Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process...

87

Transcript of Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process...

Page 1: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Contact process with aging

Aurelia Deshayes

under the direction of Olivier Garet and Régine MarchandInstitut Elie Cartan - Université de Lorraine

Young Women In ProbabilityBonn

26 may 2014

Page 2: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Contact Process

De�nition

The contact process {ξt , t ≥ 0} is a continuous-time Markov process with values

in {0, 1}Zd

. Let z ∈ Zd :

z is dead (or healthy) if ξt(z) = 0.

z is alive (or infected) if ξt(z) = 1.

Rules of evolution :

if a site is alive then it dies at rate 1 ;

if a site is dead then it turns alive at rate λ times the number of its livingneighbors.

The initial set ξ0 is a �nite set of Zd , we often take {0}.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 2 / 18

Page 3: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 4: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death

↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 5: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 6: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 7: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 8: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 9: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 10: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 11: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

0 1 0 1 0 1 1 0

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 12: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 13: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

1 1 1 1 0 1 1 1

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 14: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

1 1 0 1 0 1 1 0

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 15: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 16: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 17: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 18: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 19: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 20: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 21: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 22: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 23: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 24: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

0 1 0 2 0 0 0 0

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 25: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 26: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 27: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 28: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

0 2 0 2 0 1 2 0

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 29: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

×

× death↔ birth (λ)

••

• maturation(γ)

young

adult

••

2 2 0 2 0 1 2 0

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 3 / 18

Page 30: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 31: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 32: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation

↔ birthage 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 33: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 34: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 35: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 36: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 37: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 38: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 39: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 40: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 41: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 42: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 43: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 0

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 44: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 45: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

t

z0

Z× R+

×

×

×

×

×

×

×

×

×

×

×

×

× death

••

••

• maturation↔ birth

age 1age 2age 3

↔ 1, 2, 3↔ 2, 3↔ 3

3 0 0 2 0 1 2 1

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 4 / 18

Page 46: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Let Λ = (λi )i∈N be the sequence of birth parameters :

1 ∀i , λi ∈ R+ and λ0 = 0,

2 (λi )i is non decreasing,

3 limi→∞ λi = λ∞ <∞.

Let γ > 0 be the maturation parameter and f : Zd → N with �nite support.

De�nition

A CPA {ξft , t ≥ 0} is a continuous-time Markov process with values in NZd

andξ0 = f . Let z ∈ Zd and k ∈ N∗ :

z is dead if ξft (z) = 0,

z is alive with age k if ξft (z) = k.

Evolution rules :

a living site dies at rate 1 independently of its age,

a dead site z turns alive at rate∑

z′,‖z′−z‖1=1 λξt(z′) (with λ0 = 0),

each new o�spring has age one,

the transition from age n to age n + 1 occurs at rate γ > 0, independently ofits age.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 5 / 18

Page 47: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Let Λ = (λi )i∈N be the sequence of birth parameters :

1 ∀i , λi ∈ R+ and λ0 = 0,

2 (λi )i is non decreasing,

3 limi→∞ λi = λ∞ <∞.

Let γ > 0 be the maturation parameter and f : Zd → N with �nite support.

De�nition

A CPA {ξft , t ≥ 0} is a continuous-time Markov process with values in NZd

andξ0 = f . Let z ∈ Zd and k ∈ N∗ :

z is dead if ξft (z) = 0,

z is alive with age k if ξft (z) = k.

Evolution rules :

a living site dies at rate 1 independently of its age,

a dead site z turns alive at rate∑

z′,‖z′−z‖1=1 λξt(z′) (with λ0 = 0),

each new o�spring has age one,

the transition from age n to age n + 1 occurs at rate γ > 0, independently ofits age.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 5 / 18

Page 48: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

For t ≥ 0 we denote by :

Aft = supp ξft = {x ∈ Zd ; ξft (x) 6= 0},

= set of living points at time t.

Some properties

Let f , g : Zd → N and t > 0.

The CPA is attractive i.e f ≤ g =⇒ ξft ≤ ξgt and Af

t ≤ Agt ;

The CPA is additive i.e ξf∨gt = ξft ∨ ξgt ;

The CPA is monotone with respect to its parameters : non decreasing withrespect to birth and maturation parameters

Goal : Prove an asymptotic shape theorem for⋃

s≤t At .

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 6 / 18

Page 49: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival of contact process

De�nition

We have survival if P(∀t, ξ{0}t 6≡ 0) > 0.

extinction if P(∀t, ξ{0}t 6≡ 0) = 0.

Critical value for the PC : λc(Zd)

Zd

t

0

Phase transition [Harris, 74]

There exists a critical value λc ∈ (0,+∞) such that :

if λ < λc , then extinction,

if λ > λc , then survival.

At the critical point ? [Bezuidenhout and Grimmett, 91]

λ = λc , extinction.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 7 / 18

Page 50: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival of contact process

De�nition

We have survival if P(∀t, ξ{0}t 6≡ 0) > 0.

extinction if P(∀t, ξ{0}t 6≡ 0) = 0.

Critical value for the PC : λc(Zd) Zd

t

0

Phase transition [Harris, 74]

There exists a critical value λc ∈ (0,+∞) such that :

if λ < λc , then extinction,

if λ > λc , then survival.

At the critical point ? [Bezuidenhout and Grimmett, 91]

λ = λc , extinction.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 7 / 18

Page 51: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival of contact process

De�nition

We have survival if P(∀t, ξ{0}t 6≡ 0) > 0.

extinction if P(∀t, ξ{0}t 6≡ 0) = 0.

Critical value for the PC : λc(Zd) Zd

t

0

Phase transition [Harris, 74]

There exists a critical value λc ∈ (0,+∞) such that :

if λ < λc , then extinction,

if λ > λc , then survival.

At the critical point ? [Bezuidenhout and Grimmett, 91]

λ = λc , extinction.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 7 / 18

Page 52: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival of contact process

De�nition

We have survival if P(∀t, ξ{0}t 6≡ 0) > 0.

extinction if P(∀t, ξ{0}t 6≡ 0) = 0.

Critical value for the PC : λc(Zd) Zd

t

0

Phase transition [Harris, 74]

There exists a critical value λc ∈ (0,+∞) such that :

if λ < λc , then extinction,

if λ > λc , then survival.

At the critical point ? [Bezuidenhout and Grimmett, 91]

λ = λc , extinction.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 7 / 18

Page 53: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Krone introduces a similar quantity :

λc(γ) = inf{λ : Pλ,γ

(∀t > 0, ξ

{0(2)}t 6= ∅

)> 0}.

0(2) : the site 0 in adult state.

λ

γ

survival

γ∗

λ∗

.

λ∗ = λc(PC ). γ∗ > 0 for d = 1 by Krone (1999) and d ≥ 1 by Foxall (2014).

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 8 / 18

Page 54: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Krone introduces a similar quantity :

λc(γ) = inf{λ : Pλ,γ

(∀t > 0, ξ

{0(2)}t 6= ∅

)> 0}.

0(2) : the site 0 in adult state.

λ

γ

survival

γ∗

λ∗

.

λ∗ = λc(PC ). γ∗ > 0 for d = 1 by Krone (1999) and d ≥ 1 by Foxall (2014).A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 8 / 18

Page 55: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival region of CPA :

Sγ ={

Λ ∈ RN, non decreasing / PΛ,γ (∀t > 0, ξt 6≡ 0) > 0}.

If Λ ∈ Sγ , then one has survival.

If Λ /∈ Sγ , then one has extinction.

Let Λ = (λi )i and Λ′ = (λ′i )i . If Λ ∈ Sγ and ∀i , λi ≤ λ′i then Λ′ ∈ Sγ .

If ∀i , λi > λc(PC ), then one has survival.

About the survival region...

For m ∈ N and λ1, . . . , λm �xed, there exist λm+1 and γ such thatPΛ,γ (∀t > 0, ξt 6≡ 0) > 0.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 9 / 18

Page 56: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival region of CPA :

Sγ ={

Λ ∈ RN, non decreasing / PΛ,γ (∀t > 0, ξt 6≡ 0) > 0}.

If Λ ∈ Sγ , then one has survival.

If Λ /∈ Sγ , then one has extinction.

Let Λ = (λi )i and Λ′ = (λ′i )i . If Λ ∈ Sγ and ∀i , λi ≤ λ′i then Λ′ ∈ Sγ .

If ∀i , λi > λc(PC ), then one has survival.

About the survival region...

For m ∈ N and λ1, . . . , λm �xed, there exist λm+1 and γ such thatPΛ,γ (∀t > 0, ξt 6≡ 0) > 0.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 9 / 18

Page 57: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Survival region of CPA :

Sγ ={

Λ ∈ RN, non decreasing / PΛ,γ (∀t > 0, ξt 6≡ 0) > 0}.

If Λ ∈ Sγ , then one has survival.

If Λ /∈ Sγ , then one has extinction.

Let Λ = (λi )i and Λ′ = (λ′i )i . If Λ ∈ Sγ and ∀i , λi ≤ λ′i then Λ′ ∈ Sγ .

If ∀i , λi > λc(PC ), then one has survival.

About the survival region...

For m ∈ N and λ1, . . . , λm �xed, there exist λm+1 and γ such thatPΛ,γ (∀t > 0, ξt 6≡ 0) > 0.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 9 / 18

Page 58: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

To prove an asymptotic shape theorem...

Let Λ, γ such that PΛ,γ

(∀t > 0, ξδ0t 6≡ 0

)> 0. Pλ,γ

(• |∀t > 0, ξδ0t 6≡ 0

).

We want to prove an asymptotic shape theorem for the CPA :

Theorem

There exists a norm µ on Rd such that for every ε > 0 and every function

f : Zd → N with �nite support, P-almost surely,

∀t > 0, (1− ε)Bµ ⊂H̃ ft

t⊂ (1 + ε)Bµ,

where H̃ ft = ∪s≤tAf

s + [0, 1]d and Bµ the unit ball of µ.

H̃ ft is (almost) the set of points born before t.

1 At most linear growth

2 At least linear growth

3 Exactly linear growth

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 10 / 18

Page 59: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

To prove an asymptotic shape theorem...

Let Λ, γ such that PΛ,γ

(∀t > 0, ξδ0t 6≡ 0

)> 0. Pλ,γ

(• |∀t > 0, ξδ0t 6≡ 0

).

We want to prove an asymptotic shape theorem for the CPA :

Theorem

There exists a norm µ on Rd such that for every ε > 0 and every function

f : Zd → N with �nite support, P-almost surely,

∀t > 0, (1− ε)Bµ ⊂H̃ ft

t⊂ (1 + ε)Bµ,

where H̃ ft = ∪s≤tAf

s + [0, 1]d and Bµ the unit ball of µ.

H̃ ft is (almost) the set of points born before t.

1 At most linear growth

2 At least linear growth

3 Exactly linear growth

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 10 / 18

Page 60: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

To prove an asymptotic shape theorem...

Let Λ, γ such that PΛ,γ

(∀t > 0, ξδ0t 6≡ 0

)> 0. Pλ,γ

(• |∀t > 0, ξδ0t 6≡ 0

).

We want to prove an asymptotic shape theorem for the CPA :

Theorem

There exists a norm µ on Rd such that for every ε > 0 and every function

f : Zd → N with �nite support, P-almost surely,

∀t > 0, (1− ε)Bµ ⊂H̃ ft

t⊂ (1 + ε)Bµ,

where H̃ ft = ∪s≤tAf

s + [0, 1]d and Bµ the unit ball of µ.

H̃ ft is (almost) the set of points born before t.

1 At most linear growth

2 At least linear growth

3 Exactly linear growth

Z2

BMt

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 10 / 18

Page 61: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

To prove an asymptotic shape theorem...

Let Λ, γ such that PΛ,γ

(∀t > 0, ξδ0t 6≡ 0

)> 0. Pλ,γ

(• |∀t > 0, ξδ0t 6≡ 0

).

We want to prove an asymptotic shape theorem for the CPA :

Theorem

There exists a norm µ on Rd such that for every ε > 0 and every function

f : Zd → N with �nite support, P-almost surely,

∀t > 0, (1− ε)Bµ ⊂H̃ ft

t⊂ (1 + ε)Bµ,

where H̃ ft = ∪s≤tAf

s + [0, 1]d and Bµ the unit ball of µ.

H̃ ft is (almost) the set of points born before t.

1 At most linear growth

2 At least linear growth

3 Exactly linear growth

Z2

BMt

Bct

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 10 / 18

Page 62: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

To prove an asymptotic shape theorem...

Let Λ, γ such that PΛ,γ

(∀t > 0, ξδ0t 6≡ 0

)> 0. Pλ,γ

(• |∀t > 0, ξδ0t 6≡ 0

).

We want to prove an asymptotic shape theorem for the CPA :

Theorem

There exists a norm µ on Rd such that for every ε > 0 and every function

f : Zd → N with �nite support, P-almost surely,

∀t > 0, (1− ε)Bµ ⊂H̃ ft

t⊂ (1 + ε)Bµ,

where H̃ ft = ∪s≤tAf

s + [0, 1]d and Bµ the unit ball of µ.

H̃ ft is (almost) the set of points born before t.

1 At most linear growth

2 At least linear growth

3 Exactly linear growthZ2

BMt

Bct

tBµ

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 10 / 18

Page 63: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

1. At most linear growth

Let Λ, γ such that PΛ,γ

(∀t > 0 ξδ0t 6≡ 0

)> 0.

H ft = ∪s≤tAf

s the set of points born before t

Let (ηt)t a Richardson process (contact process without death)

BR = {y ∈ Zd : ‖y‖∞ ≤ R}

There exist M,A,B such that for all t > 0, P(ηt * BMt) ≤ A exp(−Bt).

Lemma

There exist A,B,M such that for every f : Zd → N with �nite support and every

t > 0P(H f

t * BMt) ≤ P(ηt * BMt) ≤ A exp(−Bt).

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 11 / 18

Page 64: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

1. At most linear growth

Let Λ, γ such that PΛ,γ

(∀t > 0 ξδ0t 6≡ 0

)> 0.

H ft = ∪s≤tAf

s the set of points born before t

Let (ηt)t a Richardson process (contact process without death)

BR = {y ∈ Zd : ‖y‖∞ ≤ R}

There exist M,A,B such that for all t > 0, P(ηt * BMt) ≤ A exp(−Bt).

Lemma

There exist A,B,M such that for every f : Zd → N with �nite support and every

t > 0P(H f

t * BMt) ≤ P(ηt * BMt) ≤ A exp(−Bt).

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 11 / 18

Page 65: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

If we have survival then ∀ε > 0 ∃n, a, b, such that the probability of the blockevent is at least 1− ε.

x

t

0−a +a

b

5a

6b

(x , s)

(y , t)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 12 / 18

Page 66: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

If we have survival then ∀ε > 0 ∃n, a, b, such that the probability of the blockevent is at least 1− ε.

x

t

0−a +a

b

5a

6b

(x , s)

(y , t)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 12 / 18

Page 67: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

Zd

R+

• ZN

• • •

••

(j − 2, k) (j , k) (j + 2, k)

(j + 1, k + 1)(j − 1, k + 1)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 13 / 18

Page 68: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

Zd

R+

• ZN

• • •

••

(j − 2, k) (j , k) (j + 2, k)

(j + 1, k + 1)(j − 1, k + 1)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 13 / 18

Page 69: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

τ f = inf{t ≥ 0 : ξft ≡ 0} extinction time of CPA,

Theorem

There exist A,B,C such that for all t > 0, x ∈ Zd and f : Zd → N :

P(t < τ f <∞

)≤ A exp(−Bt),

P(Bct 6* H ft , τ

f =∞) ≤ A exp(−Bt).

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 14 / 18

Page 70: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

2. At least linear growth

1 Block event

2 Percolation background

3 Expected estimates

τ f = inf{t ≥ 0 : ξft ≡ 0} extinction time of CPA,

Theorem

There exist A,B,C such that for all t > 0, x ∈ Zd and f : Zd → N :

P(t < τ f <∞

)≤ A exp(−Bt),

P(Bct 6* H ft , τ

f =∞) ≤ A exp(−Bt).

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 14 / 18

Page 71: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t f (x) = inf{t ≥ 0 : ξft (x) 6= 0} hitting time of x .

H ft = {x ∈ Zd : t f (x) ≤ t}.

Let x ∈ Zd . We want to prove the convergence of t(nx)n

when n→∞.

Ergodic subadditive theorem of Kingman :

t((n + p)x) ≤ t(nx) + t(px) ◦ space-time translation.

t(x) t(x) under Pintegrability NO YES

stationarity YES NO

subadditivity YES NO

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 15 / 18

Page 72: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t f (x) = inf{t ≥ 0 : ξft (x) 6= 0} hitting time of x .

H ft = {x ∈ Zd : t f (x) ≤ t}.

Let x ∈ Zd . We want to prove the convergence of t(nx)n

when n→∞.

Ergodic subadditive theorem of Kingman :

t((n + p)x) ≤ t(nx) + t(px) ◦ space-time translation.

t(x) t(x) under Pintegrability NO YES

stationarity YES NO

subadditivity YES NO

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 15 / 18

Page 73: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 74: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 75: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 76: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 77: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 78: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 79: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 80: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 81: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Zd

t

0 x

C

t(x)D

A

B

∞∞

σ(x)

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 16 / 18

Page 82: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t(x) t(x) under P σ under Pintegrability NO YES YES

stationarity YES NO YES

(almost) subadditivity YES NO YES

σ satis�es also growth of linear order (as t).

directional convergence for σ thanks to the almost subadditive theorem ofKesten- Hammersley(74) :

σ((n+ p)x) ≤ σ(nx) +σ(px) ◦ space-time translation+ some controlled quantity.

� Uniform continuity � of σ → shape theorem for σ.

Control of the di�erence between t and σ → shape theorem for t.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 17 / 18

Page 83: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t(x) t(x) under P σ under Pintegrability NO YES YES

stationarity YES NO YES

(almost) subadditivity YES NO YES

σ satis�es also growth of linear order (as t).

directional convergence for σ thanks to the almost subadditive theorem ofKesten- Hammersley(74) :

σ((n+ p)x) ≤ σ(nx) +σ(px) ◦ space-time translation+ some controlled quantity.

� Uniform continuity � of σ → shape theorem for σ.

Control of the di�erence between t and σ → shape theorem for t.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 17 / 18

Page 84: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t(x) t(x) under P σ under Pintegrability NO YES YES

stationarity YES NO YES

(almost) subadditivity YES NO YES

σ satis�es also growth of linear order (as t).

directional convergence for σ thanks to the almost subadditive theorem ofKesten- Hammersley(74) :

σ((n+ p)x) ≤ σ(nx) +σ(px) ◦ space-time translation+ some controlled quantity.

� Uniform continuity � of σ → shape theorem for σ.

Control of the di�erence between t and σ → shape theorem for t.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 17 / 18

Page 85: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t(x) t(x) under P σ under Pintegrability NO YES YES

stationarity YES NO YES

(almost) subadditivity YES NO YES

σ satis�es also growth of linear order (as t).

directional convergence for σ thanks to the almost subadditive theorem ofKesten- Hammersley(74) :

σ((n+ p)x) ≤ σ(nx) +σ(px) ◦ space-time translation+ some controlled quantity.

� Uniform continuity � of σ → shape theorem for σ.

Control of the di�erence between t and σ → shape theorem for t.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 17 / 18

Page 86: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

3. Exactly linear growth → shape theorem

t(x) t(x) under P σ under Pintegrability NO YES YES

stationarity YES NO YES

(almost) subadditivity YES NO YES

σ satis�es also growth of linear order (as t).

directional convergence for σ thanks to the almost subadditive theorem ofKesten- Hammersley(74) :

σ((n+ p)x) ≤ σ(nx) +σ(px) ◦ space-time translation+ some controlled quantity.

� Uniform continuity � of σ → shape theorem for σ.

Control of the di�erence between t and σ → shape theorem for t.

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 17 / 18

Page 87: Contact process with aging - iam.uni-bonn.de€¦ · Contact Process De nition The contact process f˘t;t 0 gis a continuous-time Markov process with values in f0 ;1 gZd.Let z 2Zd:

Thanks for your attention

A.Deshayes (IECL) Contact process with aging YWIP 2014 Bonn 18 / 18