Recurrence of multi-dimensional diffusion processes in Brownian environments

Let B 1,B 2,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W n (t)=B n (B n−1(⋯(B 2(B 1(t)))⋯)). Although the sequence of processes (W n ) n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W n converge to a random probability measure μ ∞. We then prove that μ ∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W ∞ of independent Brownian motions. We also prove that the collection of random variables (W ∞(t),t∈ℝ∖{0}) is exchangeable with directing measure μ ∞.

For $i = 1, \cdots ,d$, let $B_{s_i }^i $ be a one-dimensional Brownian motion on the interval $[0,a_i ]$ with absorption at the endpoints. At each instant in time, one must decide to run some subset of these d Brownian motions while holding the others fixed at their current state. The resulting process evolves in the rectangle $D = [0,a_1 ] \times \cdots \times [0,a_d ]$. If, at some instant, one decides to freeze all of the Brownian motions, then a reward is received in accordance with this final position. Two types of reward functions are considered. First, it is assumed that the reward is zero everywhere in D, except along the d edges that correspond to the coordinate axes. Along these edges, it is given by $C^3 $ strictly concave functions $\gamma _i (x_i )$, which are zero at the endpoints 0 and $a_i $ of their domains. The optimal control for this problem has a simple description. Let \[\Gamma _i \left( {x_i } \right) = - \int_0^{x_i } {u\gamma ''_i } (u)du\] and put \[M_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\max }\limits_j \Gamma _j \left( {x_i } \right)} \right\}.\] It is proved that the optimal control is: On $M_i $ run any Brownian motion except the ith and stop the first time an edge is reached. The second class of reward functions are assumed to be zero everywhere except on the facets of D that meet at the origin. On the ith such facet (i.e., where $x_i = 0$), the reward function is the product of $\gamma _j (x_j )$ for $j \ne i$. Put \[N_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\min }\limits_j \Gamma _j \left( {x_j } \right)} \right\}.\] The optimal control is: On $N_i $ run the ith Brownian motion and stop when a facet of D is reached.

Let be the first exit time of iterated Brownian motion from a domain started at and let be its distribution. In this paper we establish the exact asymptotics of over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob. 14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl. 116 (2006) 905–916], for where . Here λD is the first eigenvalue of the Dirichlet Laplacian in D , and ψ is the eigenfunction corresponding to λD . We also study lifetime asymptotics of Brownian-time Brownian motion, , where X t and Y t are independent one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.

Markov processes conditioned on their location at large exponential times

Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.

Pitman transforms and Brownian motion in the interval viewed as an affine alcove

On the Fourier structure of the zero set of fractional Brownian motion

The previous chapter considered in detail an example driven by a Brownian sheet, namely a stochastic reaction–diffusion equation. In this chapter we consider an application of one of the other formulations of infinite dimensional Brownian motion, which is the infinite sequence of independent one-dimensional Brownian motions. Such a collection will be used to define a general class of Brownian flows of diffeomorphisms [178], which are a special case of the stochastic flows of diffeomorphisms studied in [16, 25, 124, 178]. We will consider small noise asymptotics, prove the corresponding LDP, and then use it to give a Bayesian interpretation of an estimator used for image matching.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

Upper estimates for capacities are studied for symmetric Dirichlet forms which do not necessarily admit intrinsic metric. Firstly, a general method of obtaining a sharp estimate from upper estimates for cut-off functions are provided in the local and non-local cases. Secondly, a capacitary upper estimate is established for the skew product of two symmetric Dirichlet forms for which suitable capacitary estimates are given. Several examples of capacitary upper inequalities in the local and non-local cases are also given. Especially, an upper estimate for the capacity is proved for the symmetric Dirichlet form associated to the Markov process subordinated (in the Bochner sense) to the skew product of two one-dimensional Brownian motions with respect to the local time of the first Brownian motion at the origin. This estimate is used in the recent work by the present author to establish the sharp criteria for recurrence and transience of the above-mentioned process.

Let B be a one-dimensional Brownian motion and \(f: {\mathbb{R}}{\longrightarrow}{\mathbb{R}}\) be a Borel function that is locally integrable on \(\mathbb{R}\backslash \{0\}\). We present necessary and sufficient conditions (in terms of the function f) for the existence of the limit $$\lim_{\varepsilon\downarrow 0}\int_0^t f(B_s)I(\vert B_s\vert > \varepsilon)ds$$ in probability and almost surely. This limit (if it exists) can be called the principal value of the integral \(\int_0^tf(B_s)ds\).

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

We propose a pseudoratchet mechanism based on one-dimensional (1D) quantum Brownian motion, offering a route to energy-efficient Brownian computing without relying on conventional ratchet structures. Quantum resonance effects in 1D system naturally partition the momentum space into dynamically independent subspaces, each of which lacks inversion symmetry. This leads to spontaneous unidirectional particle transport near thermal equilibrium, without external forces or asymmetric potential gradients. We analyze this pseudoratchet mechanism from kinetic and entropic perspectives and show that the directional transport is sustained without energy dissipation, remaining fully consistent with the second law of thermodynamics. Our analysis provides a microscopic and physically grounded foundation for reversible information transport in Brownian computing.

Let X(t) be a controlled one-dimensional standard Brownian motion starting from x∈(−d,d). The problem of optimally controlling X(t) until |X(t)|=d for the first time is solved explicitly in a particular case. The maximal value that the instantaneous reward given for survival in (−d,d) can take is determined.