One-dimensional Brownian motion and the three-dimensional Bessel process

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.

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

Markov processes conditioned on their location at large exponential times

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.

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.

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 \(\int_0^tf(B_s)ds\).

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.

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.

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.

Let Wt be a one‐dimensional Brownian motion on the probability space (Ω,F,P), and let dxt = a(xt)dt + b(xt)dwt, b2(x) > 0, be a one‐dimensional Ito stochastic differential equation. For a(x) = a0 + a1x + … + anxn on a bounded interval we obtain a lower bound for p(t,x,y), the transition density function of the homogeneous Markov process xt, depending directly on the coefficients a0,a1, …, an, and b(x).

We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.

We give best tests and corresponding power functions for testing the hypothesis of no drift versus the alternative of positive linear drift in a one-dimensional Brownian motion process. The tests are based on a simple random sample from a first passage time distribution associated with the Brownian motion process. In reliability applications the first passage times would be the lifetimes of objects which fail when some noisy physical parameter reaches a critical value. In this case the life distribution is determined by the assumption that the physical parameter behaves like a Brownian motion and then the problem is equivalent to testing the hypothesis of infinite mean life versus the alternative of finite mean life. These tests are also applicable (and remain best) in the case of a single realization of Brownian motion for which only certain crossing time data is available.