}^{H,\alpha }_t(a):=- \frac{2}{\alpha (1-\alpha )}\left( \left( B^H_t-a\right) _{+}^{1-\alpha }- (-a)_{+}^{1-\alpha }\right) + \frac{2}{\alpha }\int _0^t\left( B^H_s-a\right) _{+}^{-\alpha }dB^H_s \end{aligned}$$ with \(t\ge 0\) and \(a\in {\mathbb R}\), where \(0<\alpha <\frac{1-H}{2H}\) and the integral is the Skorohod integral. We show that \(-\frac{\alpha }{\Gamma (1-\alpha )}{\mathcal S}^{H,\alpha }_t(\cdot )\) coincides with the left-handed Marchaud fractional derivative \({\mathscr {D}}^\alpha _{-}{\mathscr {L}}^H(\cdot ,t)\) of order \(\alpha \), and the “occupation formula” $$\begin{aligned} \int _{\mathbb R}{\mathcal S}^{H,\alpha }_t(a)f(a)da=2H\alpha ^{-1}\Gamma (1-\alpha ) \int _0^t({\mathscr {D}}_{+}^\alpha f)\left( B^H_s\right) s^{2H-1}ds \end{aligned}$$ holds for all Holder continuous functions f of order \(\beta >\alpha \) with compact support, where \({\mathscr {D}}^\alpha _{+}\) denotes the right-handed Marchaud fractional derivative of order \(\alpha \).

On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes

In papers by Yor, a remarkable class $$({\varSigma })$$ of submartingales is introduced, which, up to technicalities, are submartingales $$(X_{t})_{t\ge 0}$$ whose increasing process is carried by the times t such that $$X_{t}=0$$ . These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class $$({\varSigma })$$ have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le theoreme de Paul Levy pour des mesures signees. Seminaire de probabilites (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signees. Seminaire de probabilites (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class $$({\varSigma })$$ to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes $$\sum (H)$$ and $$\sum _\mathrm{s}(H)$$ where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.