Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential.
We construct a stochastic process whose drift is a function of the process's local time at a reflecting barrier. The process arose as a model of the interactions of a Brownian particle and an inert particle in (Knight, 2001). Interesting asymptotic results are obtained for two different arrangements of inert particles and Brownian particles. A version of the process in $\Re^d$ is also constructed.
We analyze jump processes $Z$ with ``inert drift'' determined by a ``memory'' process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.
Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.
We analyze a system of stochastic differential equations describing the joint motion of a massive (inert) particle in a viscous fluid in the presence of a gravitational field and a Brownian particle impinging on it from below, which transfers momentum proportional to the local time of collisions. We study the long-time fluctuations of the velocity of the inert particle and the gap between the two particles, and we show convergence in total variation to the stationary distribution is exponentially fast. We also produce matching upper and lower bounds on the tails of the stationary distribution …
Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z…
We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.
The paper deals with the regression model $X_t = θt + B_t$, $t\in[0, T ]$, where $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $θ$ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.
The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed. Our aim is to give an approximation's method of this limit, when the process is a 1-dimensional Itô diffusion whose drift is allowed to explode at the boundary. In a first step, we show how to restrict the study to the case of a diffusion with values in a bounded interval and whose drift is bounded. In a second step, we show an approximation method of the limiting conditional distribution of such diffusions, based on a Fleming-Viot…
Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brow…
Let $a\in\mathbb{R}$ denote an unknown stationary target with a known distribution $μ\in\mathcal{P(\mathbb{R}})$, the space of probability measures on $\mathbb{R}$. A diffusive searcher $X(\cdot)$ sets out from the origin to locate the target. The time to locate the target is $T_a=\inf\{t\ge0: X(t)=a\}$. The searcher has a given constant diffusion rate $D>0$, but its drift $b$ can be set by the search designer from a natural admissible class $\mathcal{D}_μ$ of drifts. Thus, the diffusive searcher is a Markov process generated by the operator $L=\frac D2\frac{d^2}{dx^2}+b(x)\frac d{dx}$. % equi…
We study a class of Markov processes with finite state space and continuous time that have product form stationary distributions. We obtain a number of examples that can generate conjectures for diffusions with inert drift.
We prove central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the ${\mathcal H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor be stationary and square integrable. This improves the best existing result of Komorowski, Landim and Olla (2012), where it is assumed that the stream tensor be in ${\mathcal L}^{\max\{2+δ,d\}}$, with $δ>0$. Our proof relies on the relaxed sector condition of Horváth, Tóth and Vető (2012), and is technical…
We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We als…
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process in a stationary component and a component which vanishes asymptotically for large times in the $L^p-$sense, $p\geq1$. The law of the stationary component is identified with the unique invariant proba…
This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the discrete or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stocha…
This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time $t$ is uniformly bounded by $\mathcal{O}(1/\barγ^2)$ where $\barγ$ is the average drift of the process. The method can be used as a guide for planning simulations of complex queueing systems with non-stationary arrival rates and/or service time.
Entropy Production for Stationary Diffusions on Hilbert Spaces
Institute for Functional Intelligent Materials, National University of Singapore（新加坡国立大学功能智能材料研究所）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
This work is devoted to long-time properties of the Arratia flow with drift -- a stochastic flow on $\mathbb{R}$ whose one-point motions are weak solutions to a stochastic differential equation $dX(t)=a(X(t))dt+dw(t)$ that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow (constructed in \cite{Riabov}) that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullba…
We consider a non-stationary Cox-Ingersoll-Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.
A stochastic differential equation with infinite memory is considered. The drift coefficient of the equation is a nonlinear functional of the past history of the solution. Sufficient conditions for existence and uniqueness of stationary solution are given.
We prove that every probability measure $μ$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $μ$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $γ$ and for the density $f=dμ/dγ$ the integral of $f |\log (f+1)|^α$ against $γ$ is estimated via $\|v\|_{L^1(μ)}$ for all $α<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolute…
