The Carrying Dimension of a Stochastic Measure Diffusion

Dawson, Donald A.; Hochberg, Kenneth J.

doi:10.1214/aop/1176994991

Search citation statements

Order By: Relevance

Paper Sections

Select...

A Singular Pde1

Definition1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

1986

2008

Publication Types

Select...

Article7

Other1

Relationship

Self Cite0

Independent8

Authors

Journals

Cited by 75 publications

(28 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3) converges to that of (3.4). We shall also indicate some extensions and establish a regularity theorem for the solution of (3.4).…”

Section: A Singular Pdementioning

confidence: 75%

Ergodic theory and a local occupation time for measure-valued critical branching brownian motion

Iscoe

1986

Stochastics

View full text Add to dashboard Cite

show abstract

“…3) converges to that of (3.4). We shall also indicate some extensions and establish a regularity theorem for the solution of (3.4).…”

Section: A Singular Pdementioning

confidence: 75%

Ergodic theory and a local occupation time for measure-valued critical branching brownian motion

Iscoe

1986

Stochastics

View full text Add to dashboard Cite

show abstract

“…(a) (Dawson and Hochberg [1979]) X, is a singular measure and in fact is supported by a random Borel set, A,(id), such that dim(A,(<o)) < a Qm-a.s.…”

mentioning

confidence: 99%

A Space-Time Property of a Class of Measure-Valued Branching Diffusions

Perkins¹

1988

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. If d > a, it is shown that the ¿-dimensional branching diffusion of index a, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff "-measure, where "(x) = "loglogl/x. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case d = a. In particular, the process is singular at all positive times a.s. for d> a.

show abstract

“…In dimension d = 1 the random measure X t is for each t absolutely continuous relative to Lebesgue measure [18], and the Radon-Nikodym derivative X (t, x) is jointly continuous in t, x (for t > 0). In dimensions d ≥ 2 the measure X t is almost surely singular, and is supported by a Borel set of Hausdorff dimension 2 [7].…”

Section: Definitionmentioning
confidence: 99%

Spatial epidemics: critical behavior in one dimension
Lalley
¹

2008
Probab. Theory Relat. Fields
16
0
53
0
View full text Add to dashboard Cite

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson-Watanabe processes with killing, respectively.

show abstract

The Carrying Dimension of a Stochastic Measure Diffusion

Cited by 75 publications

References 0 publications

Ergodic theory and a local occupation time for measure-valued critical branching brownian motion

Ergodic theory and a local occupation time for measure-valued critical branching brownian motion

A Space-Time Property of a Class of Measure-Valued Branching Diffusions

Spatial epidemics: critical behavior in one dimension

Contact Info

Product

Resources

About