Conditional Brownian Motion Research Articles

The mean of the running maximum of an integrated Gauss–Markov process and the connection with its first-passage time

ABSTRACTWe find explicit formulae for the mean of the running maximum of conditional and unconditional Brownian motion; they are used to obtain the mean, a(t), of the running maximum of an integrated Gauss–Markov process. Then, we deal with the connection between the moments of its first-passage-time and a(t). As explicit examples, we consider integrated Brownian motion and integrated Ornstein–Uhlenbeck process.

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.

Hitting probabilities of conditional Brownian motion and polarisation

We study the behaviour of the hitting probabilities of conditional Brownian motion in a domain D in Euclidean space when we apply polarization to D. We also study how polarization affects the probability that conditional Brownian motion meets a subset of D.

Towards an Innovation Theory of Spatial Brownian Motion under Boundary Conditions

Abstract Set-parametric Brownian motion b in a star-shaped set G is considered when the values of b on the boundary of G are given. Under the conditional distribution given these boundary values the process b becomes some set-parametrics Gaussian process and not Brownian motion. We define the transformation of this Gaussian process into another Brownian motion which can be considered as “martingale part” of the conditional Brownian motion b and the transformation itself can be considered as Doob–Meyer decomposition of b. Some other boundary conditions and, in particular, the case of conditional Brownian motion on the unit square given its values on the whole of its boundary are considered.

Diffusion processes on complete riemannian manifolds

In this paper, a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established. Applying it to the heat kernel estimate of the operator 1/2 Δ +b, we obtain the Aronson's estimate for the operator 1/2 Δ +b, which can be regarded as an extension of Peter Li and S.T. Yau's heat kernel estimate for the Laplace-Beltrami operator.

Conditional transformation of drift formula and potential theory for 1/2?+b(�)�?

Using conditional Brownian motion and the transformation of drift formula (of Cameron-Martin, Girsarov, Maruyama) we give integral conditions on a vector fieldb which imply the harmonic measures and Green functions for 1/2Δ and 1/2Δ+b(·)·∇ on a bounded Lipschitz domainD are equivalent. By equivalent we mean there exist two-sided inequalities with constants depending only onb andD. This enables one to conclude the potential theory for 1/2Δ+b(·)·∇ onD and 1/2Δ onD are the same.

Conditional Brownian Motion in Rapidly Exhaustible Domains

Let $D$ be a domain in $\mathbb{R}^d$ and let $\Delta_1$ be the set of minimal points of the Martin boundary of $D$. For $x \in D$ and $z \in \Delta_1$, let $(X_t)$ under the law $P^{x; z}$ be Brownian motion in $D$, starting at $x$ and conditioned to converge to $z$. Let $\tau$ be the lifetime of $(X_t)$, so $X_{\tau-} = z P^{x; z}$ a.s. Let $q \in L^p(D)$ for some $p > d/2$. Under the assumption that $D$ is what we call rapidly exhaustible, which is essentially a very weak boundary smoothness condition, we show that if the quantity $E^{x;z}\big\{\exp\big\lbrack\int^\tau_0 q(X_s) ds\big\rbrack\big\}$ is finite for one $x \in D$ and one $z \in \Delta_1$, then this quantity is bounded on $D \times \Delta_1$. This result may be viewed as saying, in a fairly strong sense, that the amount of time $(X_t)$ spends in each part of $D$ does not depend very much on the minimal Martin boundary point $z$ to which $(X_t)$ is conditioned to converge.

Schrödinger conditional brownian motion and stochastic calculus of variations

We present a new concept of Schrodinger conditional Brownian motion, which may be considered as a generalization of the h-process given by Doob in the case that the harmonic function h is replaced by a positive solution u of Schrodinger equation M = 0.The potential term V is in the basic class K which includes unbounded functions such as the Coulomb potential.We prove that the change from the Brownian motion to the “Schrodinger process” corresponds to the change from zero drift to the drift V(lnu). Moreover, we prove that the Schrodinger process and drift V(ln u)provide the solution of the stochastic calculus of variations corresponding to a problem of least average action. This is the same type as the stochastic optimal control problem originated by Fleming.

Conditional brownian motion and the boundary limits of harmonic functions

Conditional brownian motion and the boundary limits of harmonic functions