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• Non-linear PDEs and measure-valued branching Markov processes

Lucian Beznea

Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764, RO-014700 Bucharest, Romania. E-mail: lucian.beznea@imar.ro

July 11, 2012, BCAM Seminar, Bilbao

13 iulie 2012

• General frame

E : a Lusin topological space with Borel σ-algebra B

L : the generator of a right Markov process X = (Ω,F ,Ft ,Xt , θt ,Px ) with state space E :

– (Ω,F) is a measurable space, Px is a probability measure on (Ω,F) for every x ∈ E

– (Ft )t≥0: filtration on Ω

– The mapping [0,∞)× Ω 3 t 7−→ Xt (ω) ∈ E is B([0,∞))×F-measurable

– Xt is Ft/B–measurable for all t

• • Transition function: a semigroup of kernels (Pt )t≥0 on (E ,B), such that for all t ≥ 0, x ∈ E and A ∈ B one has

Px (Xt ∈ A) = Pt (x ,A)

[ If f ∈ pB then Ex (f ◦ Xt ) = Pt f (x) ]

– For each ω ∈ Ω the mapping

[0,∞) 3 t 7−→ Xt (ω) ∈ E

is right continuous

• Path regularity: càdlàg trajectories Let µ be a finite measure on E . The right process X has càdlàg trajectories Pµ-a.e. if it possesses left limits in E Pµ-a.e. on [0, ζ); ζ is the life time of X

• The resolvent of the process X

U = (Uq)q>0,

Uqf (x) := Ex ∫ ∞

0 e−qt f ◦ Xtdt =

∫ ∞ 0

e−qtPt f (x)dt ,

x ∈ E ,q > 0, f ∈ pB

L is the infinitesimal generator of (Uq)q>0 , [ Uq = (q − L)−1

]

• L-superharmonic function

The following properties are equivalent for a function v : E −→ R+:

(i) v is (L − q)-superharmonic

(ii) There exists a sequence (fn)n of positive, bounded, Borel measurable functions on E such that Uqfn ↗ v

(iii) αUq+αv ≤ v for all α > 0 and αUq+αv ↗ v when α↗∞

(iv) e−qtPtv ≤ v for all t > 0 and limt→0 Ptv = v

S(L − q) : the set of all (L − q)-superharmonic functions

• Applications: Construction of measure-valued branching processes associated to some nonlinear PDEs

A1. Continuous branching processes

A2. Discrete branching processes

A3. A nonlinear Dirichlet problem

• References

• M. Nagasawa ([Sémin. de probab. (Strasbourg) 10, 1976, pp. 184-193]) related this nonlinear problem to a branching Markov process. • M. Silverstein: Markov processes with creation of particles, Z. Wahrscheinliehkeitstheorie verw. Geb. 9 (1968), 235–257 • N. Ikeda, M. Nagasawa, S. Watanabe: Branching Markov processes, I,II, J. Math. Kyoto Univ. 8(1968),365-410, 233-278 • P.J. Fitzsimmons: Construction and regularity of measure-valued Markov branching processes, Israel J. Math. 64, 337-361, 1988 • E.B. Dynkin: Diffusions, superdiffusions and partial differential equations, Colloq. publications (Amer. Math. Soc.), 50, 2002 • E. Pei Hsu: Branching Brownian motion and the Dirichlet problem of a nonlinear equation, In: Seminar on Stoch. Proc., 1986, Birkhäuser 1987 • K. Janssen:Rev. Roum. Math. Pures Appl. 51(2006),655-664 • Li, Zenghu Measure-Valued Branching Markov Processes. (Probab. Appl.), Springer 2011

• Space of measures

M(E): the space of all positive finite measures on (E ,B) endowed with the weak topology.

For a function f ∈ bpB consider the mappings

lf : M(E) −→ R,

lf (µ) := 〈µ, f 〉 := ∫

fdµ, µ ∈ M(E),

ef : M(E) −→ [0,1] ef := exp(−lf ).

M(E):= the σ-algebra on M(E) generated by {lf | f ∈ bpB}, the Borel σ-algebra on M(E)

• The space of finite configurations of E

S: the set of all positive measures µ on E which are finite sums of Dirac measures, µ =

∑m k=1 δxk , where x1, . . . , xm ∈ E .

S is identified with the direct sum of all symmetric m-th powers E (m) of E , hence

S = ⊕ m≥1

E (m),

and it is equipped with the canonical topological structure.

B(S): the Borel σ-algebra on S.

• Multiplicative functions

• Let ϕ ∈ pB, ϕ ≤ 1. Consider the function ϕ̂ : S −→ R, called multiplicative, defined as

ϕ̂(x) := ϕ(x1) · . . . · ϕ(xm) for x = (x1, . . . , xm) ∈ E (m).

• A multiplicative function ϕ̂ is the restriction to S of an exponential function on M(E),

ϕ̂ = e−lnϕ.

• Branching kernel

Let p1,p2 be two finite measures on M(E).

• The convolution p1 ∗ p2: the finite measure on M(E) defined for every bounded Borel function F on M(E) by

∫ p1 ∗ p2(dν)F (ν) :=

∫ p1(dν1)

∫ p2(dν2)F (ν1 + ν2).

• If p1 and p2 are concentrated on S then p1 ∗ p2 has the same property and

p1 ∗ p2(ϕ̂) = p1(ϕ̂)p2(ϕ̂).

• Branching kernel: a kernel K on M(E) such that for all µ, ν ∈ M(E) we have

Kµ+ν = Kµ ∗ Kν .

• Branching process

A Markov process X with state space M(E) is called branching process provided that for all µ1, µ2 ∈ M(E), the process Xµ1+µ2 starting from µ1 + µ2 and the sum Xµ1 + Xµ2 are equal in distributions, i.e., for all t ≥ 0 and F ∈ bpM(E) we have∫

F (X t (ω))P µ1+µ2(dω) =

∫ ∫ (F (X t (ω)+X t (ω′))P

µ1(dω)P µ2(dω′)

X is a branching process⇐⇒ P t is a branching kernel for all t .

• A1. Nonlinear evolution equation

(∗)

 d dt vt (x) = Lvt (x) + Φ(x , vt (x))

v0 = f ,

where f ∈ pbB.

Aim: To give a probabilistic treatment of the equation (∗).

• L is the infinitesimal generator of a right Markov process with state space E , called spatial motion.

• Branching mechanism

A function Φ : E × [0,∞) −→ R of the form

Φ(x , λ) = −b(x)λ− c(x)λ2 + ∫ ∞

0 (1− e−λs−λs)N(x ,ds)

• c ≥ 0 and b are bounded B-measurable functions

• N : pB((0,∞)) −→ pB(E) is a kernel such that

N(u ∧ u2) ∈ bpB

Examples of branching mechanisms

Φ(λ) = −λα if 1 < α ≤ 2

• Construction of the nonlinear semigroup ([Fitzsimmons 88])

The equation

(∗)

 d dt vt (x) = Lvt (x) + Φ(x , vt (x))

v0 = f ,

is formally equivalent with

(∗∗) vt (x) = Pt f (x) + ∫ t

0 Ps(x ,Φ(·, vt−s))ds,

t ≥ 0, x ∈ E

• The following assertions hold.

i) For every f ∈ bpB the equation (∗∗) has a unique solution (t , x) 7−→ Vt f (x) jointly measurable in (t , x) such that sup0≤s≤t ||Vsf ||∞ 0.

ii) For all t ≥ 0 and x ∈ E we have 0 ≤ Vt f (x) ≤ eβt ||f ||∞.

iii) If t 7−→ Pt f (x) is right continuous on [0,∞) for all x ∈ E then so is t 7−→ Vt f (x).

iv) The mappings f 7−→ Vt f form a nonlinear semigroup of operators on bpB.

v) For all t ≥ 0 and µ ∈ M(E) the map f 7−→ 〈µ,Vt f 〉 is negative definite on the semigroup bpB.

vi) If (fn)n ⊂ bpB is a decreasing sequence, fn ↘ f , then Vt fn ↘ Vt f for every t ≥ 0.

• The branching semigroup on the space of measures

Let (Vt )t≥0 be the nonlinear semigroup of operators on bpB. Then there exists a unique Markovian semigroup of branching kernels (Qt )t≥0 on (M(E),M(E)) such that for all f ∈ bpB and t > 0 we have

Qt (ef ) = eVt f .

• The infinitesimal generator of the forthcoming branching process

If L is the infinitesimal generator of the semigroup (Qt )t≥0 on M(E) and

F = ef

with f ∈ bpB, then

LF (µ) = ∫

E µ(dx)c(x)F ′′(µ, x)+∫

E µ(dx)[LF ′(µ, ·)(x)− b(x)F ′(µ, x)] +∫

E µ(dx)

∫ ∞ 0

N(x ,ds)[F (µ+ sδx )− F (µ)− sF ′(µ, x)]

where F ′(µ, x) and F ′′(µ, x) are the first and second variational derivatives of F [F ′(µ, x) = limt→0 1t (F (µ+ tδx )− F (µ))].

• Linear and exponential type superharmonic functions for the branching process

Let β := ||b−||∞,

β′ ≥ β and b′ := b + β′.

If u ∈ bpB then the following assertions are equivalent.

i) u ∈ S(L − b′)

ii) lu ∈ S(L − β′)

iii) For every α > 0 we have 1− eαu ∈ S(L − β′).

• Reduced function and the induced capacity

If M ∈ B, q > 0, and u ∈ S(L − q) then the reduced function of u on M (with respect to L− q) is the function RMd u defined by

RMq u := inf {

v ∈ S(L − q) : v ≥ u on M } .

• The reduced function RMq u is universally B-measurable.

• Let p := Uq1. The functional M 7−→ cµ(M), M ⊂ E , defined by

cµ(M) := inf{ ∫

E RGq p dµ : G open , M ⊂ G}

is a Choquet capacity on E .

• RMq f (x) = Ex (f (XDM )) [Hunt’s Theorem],

where DM := inf{t ≥ 0 : Xt ∈ M}.

• Tightness property of the capacity

The capacity cµ is tight provided that there exists an increasing sequence (Kn)n of compact sets such that

inf n

cµ(E \ Kn) = 0

or equivalently, Pµ(lim

n DE\Kn < ζ) = 0.

"The process lies in ⋃

n Kn P µ-a.s. up to the life time."

• Compact Lyapunov function

A (L − q)-superharmonic function v is called c