Non-linear PDEs and measure-valuedbranching Markov processes

Lucian Beznea

Simion Stoilow Institute of Mathematics of the Romanian AcademyP.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 processX = (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 isB([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 trajectoriesLet µ be a finite measure on E .The right process X has càdlàg trajectories Pµ-a.e. if itpossesses 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∫ ∞

0e−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 functionv : 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 processesassociated 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 Markovprocess.• M. Silverstein: Markov processes with creation of particles, Z.Wahrscheinliehkeitstheorie verw. Geb. 9 (1968), 235–257• N. Ikeda, M. Nagasawa, S. Watanabe: Branching Markovprocesses, I,II, J. Math. Kyoto Univ. 8(1968),365-410, 233-278• P.J. Fitzsimmons: Construction and regularity ofmeasure-valued Markov branching processes, Israel J. Math.64, 337-361, 1988• E.B. Dynkin: Diffusions, superdiffusions and partial differentialequations, Colloq. publications (Amer. Math. Soc.), 50, 2002• E. Pei Hsu: Branching Brownian motion and the Dirichletproblem 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, theBorel 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 sumsof Dirac measures, µ =

∑mk=1 δxk , where x1, . . . , xm ∈ E .

S is identified with the direct sum of all symmetric m-th powersE (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 ϕ : S −→ R,called multiplicative, defined as

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

• A multiplicative function ϕ is the restriction to S of anexponential function on M(E),

ϕ = 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 thesame property and

p1 ∗ p2(ϕ) = p1(ϕ)p2(ϕ).

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

Kµ+ν = Kµ ∗ Kν .

Branching process

A Markov process X with state space M(E) is called branchingprocess provided that for all µ1, µ2 ∈ M(E), the process Xµ1+µ2

starting from µ1 + µ2 and the sum Xµ1 + Xµ2 are equal indistributions, i.e., for all t ≥ 0 and F ∈ bpM(E) we have∫

F (X t (ω))Pµ1+µ2(dω) =

∫ ∫(F (X t (ω)+X t (ω

′))Pµ1(dω)P

µ2(dω′)

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

A1. Nonlinear evolution equation

(∗)

ddt 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 processwith 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

(∗)

ddt vt (x) = Lvt (x) + Φ(x , vt (x))

v0 = f ,

is formally equivalent with

(∗∗) vt (x) = Pt f (x) +

∫ t

0Ps(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 thatsup0≤s≤t ||Vsf ||∞ <∞, for all t > 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 thenso is t 7−→ Vt f (x).

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

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

vi) If (fn)n ⊂ bpB is a decreasing sequence, fn f , thenVt 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 branchingkernels (Qt )t≥0 on (M(E),M(E)) such that for all f ∈ bpB andt > 0 we have

Qt (ef ) = eVt f .

The infinitesimal generator of the forthcomingbranching process

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

F = ef

with f ∈ bpB, then

LF (µ) =

∫Eµ(dx)c(x)F ′′(µ, x)+∫

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

Eµ(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 variationalderivatives of F [F ′(µ, x) = limt→0

1t (F (µ+ tδx )− F (µ))].

Linear and exponential type superharmonic functionsfor the branching process

Letβ := ||b−||∞,

β′ ≥ β andb′ := 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 functionof u on M (with respect to L− q) is the function RM

d 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 , definedby

cµ(M) := inf∫

ERG

q p dµ : G open , M ⊂ G

is a Choquet capacity on E .

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

where DM := inft ≥ 0 : Xt ∈ M.

Tightness property of the capacity

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

infn

cµ(E \ Kn) = 0

or equivalently,Pµ(lim

nDE\Kn < ζ) = 0.

"The process lies in⋃

n Kn Pµ-a.s. up to the life time."

Compact Lyapunov function

A (L − q)-superharmonic function v is called compactLyapunov function provided that it is finite µ-a.e. and the set[v ≤ α] is relatively compact for all α > 0.

Proposition

Assume that (Pt )t≥0 is Markovian, i.e., Pt1 = 1 for all t > 0.Then the following assertions are equivalent.

(a) The capacity cµ is tight.

(b) There exists a compact Lyapunov function.

[L.B. & M. Röckner, Bull. Sci. Math. 2011]

[L.B. & M. Röckner, Complex Analysis and Op. Th. 2011]

Existence of Lyapunov functions for the superprocess

Assume that the spatial motion X is a Hunt process (i.e., it isquasi-left-continuous on [0,∞)).

Then for every λ ∈ M(E) there exists a compact Lyapunovfunction F with respect to the (X ,Φ)-superprocess, such thatF (λ) <∞.

Sketch of the proof

Since the spatial motion X has càdlàg trajectories it follows thatthere exists a λ-nest of compact sets of E

=⇒

there exists a Lyapunov function v ∈ L1(E , λ) ∩ S(L − b′).

=⇒

F := lv ∈ S(L − β′) and it has compact level sets

(cf. [V. Bogachev, Springer 2007]).

Remark

• The existence of the compact Lyapunov functions is themain step for the proof of the càdlàg property of the paths ofthe measure-valued (X ,Φ)-superprocess.

• Zenghu Li (Springer 2011) proved that in order to get thequasi left continuity of the branching process, the hypothesis"X is a Hunt process" is necessary.

• Probabilistic description of the (X ,Φ)-superprocess:

"A measure-valued Markov process describes the evolution ofa random cloud. The branching property means that anyparts of the cloud at time t do not interact after t. "

[E.B. Dynkin , S.E. Kuznetsov , A.V. Skorokhod, Probab.Theory Relat, Fields 99, 55-96 (1994)]

A2. Discrete branching;Probabilistic description

• An initial particle starts at a point of E and moves accordingto a base process X (with state space E) until a random time(defined by killing X ) when it splits into a random number m ofnew particles, its direct descendants, placed in E .

• Each direct descendant starts at the terminal position of theparent particle and moves on according to the m independentcopies of X an so on.

Construction of branching kernels on S

Proposition

(i) For every sub-Markovian kernel B : pB(S) −→ pB thereexists a branching kernel B on (S,B(S)) such that for everyϕ ∈ pB, ϕ ≤ 1, we have

Bϕ = Bϕ.

(ii) Conversely, if H is a branching kernel on (S,B(S)) thenthere exists a unique sub-Markovian kernel B : pB(S) −→ pBsuch that H = B.

Example of branching kernel on S

Let qk ∈ pB for all k ≥ 1, satisfying∑

k≥1 qk ≤ 1.

Consider the kernel B, where B : pB(S) −→ pB is defined as

(∗) Bg(x) :=∑k≥1

qk (x)gk (x , . . . , x), g ∈ bpB(S), x ∈ E

with gk := g|E (k) .

In particular, for all ϕ ∈ pB, ϕ ≤ 1, we have

Bϕ =∑

k≥1 qkϕk .

Branching processes on the finite configurations

• Base process: X = (Ω,F ,Ft ,Xt , θt ,Px ), a right (Markov)process with state space E

• Branching kernel: a sub-Markovian kernelB : bpB(S) −→ bpB such that

supx∈E

Bl1(x) <∞.

If B is given by (∗) then the above condition is equivalent with

supx∈E

∑k≥1

kqk (x) <∞.

• Killing kernel: c ∈ bpB

Theorem

There exists a branching semigroup (Ht )t≥0 on S, having thefollowing property: If u ∈ pB, u ≤ 1, is such that u is aninvariant function with respect to this semigroup, then u belongsto the domain of L (the infinitesimal generator of the baseprocess X) and

(L − c)u + cBu = 0.

In particular, if B is given by (∗), then u is the solution of thenonlinear equation

(L − c)u + c∑k≥1

qkuk = 0.

Remark

Let (Ht )t≥0 be the branching semigroup on S constructed in theabove theorem. Then there exists a family (Vt )t≥0, a (nonlinear)semigroup on bpB, such that

Ht (ef ) = eVt f , f ∈ bpB.

Theorem

If X is Hunt process with state space E, then (Ht )t≥0 is thetransition function of a càdlàg branching process with statespace S.

[L.B.& O. Lupascu, in preparation]

Remark

(i) If u ∈ pB is an excessive function with respect to the baseprocess X , then lu is excessive with respect to the branchingsemigroup (Ht )t≥0.

(ii) If u is a compact Lyapunov function for the base process X ,then lu is a compact Lyapunov function for the branchingsemigroup (Ht )t≥0. In particular, condition (B) holds for thebranching semigroup.

(iii) One can take a superprocess (X ,Φ) as base process. Thefinal result of the construction will be a Markov process on thefinite configurations of positive finite measures on E , describingthe evolution of a random cloud controlled not only by abranching mechanism Φ but also by a discrete branching, in anew dimension, along which the splitting into a random numberof clouds takes place, commanded by a "killing kernel" and a"branching kernel".

A3. A nonlinear Dirichlet problem

• D : a bounded open subset of Rd

• (qn)n≥2 ⊂ C+b (D) such that∑n≥2

qn ≤ 1, c ∈ C+b (D)

• ϕ: a bounded measurable function on the boundary ∂D of D

• Let L be the infinitesimal generator of a Feller semigroupon D and consider the following nonlinear Dirichlet problem:

Lu + c(∑n≥2

qnun − u) = 0 on D,

u = ϕ on ∂D.

References

A1. Continuous branching processes

[L.B., J. Euro. Math. Soc. 2011]

A2. Discrete branching processes

A3. A nonlinear Dirichlet problem

[L.B. & A. Oprina, J. Math. Anal. Appl. 2011]

[L.B., O. Lupascu, and A. Oprina, in: CRM Proceedings andLecture Notes 55, AMS 2012] – a unifying construction