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: [email protected]

July 11, 2012, BCAM Seminar, Bilbao

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 , P x ) with state space E: – (Ω, F) is a measurable space, P x 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 P x (Xt ∈ A) = Pt (x, A) [ If f ∈ pB then E x (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 P µ -a.e. if it possesses left limits in E P µ -a.e. on [0, ζ); ζ is the life time of X

The resolvent of the process X

U = (Uq )q>0 ,

Z

Uq f (x) := E

x

∞

e 0

−qt

Z f ◦ Xt dt =

∞

e−qt Pt f (x)dt ,

0

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 Uq fn % v

(iii)

αUq+α v ≤ v for all α > 0 and αUq+α v % v when α % ∞

(iv )

e−qt Pt v ≤ v for all t > 0 and limt→0 Pt v = 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, Z lf (µ) := hµ, f i :=

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 P 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 M S= E (m) , m≥1

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 b : S −→ R, called multiplicative, defined as ϕ(x) b := ϕ(x1 ) · . . . · ϕ(xm ) for x = (x1 , . . . , xm ) ∈ E (m) . • A multiplicative function ϕ b is the restriction to S of an exponential function on M(E), ϕ b = 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 Z

Z p1 ∗ p2 (dν)F (ν) :=

Z 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 (ϕ) b = p1 (ϕ)p b 2 (ϕ). b

• 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 X µ1 +µ2 starting from µ1 + µ2 and the sum X µ1 + X µ2 are equal in distributions, i.e., for all t ≥ 0 and F ∈ bpM(E) we have Z Z Z µ +µ µ µ F (X t (ω))P 1 2 (dω) = (F (X t (ω)+X t (ω 0 ))P 1 (dω)P 2 (dω 0 )

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

A1. Nonlinear evolution equation

(∗)

  

d dt vt (x)

 

v0 = f ,

= Lvt (x) + Φ(x, vt (x))

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

Z ∞ + (1 − e−λs −λs)N(x, ds) 0

•

c ≥ 0 and b are bounded B-measurable functions

•

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

N(u ∧ u 2 ) ∈ bpB Examples of branching mechanisms Φ(λ) = −λα

if

1 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−→ hµ, Vt f i 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 Z LF (µ) = µ(dx)c(x)F 00 (µ, x)+ E Z µ(dx)[LF 0 (µ, ·)(x) − b(x)F 0 (µ, x)] + E Z Z ∞ µ(dx) N(x, ds)[F (µ + sδx ) − F (µ) − sF 0 (µ, x)] E

0

where F 0 (µ, x) and F 00 (µ, x) are the first and second variational derivatives of F [F 0 (µ, x) = limt→0 1t (F (µ + tδx ) − F (µ))].

Linear and exponential type superharmonic functions for the branching process

Let β := ||b− ||∞ , β 0 ≥ β and b0 := b + β 0 . If u ∈ bpB then the following assertions are equivalent. i) u ∈ S(L − b0 ) ii) lu ∈ S(L − β 0 ) iii) For every α > 0 we have 1 − eαu ∈ S(L − β 0 ).

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 RdM u defined by  RqM u := inf v ∈ S(L − q) : v ≥ u on M . •

The reduced function RqM u is universally B-measurable.

• Let p := Uq 1. The functional M − 7 → cµ (M), M ⊂ E, defined by Z cµ (M) := inf{ RqG p dµ : G open , M ⊂ G} E

is a Choquet capacity on E. •

RqM f (x) = E x (f (XDM ))

where DM := inf{t ≥ 0 : Xt ∈ M}.

[Hunt’s Theorem],

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 cµ (E \ Kn ) = 0 n

or equivalently, P µ (lim DE\Kn < ζ) = 0. n

"The process lies in

S

n

Kn P µ -a.s. up to the life time."

Compact Lyapunov function A (L − q)-superharmonic function v is called compact Lyapunov 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., Pt 1 = 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 is quasi-left-continuous on [0, ∞)). Then for every λ ∈ M(E) there exists a compact Lyapunov function F with respect to the (X , Φ)-superprocess, such that F (λ) < ∞.

Sketch of the proof

Since the spatial motion X has càdlàg trajectories it follows that there exists a λ-nest of compact sets of E =⇒ there exists a Lyapunov function v ∈ L1 (E, λ) ∩ S(L − b0 ). =⇒ F := lv ∈ S(L − β 0 ) and it has compact level sets (cf. [V. Bogachev, Springer 2007]).

Remark • The existence of the compact Lyapunov functions is the main step for the proof of the càdlàg property of the paths of the measure-valued (X , Φ)-superprocess. • Zenghu Li (Springer 2011) proved that in order to get the quasi 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 of a random cloud. The branching property means that any parts 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 according to a base process X (with state space E) until a random time (defined by killing X ) when it splits into a random number m of new particles, its direct descendants, placed in E. • Each direct descendant starts at the terminal position of the parent particle and moves on according to the m independent copies of X an so on.

Construction of branching kernels on S

Proposition (i) For every sub-Markovian kernel B : pB(S) −→ pB there b on (S, B(S)) such that for every exists a branching kernel B ϕ ∈ pB, ϕ ≤ 1, we have bϕ B b = Bcϕ. b (ii) Conversely, if H is a branching kernel on (S, B(S)) then there exists a unique sub-Markovian kernel B : pB(S) −→ pB b such that H = B.

Example of branching kernel on S

Let qk ∈ pB for all k ≥ 1, satisfying

P

k ≥1 qk

≤ 1.

b where B : pB(S) −→ pB is defined as Consider the kernel B, X qk (x)gk (x, . . . , x), g ∈ bpB(S), x ∈ E (∗) Bg(x) := k ≥1

with gk := g|E (k ) . In particular, for all ϕ ∈ pB, ϕ ≤ 1, we have Bϕ b=

k k ≥1 qk ϕ .

P

Branching processes on the finite configurations • Base process: X = (Ω, F, Ft , Xt , θt , P x ), a right (Markov) process with state space E • Branching kernel: a sub-Markovian kernel B : bpB(S) −→ bpB such that sup Bl1 (x) < ∞. x∈E

If B is given by (∗) then the above condition is equivalent with X sup kqk (x) < ∞. x∈E k ≥1

•

Killing kernel: c ∈ bpB

Theorem b t )t≥0 on S, having the There exists a branching semigroup (H b is an following property: If u ∈ pB, u ≤ 1, is such that u invariant function with respect to this semigroup, then u belongs to the domain of L (the infinitesimal generator of the base process X ) and b = 0. (L − c)u + cB u In particular, if B is given by (∗), then u is the solution of the nonlinear equation X (L − c)u + c qk u k = 0. k ≥1

Remark

b t )t≥0 be the branching semigroup on S constructed in the Let (H above theorem. Then there exists a family (Vt )t≥0 , a (nonlinear) semigroup on bpB, such that b t (ef ) = eV f , H t

f ∈ bpB.

Theorem b t )t≥0 is the If X is Hunt process with state space E, then (H transition function of a càdlàg branching process with state space S. [L.B.& O. Lupascu, in preparation]

Remark (i) If u ∈ pB is an excessive function with respect to the base process X , then lu is excessive with respect to the branching b t )t≥0 . semigroup (H (ii) If u is a compact Lyapunov function for the base process X , then lu is a compact Lyapunov function for the branching b t )t≥0 . In particular, condition (B) holds for the semigroup (H branching semigroup. (iii) One can take a superprocess (X , Φ) as base process. The final result of the construction will be a Markov process on the finite configurations of positive finite measures on E, describing the evolution of a random cloud controlled not only by a branching mechanism Φ but also by a discrete branching, in a new dimension, along which the splitting into a random number of clouds takes place, commanded by a "killing kernel" and a "branching kernel".

A3. A nonlinear Dirichlet problem • •

D : a bounded open subset of Rd X (qn )n≥2 ⊂ Cb+ (D) such that qn ≤ 1, c ∈ Cb+ (D) n≥2

• ϕ: a bounded measurable function on the boundary ∂D of D • Let L be the infinitesimal generator of a Feller semigroup on D and consider the following nonlinear Dirichlet problem:  X  Lu + c( qn u n − u) = 0 on D,    n≥2

    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 and Lecture Notes 55, AMS 2012] – a unifying construction