One-dimensional Brownian Motion Research Articles

On the Identification of Noncausal Wiener Functionals from the Stochastic Fourier Coefficients

Let $$(B_t)_{t\in [0,1]}$$ be a one-dimensional Brownian motion starting at the origin and $$(e_n)_{n\in \mathbb {N}}$$ a complete orthonormal system of $$L^2([0,1],\mathbb {C})$$ . Ogawa and Uemura (J Theor Probab 27:370–382, 2014) defined the stochastic Fourier coefficient $$\hat{a}_{n}(\omega )$$ of a random function $$a(t,\omega )$$ by $$\hat{a}_{n}(\omega ):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t$$ , where $$\int \,\delta B$$ stands for the Skorokhod integral, and considered the question of whether the random function $$a(t,\omega )$$ can be identified from the totality of $$\hat{a}_{n}$$ , $$n\in \mathbb {N}$$ . Ogawa and Uemura (Bull Sci Math 138:147–163, 2014) discussed the identification problem for the stochastic Fourier coefficient $$\mathcal {F}_{n}(\delta X)$$ of a Skorokhod-type stochastic differential $$\delta X_t=a(t,\omega )\,\delta B_t+b(t,\omega )\,\hbox {d}t$$ , where $$\mathcal {F}_{n}(\delta X)$$ is defined by $$\mathcal {F}_{n}(\delta X):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t +\int _0^1 \overline{e_n(t)} b(t,\omega ) \, \hbox {d}t$$ . They obtained affirmative answers to these questions under certain conditions. In this paper, we ask similar questions and we obtain affirmative answers under weaker conditions. Moreover, we consider the identification problem for the stochastic Fourier coefficients based on Ogawa integral.

Scaling limit for a neuron-firing network model

The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in ℝ and at all points of ℝ. In this paper, we only consider starting times which are nonnegative, that is, all the space-time points in ℝ× [0, ∞]; accordingly, we call it nonnegative Brownian web. Moreover, we introduce a model named neuron-firing network model and show that under diffusive scaling this model converges in distribution to the nonnegative Brownian Web.

The first hitting time of the integers by symmetric Lévy processes

For one-dimensional Brownian motion, the exit time from an interval has finite exponential moments and its probability density is expanded in exponential terms. In this note we establish its counterpart for certain symmetric Lévy processes. Applying the theory of Pick functions, we study properties of the Laplace transform of the first hitting time of the integer lattice as a meromorphic function in detail. Its density is expanded in exponential terms and the poles and the zeros of a Pick function play a crucial role.Intermediate results concerning finite exponential moments are also obtained for a class of nonsymmetric Lévy processes.

Markov processes conditioned on their location at large exponential times

Suppose that (Xt)t≥0 is a one-dimensional Brownian motion with negative drift −μ. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like X conditioned to hit 0, after which time it behaves as X killed at the last time X visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift +μ when it is negative, like Brownian motion with negative drift −μ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state a at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the h-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.

English

We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small L2-perturbations.

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.

The flashing Brownian ratchet and Parrondo's paradox.

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

The orthogonal complements of $\boldsymbol{H^1(\mathbb{R})}$ in its regular Dirichlet extensions

Consider the regular Dirichlet extension $({\mathcal{E}},{\mathcal{F}})$ for one-dimensional Brownian motion, that $H^1(\mathbb{R})$ is a subspace of ${\mathcal{F}}$ and ${\mathcal{E}}(f,g)=\frac12{D}(f,g)$ for $f,g\in~H^1(\mathbb{R})$. Both $H^1(\mathbb{R})$ and ${\mathcal{F}}$ are Hilbert spaces under ${\mathcal{E}}_\alpha$ and hence there is $\alpha$-orthogonal compliment $\mathcal{G}_\alpha$. We give the explicit expression for functions in $\mathcal{G}_\alpha$ which then can be described by other two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.

Electrical autonomous Brownian gyrator.

We study experimentally and theoretically the steady-state dynamics of a simple stochastic electronic system featuring two resistor-capacitor circuits coupled by a third capacitor. The resistors are subject to thermal noises at real temperatures. The voltage fluctuation across each resistor can be compared to a one-dimensional Brownian motion. However, the collective dynamical behavior, when the resistors are subject to distinct thermal baths, is identical to that of a Brownian gyrator, as first proposed by Filliger and Reimann [Phys. Rev. Lett. 99, 230602 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.230602]. The average gyrating dynamics is originated from the absence of detailed balance due to unequal thermal baths. We look into the details of this stochastic gyrating dynamics, its dependences on the temperature difference and coupling strength, and the mechanism of heat transfer through this simple electronic circuit. Our work affirms the general principle and the possibility of a Brownian ratchet working near room temperature scale.

Transition path times reveal memory effects and anomalous diffusion in the dynamics of protein folding.

Recent single-molecule experiments probed transition paths of biomolecular folding and, in particular, measured the time biomolecules spend while crossing their free energy barriers. A surprising finding from these studies is that the transition barriers crossed by transition paths, as inferred from experimentally observed transition path times, are often lower than the independently determined free energy barriers. Here we explore memory effects leading to anomalous diffusion as a possible origin of this discrepancy. Our analysis of several molecular dynamics trajectories shows that the dynamics of common reaction coordinates used to describe protein folding is subdiffusive, at least at sufficiently short times. We capture this effect using a one-dimensional fractional Brownian motion (FBM) model, in which the system undergoes a subdiffusive process in the presence of a potential of mean force, and show that this model yields much broader distributions of transition path times with stretched exponential long-time tails. Without any adjustable parameters, these distributions agree well with the transition path times computed directly from protein trajectories. We further discuss how the FBM model can be tested experimentally.

On structure of regular Dirichlet subspaces for one-dimensional Brownian motion

The main purpose of this paper is to explore the structure of regular Dirichlet subspaces of one-dimensional Brownian motion. As stated in [Osaka J. Math. 42 (2005) 27–41], every such regular Dirichlet subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^{c}$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular Dirichlet subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular Dirichlet subspace on $F$ is still a regular Dirichlet subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover, we shall prove that the trace of Brownian motion on $F$ may be decomposed into two parts; one is the trace of the regular Dirichlet subspace on $F$, which has only the nonlocal part and the other comes from the orthogonal complement of the regular Dirichlet subspace, which has only the local part. Actually the orthogonal complement of regular Dirichlet subspace corresponds to a time-changed absorbing Brownian motion after a darning transform.

Stochastic Operational Matrix Method for Solving Stochastic Differential Equation by a Fractional Brownian Motion

This article proposes an effective method for solving stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter \(H\in (0,{1\over 2})\) and n independent one-dimensional standard Brownian motion. Hat basis functions and their stochastic operational matrix, convert the SDE into a linear lower triangular system. Also, the error analysis of the proposed method is investigated and we prove that the order of convergence is \(O(h^2)\). Then, numerical examples affirm the efficiency of the method.

Brownian Random Walk

We construct a model of one-dimensional Brownian motion in the form of a random walk as an alternative to the Wiener random process.

Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. This complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity. We also show that the end point of the process conditioned on survival up to time $t$ rescaled by $\sqrt{t} $ converges in distribution to a non-trivial random variable, as $t$ tends to infinity, which is in fact invariant with respect to the drift $h>0$. We thus prove that it is sub-ballistic and estimate the speed of escape. The latter is in a sharp contrast with discrete models of dimension larger or equal to $2$ when the behaviour at criticality is ballistic, see [7], and even to many one dimensional models which exhibit ballistic behaviour at criticality, see [8].

Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments

We consider $d$-dimensional diffusion processes in multi-parameter random environments which are given by values at different $d$ points of one-dimensional $\alpha $-stable or $(r, \alpha )$-semi-stable Levy processes. From the model, we derive some conditions of random environments that imply the dichotomy of recurrence and transience for the $d$-dimensional diffusion processes. The limiting behavior is quite different from that of a $d$-dimensional standard Brownian motion. We also consider the direct product of a one-dimensional diffusion process in a reflected non-positive Brownian environment and a one-dimensional standard Brownian motion. For the two-dimensional diffusion process, we show the transience property for almost all reflected Brownian environments.

Exact distributions of cover times for N independent random walkers in one dimension.

We study the probability density function (PDF) of the cover time t_{c} of a finite interval of size L by N independent one-dimensional Brownian motions, each with diffusion constant D. The cover time t_{c} is the minimum time needed such that each point of the entire interval is visited by at least one of the N walkers. We derive exact results for the full PDF of t_{c} for arbitrary N≥1 for both reflecting and periodic boundary conditions. The PDFs depend explicitly on N and on the boundary conditions. In the limit of large N, we show that t_{c} approaches its average value of 〈t_{c}〉≈L^{2}/(16DlnN) with fluctuations vanishing as 1/(lnN)^{2}. We also compute the centered and scaled limiting distributions for large N for both boundary conditions and show that they are given by nontrivial N independent scaling functions.

Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process

Abstract In this paper, we consider both, the strong and weak convergence of the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time L t a ${L_{t}^{a}}$ from the stochastic differential equations of type X t = X 0 + ∫ 0 t φ ⁢ ( X s ) ⁢ 𝑑 B s + ∫ ℝ ν ⁢ ( d ⁢ a ) ⁢ L t a . $X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})\,dB_{s}+\int_{\mathbb{R}}\nu(da)L_{t}^{% a}.$ Here B is a one-dimensional Brownian motion, φ : ℝ → ℝ ${\varphi:\mathbb{R}\rightarrow\mathbb{R}}$ is a bounded measurable function, and ν is a bounded measure on ℝ ${\mathbb{R}}$ . We provide the approximation of Euler–Maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler–Maruyama X t n ${X_{t}^{n}}$ of the above mentioned equation, and we provide the rate of strong convergence Error = 𝔼 ⁢ | X T - X T n | ${\operatorname{Error}=\mathbb{E}\lvert X_{T}-X_{T}^{n}\rvert}$ , and the rate of weak convergence Error = 𝔼 ⁢ | G ⁢ ( X T ) - G ⁢ ( X T n ) | ${\operatorname{Error}=\mathbb{E}\lvert G(X_{T})-G(X_{T}^{n})\rvert}$ , for any function G : ℝ → ℝ ${G:\mathbb{R}\rightarrow\mathbb{R}}$ of bounded variation.

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

The α-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion

Roughly speaking, a regular subspace of a Dirichlet form is a subspace, which is also a regular Dirichlet form, on the same state space. In particular, the domain of regular subspace is a closed subspace of the Hilbert space induced by the domain and $\alpha$-inner product of original Dirichlet form. We shall investigate the orthogonal complement of regular subspace of 1-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the $\alpha$-harmonic equation under Neumann boundary condition.

Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance

In this paper, we address the problem of optimal management of renewable resources such as agricultural commodities and fishery production. For that purpose, we consider the population associated with such commodities and assume that its size evolves according to a logistic growth model with a predation term given by a Holling type-n functional response. Additionally, we assume that such population is subject to random fluctuations, modeled by a diffusive term driven by a one-dimensional Brownian motion and having a power-type coefficient, thus endowing the model under consideration with the property of having constant elasticity of variance. Since the stochastic differential equation associated with this model does not fit the standard assumptions in the stochastic optimal control literature, namely sublinear growth, we develop an appropriate version of the dynamic programming principle for the problem under consideration herein, proceeding also to provide a characterization of the optimal harvesting strategies and discuss some qualitative properties of the corresponding value function.