Related papers: Conditioned Martingales
In this paper we present a martingale related to the exit measures of super-Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is…
It is known that a full description of Brownian motion in the entire course of time should incorporate both kinetic and hydrodynamic effects, but a formula accounts for both effects has been established only in three dimension and only for…
When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $\xi=\sum_{j \in…
Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of…
We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a…
We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the…
When the unconditioned process is a diffusion submitted to a space-dependent killing rate $k(\vec x)$, various conditioning constraints can be imposed for a finite time horizon $T$. We first analyze the conditioned process when one imposes…
Assume that $X$ is a continuous square integrable process with zero mean, defined on some probability space $(\Omega,\mathrm {F},\mathrm {P})$. The classical characterization due to P. L\'{e}vy says that $X$ is a Brownian motion if and only…
We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the…
Let (B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t + \mu t) be a three-dimensional Brownian motion with drift \mu, starting at the origin. Then X_t = ||(B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t +\mu t)||, its distance from the starting point, is a diffusion with…
We consider a diffusion processes $\{ X_t \}$ on an interval in the natural scale. Some results are known under which $\{ X_t \}$ is a martingale, and we give simple and analytic proofs for them.
We study analytically the order and gap statistics of particles at time $t$ for the one dimensional branching Brownian motion, conditioned to have a fixed number of particles at $t$. The dynamics of the process proceeds in continuous time…
We give a probabilistic representation of a one-dimensional diffusion equation where the solution is discontinuous at $0$ with a jump proportional to its flux. This kind of interface condition is usually seen as a semi-permeable barrier.…
We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the…
We prove existence and uniqueness for semimartingale reflecting diffusions in 2-dimensional piecewise smooth domains with varying, oblique directions of reflection on each "side", under geometric, easily verifiable conditions. Our…
Sticky diffusion processes on bounded domains spend finite time (and finite mean time) on the lower-dimensional space given by the boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on…
Bessel process is defined as the radial part of the Brownian motion (BM) in the $D$-dimensional space, and is considered as a one-parameter family of one-dimensional diffusion processes indexed by $D$, BES$^{(D)}$. It is well-known that…
Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the…
The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length…
We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly…