144 research outputs found
Stationary distributions for diffusions with inert drift
Consider reflecting Brownian motion in a bounded domain in 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 Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential
Diffusion in multiscale spacetimes
We study diffusion processes in anomalous spacetimes regarded as models of
quantum geometry. Several types of diffusion equation and their solutions are
presented and the associated stochastic processes are identified. These results
are partly based on the literature in probability and percolation theory but
their physical interpretation here is different since they apply to quantum
spacetime itself. The case of multiscale (in particular, multifractal)
spacetimes is then considered through a number of examples and the most general
spectral-dimension profile of multifractional spaces is constructed.Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected,
references adde
Composition of processes and related partial differential equations
In this paper different types of compositions involving independent
fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial
differential equations governing the distributions of
I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and
J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods
and compared with those existing in the literature and with those related to
B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0
is examined in detail and its moments are calculated. Furthermore for
J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following
factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0.
A series of compositions involving Cauchy processes and fractional Brownian
motions are also studied and the corresponding non-homogeneous wave equations
are derived.Comment: 32 page
Loop-Erasure of Plane Brownian Motion
We use the coupling technique to prove that there exists a loop-erasure of a
plane Brownian motion stopped on exiting a simply connected domain, and the
loop-erased curve is the reversal of a radial SLE curve.Comment: 10 page
Bismut-Elworthy-Li formulae for Bessel processes
In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all δ ≥ 0 and T > 0, we compute the derivative of the function x↦PδTF(x), where (Pδt)t≥0 is the transition semi-group associated to the δ-dimensional Bessel process, and F is any bounded Borel function on R+. The obtained expression shows a nice interplay between the transition semi-groups of the δ—and the (δ + 2)-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula
Reduction of dimensionality in a di usion search process and kinetics of gene expression
Abstract In order to activate a gene in a DNA molecule a speciÿc protein (transcription factor) has to bind to the promoter of the gene. We formulate and partially answer the following question: how much time does a transcription factor, which activates a given gene, need in order to ÿnd this gene inside the nucleus of a cell? The estimate based on the simplest model of di usion gives a very long time of days. We discuss various mechanisms by which the time can be reduced to seconds, in particular, the reduction of dimensionality, in which di usion takes place, from three-dimensional space to two-dimensional space. The potential needed to keep the di using particle in 2D (i.e, at the surface of size L 2 in a volume of size L 3 ) should scale as U ∼ kBT ln L. For aL = 1 m and a target size a = 10 A we ÿnd U = 8kBT, i.e., it is a potential strength of the order of the strength of ionic interactions in water
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
Shy couplings, CAT(0) spaces, and the lion and man
Two random processes X and Y on a metric space are said to be ε-shy coupled if there
is positive probability of them staying at least a positive distance ε apart from each other forever.
Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling.
Previous non-existence results for co-adapted shy coupling of reflected Brownian motion required
convexity conditions; we remove these conditions by showing the non-existence of shy co-adapted
couplings of reflecting Brownian motion in any bounded CAT(0) domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded
planar domains with C2 boundary.
The proof uses a Cameron-Martin-Girsanov argument, together with a continuity property of
the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a
generalization of Gauss' Lemma is established that shows differentiability of the intrinsic distance
function for closures of CAT(0) domains with boundaries satisfying uniform exterior sphere and
interior cone conditions. By this means, the shy coupling question is converted into a Lion and
Man pursuit-evasion problem
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