Backbone decomposition for superprocesses and applications. Backbone decomposition for superprocesses and applications. A. E. Kyprianou1 Department of Mathematical Sciences, University of Bath 1/ 17 1Julien Berestycki, Rongli Liu, Antonio Murillo-Salas, Yanxia Ren To be advertised next week: Chair of the probability laboratory at Bath PROB-L@B Probability Laboratory at Bath Now receiving submissions in applications of probability: Acta Applicandae Mathematicae Backbone decomposition for superprocesses and applications. 2/ 17 PROB-L@B Probability Laboratory at Bath Now receiving submissions in applications of probability: Acta Applicandae Mathematicae Backbone decomposition for superprocesses and applications. To be advertised next week: Chair of the probability laboratory at Bath 2/ 17 PROB-L@B Probability Laboratory at Bath Backbone decomposition for superprocesses and applications. To be advertised next week: Chair of the probability laboratory at Bath Now receiving submissions in applications of probability: Acta Applicandae Mathematicae 2/ 17 Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. Total mass: The process fh1;Zti : t ≥ 0g is a continuous time Galton-Watson process. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2. R PNt Notation: hf; Zti = E f(x)Zt(dx) = i=1 f(zi(t)) when PNt Zt(·) = i=1 δzi(t)(·). 2At the moment, the word ‘nice’ means any E-valued Markov process for which the mathematics in this talk can be carried out! However this is a very large class of processes d including, for example many conservative diffusions in E = R . 3/ 17 3 Pt[1] = 1 for all t ≥ 0. Backbone decomposition for superprocesses and applications. (P;F ; E)-Branching Markov process Atomic-measure valued Markov process: Z = fZt : t ≥ 0g with Pn probabilities denoted by Pµ where µ(·) = i=1 δxi (·) with xi 2 E. Path construction: Under Pµ, from each xi 2 E initiate: iid copies of a nice2 conservative3 E-valued Markov process whose semi-group is denoted by P = fPt : t ≥ 0g, each of which have a branching generator given by 1 ! X n F (s) = q s pn − s : n=0 Markov property: Zt+s is equal in law to an independent copy of Zs under PZt . Branching property: For atomic measures µ1 and µ2, (Z; Pµ1+µ2 ) has the (1) (2) (i) same law as Z + Z where Z has law Pµi for i = 1; 2.