We discuss the distributions of three functionals of the free Brownian bridge: its $Ł^2$-norm, the second component of its signature and its Lévy area. All of these are freely infinitely divisible. We introduce two representations of the free Brownian bridge as series of free semicircular random variables are used, analogous to the Fourier representations of the classical Brownian bridge due to \ts{Lévy} and \ts{Kac}.
We introduce an infinite time horizon Brownian bridge which is determined by a stochastic Langevin equation with time dependent drift coefficient. We show that this process goes to zero almost surely when the time goes to infinity and study the existence and asymptotic behavior of its local time as well as its Hölder continuity in time variable and in location variable. The main difficulty is the lack of stationarity of the process so that the powerful tools for stationary (Gaussian) processes are not applicable. We employ the Garsia-Rodemich-Rumsey inequality to get around this type of diffic…
A class of bridges of iterated integrals of Brownian motion related to various boundary value problems involving the one-dimensional polyharmonic operator
设$(B(t))_{t\in [0,1]}$为线性布朗运动，$(X_n(t))_{t\in [0,1]}$为布朗运动的$(n-1)$-重积分，$n$为正整数：$$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$本文构造了过程$(X_n(t))_{t\in [0,1]}$的0次和1次之间的几个桥，其中包含$X_n$在0次和1次的连续导数的条件.对于这个族的桥梁，我们作出了相应的某些边值问题有关的一维多调和算子。我们还研究了经典的预测问题。我们的结果涉及各种Hermite插值多项式。
Parameter Estimation for Complex α-Fractional Brownian Bridge
School of Big Data, Baoshan University（宝山学院大数据学院）；School of Mathematics and Statistics, Jiangxi Normal University（江西师范大学数学与统计学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical and Statistical Sciences, Arizona State University（亚利桑那州立大学数学与统计科学学院）
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $μ_{A_n}=(1/n)\sum_{i}δ_{a_i}, μ_{B_n}=(1/n)\sum_i δ_{b_i}$ are discretization of probability measures $μ_A, μ_B$. Under regularity assumptions of $μ_A, μ_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
Consider a system of $N$ non-intersecting Brownian bridges in $[0,1]$, and let $\mathcal M_N(p)$ be the maximal height attained by the top path in the interval $[0,p]$, $p\in[0,1]$. It is known that, under a suitable rescaling, the distribution of $\mathcal M_N(p)$ converges, as $N\to\infty$, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to $p\to1$ and $p\to0$). It is also known that, for fixed $N$, $\mathcal M_N(1)$ is distributed as the top eigenvalue of a random m…
In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation o…
We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde μ({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study …
We consider $N$ non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large $N$ limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and to the Aztec diamond on restricted domains.
