1,875 research outputs found
Torsional rigidity for cylinders with a Brownian fracture
We obtain bounds for the expected loss of torsional rigidity of a cylinder
of length due to a Brownian
fracture that starts at a random point in and runs until the first
time it exits . These bounds are expressed in terms of the geometry
of the cross-section . It is shown that if is a
disc with radius , then in the limit as the expected
loss of torsional rigidity equals for some . We derive
bounds for in terms of the expected Newtonian capacity of the trace of a
Brownian path that starts at the centre of a ball in with radius
and runs until the first time it exits this ball.Comment: 18 page
Heat content and inradius for regions with a Brownian boundary
In this paper we consider , Brownian motion of time length , in -dimensional Euclidean space and on the -dimensional
torus . We compute the expectation of (i) the heat content at time
of for fixed and in the
limit , when is kept at temperature 1 for all and has initial temperature 0, and (ii)
the inradius of for in the
limit .Comment: 13 page
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Heat content and inradius for regions with a Brownian boundary
In this paper we consider \beta [0, s], Brownian motion of time length s > 0, in m-dimensional Euclidean space R^m and on the m-dimensional torus T^m. We compute the expectation of (i) the heat content at time t of R^m \ \beta [0, s] for fixed s and m = 2,3 in the limit t \downarrow 0, when \beta [0, s] is kept at temperature 1 for all t > 0 and R^m \ \beta [0, s] has initial temperature 0, and (ii) the inradius of T^m \ \beta [0, s] for m = 2,3,… in the limit s \rightarrow \infty. Key words and phrases. Laplacian, Brownian motion, Wiener sausage, heat content, inradius, spectrum
The renormalization transformation for two-type branching models
This paper studies countable systems of linearly and hierarchically
interacting diffusions taking values in the positive quadrant. These systems
arise in population dynamics for two types of individuals migrating between and
interacting within colonies. Their large-scale space-time behavior can be
studied by means of a renormalization program. This program, which has been
carried out successfully in a number of other cases (mostly one-dimensional),
is based on the construction and the analysis of a nonlinear renormalization
transformation, acting on the diffusion function for the components of the
system and connecting the evolution of successive block averages on successive
time scales. We identify a general class of diffusion functions on the positive
quadrant for which this renormalization transformation is well-defined and,
subject to a conjecture on its boundary behavior, can be iterated. Within
certain subclasses, we identify the fixed points for the transformation and
investigate their domains of attraction. These domains of attraction constitute
the universality classes of the system under space-time scaling.Comment: 48 pages, revised version, to appear in Ann. Inst. H. Poincare (B)
Probab. Statis
Torsional rigidity for regions with a Brownian boundary
Article / Letter to editorMathematisch Instituu
Collision local time of transient random walks and intermediate phases in interacting stochastic systems
In a companion paper, a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d = 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds for two transient but not strongly transient random walks. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments
Quenched LDP for words in a letter sequence
When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. We apply our LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d = 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments
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