A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

D. Blömker

, M. Romito

, R. Tribe

aInstitut für Mathematik, RWTH Aachen, Templergraben 55, 52052 Aachen, Germany bDipartimento di Matematica “U. Dini”, viale Morgagni 67/a, 50134 Firenze, Italia

cUniversity of Warwick, Mathematics Institute, CV4 7AL Coventry, UK

Received 26 May 2005; received in revised form 25 November 2005; accepted 7 February 2006 Available online 25 September 2006

Abstract

The solutions to a large class of non-linear parabolicPDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a comparison scalarPDE.

In cases where the representation fails to be integrable, a sequence of pruned trees is constructed, producing approximate sto- chastic representations that in some cases converge, globally in time, to the solution of the originalPDE.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Les solutions d’une latge classe EDP non linéaires paraboliques sont données sous la forme d’espérances de fonctionnelles adéquatement choisies d’un arbre de branchement. L’intégrabilité est établie par comparaison avec une EDP scalaire. Lorsque l’intégrabilité fait défaut, un procédé d’élagage permet dans certains cas de produire une suite convergeant vers la solution de l’EDP d’origine.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:primary 60J80; secondary 35Q30, 35K55, 35C99, 76M35, 60H30

Keywords:Branching processes; Pruned trees; Stochastic representation; Burgers equation; Navier–Stokes equations; Semi-implicit approximation

1. Introduction

This paper considers stochastic representations for solutions to a large class of non-linear parabolic PDEs, or systems ofPDEs, of the type

∂_tu=Au+F(u)+f, (1.1)

* Corresponding author.

E-mail addresses:bloemker@instmath.rwth-aachen.de (D. Blömker), romito@math.unifi.it (M. Romito), tribe@maths.warwick.ac.uk (R. Tribe).

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.02.001

whereA is a linear operator with a complete set of eigenfunctions, F is a polynomial nonlinearity in u and its derivatives, andf is a given driving function. In short, the solution uis expanded into a Fourier series using the eigenfunctions ofA. This yields (as in spectral Galerkin methods) a system of countably many coupledODEs for the Fourier coefficients. ThisODEsystem is then solved in a weighted^∞-space, via an expectation over a tree of branching particles. The rules for the branching and dying probabilities arise from the particularPDEbeing studied.

Moreover thePDEdetermines an evaluation operatorR_t, acting on the treeTk of particles rooted at each Fourier modek, so that, under integrability assumptions, the (suitably weighted)kth Fourier modeχ_k(t )is given by

χ_k(t )=^E R_t(Tk)

.

This is precisely the method of Le Jan and Sznitman [10,11] where they treated the Navier–Stokes equations inR³. We comment on related literature later in this introduction.

1.1. Integrability

The major drawback of the stochastic representation is that it often fails to exist for large timest, although the solution to thePDE may still exist. The problem is thatR_t(Tk)may fail to be an integrable random variable for tt₀. The standard way to check integrability is to establish a scalar real-valued comparison equation. The finiteness of this comparison equation implies the integrability needed for the stochastic representation to hold (and in some cases provides a sufficient condition as well). However, the comparison equation represents a worst case scenario with super-linear (explosive) growth and it ignores most of the structure of the non-linearity in the originalPDE. It typically leads to a representation that holds for allt0 only for small initial data.

Our main aim is to present an approach to treat cases where integrability fails. Our approach is to construct setsΩn, withP[Ω_n] ↑1 so that

χ_k(t )= lim

n→∞E

R_t(Tk)1Ω_n

. (1.2)

This treats the expectation somewhat as a singular integral, where we have to be careful how to cut out the singularity.

The method we use, explained in Section 5, is to construct a pruned branching treeT_k⁽ⁿ⁾ which will agree withTk

onΩ_n. The expectation for the pruned tree E

R_t T_k⁽ⁿ⁾

=^E

R_t(Tk)1Ωn

is always well defined and will represent thekth Fourier mode of the solution to a semi-implicit approximation scheme of the type

∂_tu⁽ⁿ⁾=Au⁽ⁿ⁾+F

u⁽ⁿ⁾, u⁽ⁿ⁻¹⁾ +f.

We then use^PDEtechniques to verify that the approximation scheme converges to a solution of the original^PDE. Although there are general results for the convergence of such approximations (see for example Bjørhus and Stuart [5]) the assumptions are usually quite restrictive. Since stronger arguments are specific to the particularPDEbeing studied, we present the arguments only in one representative case, namely for Burgers equation. In this example the limiting probabilistic representation (1.2) holds for all timest0. Currently, only in a very simple framework of small initial conditions and uniformly small forcing is it currently possible to derive such representation results for large times (for the 3D Navier–Stokes equations see Bakhtin [3] and Waymire [21]).

1.2. A toy example

We illustrate here the key idea of the pruning scheme on the simplest example possible, namely the ordinary differential equationu˙= −u+u². The solution can be given by the stochastic representationu(t )=^E[u(0)^Nt], where Nt is the number of particles at timet of a simple rate one binary branching process, starting from a single particle at time 0. It is easy to verify that the representation is well defined for all timest0 if and only if|u(0)|1, in that the variable|u(0)|^Nt becomes non-integrable for larget when|u(0)|>1, while the solutions of the equation blow up only ifu(0) >1.

We now give a modification of the branching process. Give each particle a label from the integersN. Particles still branch at rate 1 but a particle with labelnproduces two offspring, one with labelnand one with labeln−1. When

a particle of type 0 tries to branch it simply dies. Start with a single particle with labelnand letNt(n)denote the total number of particles at timet, regardless of their label. Setu_n(t )=^E[u(0)^Nt(n)]forn0 andu₋₁≡0. Then it is possible to show thatu_n(t )solves the following semi-implicit iterative scheme

˙

un= −un+un−1un, un(0)=u(0), forn0.

It is straightforward to check thatu_n(t )is well defined for allnandt. Moreover,u_nconverges to the solutionu(t )of the original problem for each initial conditionu(0)1. This yields the stochastic representation

u(t )= lim

n→∞E

u(0)^Nt(n) valid for allu(0)1 and allt0.

Remark 1.1.The seemingly simpler modification (used by Le Jan and Sznitman [10] for their uniqueness proof and by Bhattacharya et al. [4]) where a particle with labelnproduces two offspring each with labeln−1, leads to the explicit iterative schemeu˙_n= −u_n+u²_n−1. Unfortunately, the limit ofu_n(t )for larget, asn→ ∞, fails to exist for u(0) <−1.

The semi-implicit approximation scheme works for other polynomial non-linearities. For example, if one considers

˙

u= −u−u³, the approximation schemeu_n= −u_n−u²_n−1u_n, where each particle with labelnbranches into three particles, one with labelnand two with labeln−1, is convergent to the true global solution for any initial condition.

1.3. Related literature

Earlier papers connecting branching particle systems toPDEs (for instance Skorokhod [18] or Ikeda, Nagasawa and Watanabe [9], and later McKean [13]) use branching coupled with a diffusion, and the stochastic representation is derived directly without Fourier series, so that the linear operatorAis limited to generators of Markov processes. The idea of representing the Fourier modes, from Le Jan and Sznitman [10], was later extended in Bhattacharya et al. [4], Chen et al. [6], Waymire [21], and to a representation in the physical space in Ossiander [16].

When deriving a system of^ODEs in^∞space, there is considerable freedom in the choice of weights for the Fourier coefficients. See Bhattacharya et al. [4] for an extensive discussion in the case of 3D Navier–Stokes. We establish a comparison equation for our examples in Section 4. We show that for a class of equations, which includes Burgers but not 3D Navier–Stokes, that the choice of weights cannot affect the integrability of the probabilistic representation.

Pruning of the random trees also appears in the literature. In Bhattacharya et al. [4] the branching trees are pruned afterngenerations. This gives a stochastic representation of a Picard iteration scheme converging to the original^PDE, but, as stated in [4], the existence of the expectation is equivalent to the convergence of the Picard iteration scheme.

In another approach to non-integrability, Morandin [15] suggested a clever re-summation of the expectation in order to improve the convergence for large times, but he was only able to rigorously verify the global convergence of his method in a simple example where (1.1) is a one-dimensionalODE.

1.4. Layout of the paper

The paper is organised as follows. In Section 2 we present our abstract ODE setting and show how to recast three representative examples ofPDEs with quadratic non-linearities into this required form. In Section 3 we develop the notation needed for the random trees and establish the stochastic representation under the assumption that it is integrable. In Section 4 we investigate the comparison equation which typically shows the representation is integrable at small times, or, when there is no linear instability, for all times with small data. For a class of equations we obtain necessary and sufficient conditions for the integrability of the stochastic representation that are independent of the choice of weights in Fourier space. In Section 5 we develop the idea of pruning and apply it to the case of the 1D Burgers equation.

2. Abstract setting and examples

We first present an infinite system ofODEs involving a quadratic non-linearity. The system is indexed overk∈Z^d. We then discuss several examples ofPDEs on the torus[0,2π )^dand recast their Fourier transforms into our abstract

ODE setting. We do not present the highest generality possible, but focus instead only on an equation with one quadratic nonlinearity, one additional linear part and one forcing term.

2.1. The general system ofODEs

We consider solutionsχ (t ):Z^d→C^r to the following infinite dimensional system ofC^r-valuedODEs

˙ χ_k=λ_k

−χ_k+C_Rp_kχ_k+C_B

l,m∈Z^d

q_k,l,mB_k,l,m(χ_l, χ_m)+d_kγ_k (2.1)

withk∈Z^d. The constantsλ_k>0 (which will determine the rate of particle evolution),p_k,q_k,l,m,d_k∈ [0,1](which will determine the probabilities of regeneration, branching and dying), andC_R,C_B0 (the regeneration and branch- ing constants) are fixed, as are bilinear operatorsB_k,l,m:C^r×C^r→C^r satisfying

B_k,l,m(χ , χ)|χ||χ|

for allχ , χ∈C^r. The choice of these constants will arise from the Fourier transform of thePDEbeing studied. We assume throughout that

p_k+q_k+d_k=1 for allk∈Z^d, (2.2)

and

pk→0, qk→0, as|k| → ∞, (2.3)

where

q_k=

l,m∈Z^d

q_k,l,m.

The data for the equations consists of a time dependent forcingγ= {γ_k(t ): k∈Z^d, t0}and an initial condition χ (0)= {χ_k(0): k∈Z^d}. We consider the above system in its mild formulation, that is for given data we look for measurablet →χ_k(t )∈C^r satisfying, fork∈Z^d,

χ_k(t )=e^−λktχ_k(0)+ t 0

λ_ke^{−λk(t−s)}

C_Rp_kχ_k(s)+C_B

l,m∈Z^d

q_k,l,mB_k,l,m

χ_l(s), χm(s)

+d_kγ_k(s) ds.

(2.4) Note that we need some regularity ofχ_k, in order to make (2.4) well defined.

Remark 2.1.The abstract setting appears quite demanding in terms of the number and complexity of assumptions to be checked. The key assumption to verify is (2.3). However the examples provided below will show that is possible to get recast interesting examples into this framework, by simply exploiting the known characteristics of their non-linear operators. Moreover, we give in Section 2.5 a sufficient criterion which is easy to check for the underlyingPDE. Remark 2.2.There is considerable flexibility when choosing the constants in theODEsystem (2.1). For example, we can adjust the probabilitiesp_k,q_k,l,m, andd_k by adjusting the constantsC_B,C_R and considering modified forcing dataγ. In particular, in an equation where the probabilities do not add up to 1 in (2.2), it is always possible to adjust d_k and the forcing data so that this constraint holds. Similarly, an equation with C_R andC_B replaced by bounded functions ofkcan be recast into the form (2.1) by forcing thekdependence into the probabilitiesp_k,q_k, andd_k.

2.2. The d-dimensional Burgers equations

Consider solutionsu(t, x)∈R^d, fort0 andx∈ [0,2π )^dto the Burgers system ∂_tu− u+(u· ∇)u=f,

u(0)=u⁰, (2.5)

with periodic boundary conditions, wheref is an external forcing. We restrict ourselves to periodic boundary condi- tions, as the non-linearity is easy to compute in the Fourier basis.

If we expand the solution u(t, x)=

k∈Z^d

u_k(t )e^ik·x,

the equation reads in the Fourier coefficients as

˙

u_k= −|k|²u_k−i

l+m=k

(u_l·m)um+f_k.

The sum is over alll,m∈Z^d satisfyingl+m=k. Define a weight functionw_k=1∨ |k|^γ, whereγ >0 will be chosen shortly, and setχ_k=w_ku_k. Then

χ˙_k= −|k|²χ_k−i

l+m=k|m|w_k

wmwl(χ_l·_|^m_m|)χ_m+f_kw_k,

χ_k(0)=u_k(0)wk. (2.6)

Note that the modeu0has no linear dissipation. Below we will add and subtractλ0χ0to the equation forχ0, which introduces a linear instability but which allows us to write the equation in our desired abstract form. We note that in dimensiond=1 this trick is unnecessary: the equations for the zeroth mode decouples, in that it simplifies tou˙₀=f₀, and it is then possible to reduce the problem to the casef₀=u₀=0.

We now show one way to recast (2.6) into the abstract form (2.1). For givenC_R,C_B,λ₀>0 we define λ_k=

|k|², k=0,

λ₀, k=0, p_k=

0, k=0, C_R⁻¹, k=0, q_k,l,m=C_B⁻¹ |m|wk

λ_kw_lwm

, B_k,l,m(χ , χ)= −i

χ· m

|m|

χ,

wheneverl+m=k (and zero otherwise). Lemma 2.3 below ensures, provided we chooseγ >max{^d+₂¹, d−1}, that q_k=

l,mq_k,l,m<∞ and that q_k→0 as |k| → ∞. Thus by taking CB, CR sufficiently large we have that p_k+q_k <1 and it remains only to define d_k=1−p_k−q_k andγ_k=(f_kw_k/λ_kd_k)for k∈Z^d. Again, there is considerable flexibility in these choices.

Lemma 2.3.For allγ1,γ2>0withγ1+γ2> d there existsC=C(γ1, γ2) <∞so that, for allk∈Z^d, withk=0,

l+m=k l=0,m=0

1

|m|^γ1|l|^γ2 C

1+ |k|₋β

, ifγ₁=d andγ₂=d, C

1+ |k|₋β

log 1+ |k|

, ifγ₁=d orγ₂=d,

whereβ=min{γ₁, γ₂, γ₁+γ₂−d}and the sum is over all indicesl,minZ^dsatisfying the given constraints.

One way to prove this lemma, whose proof is omitted, is to compare above and below by suitable continuous integrals.

2.3. Two-dimensional Navier–Stokes equations

We briefly treat the two-dimensional Navier–Stokes in its vorticity formulation, since this will be used in Sec- tion 4 as an example where the comparison equation yields exact statements about the integrability of the stochastic representation.

In dimensiond=2 the vorticityξ=curluis a scalar and satisfies, on the torus[0,2π )²and with periodic boundary conditions,

∂_tξ− ξ+(u· ∇)ξ=f,

ξ(0)=ξ⁰, (2.7)

whereuis the solution to the Navier–Stokes equations. The Fourier coefficients satisfy the following system, ξ˙k= −|k|²ξk+

l+m=k

k·l^⊥

|l|² ξlξm+fk,

wherel^⊥=(l₂,−l₁). For simplicity we shall assume that f₀=0 and the vorticity has mean valueξ₀ zero and is omitted from the system.

We then setχ_k= |k|^γξ_kfor someγ >¹₂. ForC_B>0 we then define λk= |k|², Bk,l,m(χ , χ)= k·l^⊥

|k·l^⊥|χ χ, qk,l,m=C_B⁻¹|k|^γ−2|k·l^⊥|

|l|^γ+2|m|^γ ,

for allk,l,m∈Z² satisfyingk·l^⊥=0 andl+m=k(and zero otherwise). Lemma 2.3 ensures thatqk<∞ and thatqk→0 as|k| → ∞. TakingCB large enough we have thatqk<1 (note that here we may takepk=0). So the recasting is complete if we defineγ_k=(|k|^γ−2/d_k)f_k.

2.4. A surface growth equation

This final example illustrates the change in weights needed for a higher order equation and the need to consider linear instabilities. In particular, the linear operator does not generate a diffusion. Therefore, the Fourier transform is necessary for the stochastic representation. Consider the following scalar equation arising in some models for surface growth,

∂_tu= −a₁ ²u−a₂ u−a₃ |∇u|²+a₄|∇u|²+f,

with periodic boundary conditions on[0,2π )^d, withd=1,2 anda_i >0 fori=1,2,3. As for Navier–Stokes, there remain basic unresolved questions on existence and uniqueness. See Raible et al. [17] for the derivation of the model, and Blömker et al. [7] for a rigorous mathematical treatment usingPDEtechniques. For simplicity, we assumea4=0 and that the mean value

u(t, x)dxis zero, allowing us to omit the coefficientu₀. The equation for the Fourier coefficients is given by

˙

u_k= −a₁|k|⁴u_k+a₂|k|²u_k+a₃|k|²

l+m=k

(l·m)ulu_m+f_k.

Thek dependence in the term+C_Rp_kχ_k in (2.1) is exploited in this example to allow for the terma₂ u, which corresponds to the physically important linear instability of the equation. We set χ_k= |k|^γu_k for γ >0, with γ >max{d,1+d/2}, and then choose, for allk=0,

λ_k=a1|k|⁴, p_k=a₂

a₁C_R⁻¹|k|^γ−2 and

B_k,l,m(χ , χ)= l·m

|l·m|χ χ, q_k,l,m=C_B⁻¹a3|k|^γ−2|l·m|

a₁|l|^γ|m|^γ

withB_k,l,m andq_k,l,mequal to zero ifl·m=0 orl+m=k. Lemma 2.3 guarantees thatp_k+q_k<1 whenC_B,C_R are taken large enough and we can define

dk=1−qk−pk and γk=|k|^γ−4 a₁d_k fk

to obtain a system in the form (2.1).

2.5. Extensions

It is clear that the framework could be modified to allow several possible extensions, for example equations with other boundary conditions, or with polynomial non-linearities (or even analytic non-linearities, see Athreya and Tribe [2]), or with non-linear multiplicative forcing terms.

As an example, we indicate a fairly simple criterion that ensures the existence of the stochastic representation. It is easily checkable from the data of thePDEand covers equations with a variety of boundary conditions. Moreover, we give an explicit example with Dirichlet boundary conditions, to show that this is as simple as the periodic case.

2.5.1. A general criterion

Consider for simplicity an equation of the type

∂tu=Au+F(u, u),

whereAis a self-adjoint differential operator in a Hilbert spaceHwith a complete set of normalised eigenfunctions (e_k)_k∈N and corresponding eigenvalues (λ_k)_k∈N, and some bi-linear operator F. Thus, using u=_∞

k=1u_ke_k we derive

∂_tu_k=λ_ku_k+ ^∞

l,m=1

u_lu_m

F(e_l, e_m), e_k

H.

We suppose, for simplicity of presentation, thatAis positive, i.e.λ_k>0. Assume that there is aδ >0 such that ∞

k=1

λ⁻_k^δ<∞. (2.8)

In particular,λ_k→ ∞ask→ ∞. Finally, assume the following crucial property on the non-linear operator:

F is a continuous map fromD A^α−δ

×D A^α−δ

toD

A^α−1−ε

, (2.9)

for someαandε >0.

Choose weightsw_k =C₀λ^α_k, withC₀ large enough (according to the estimates below), and setχ_k=w_ku_k. By computations entirely similar to the ones in the previous section, we find

q_k,l,m= wk

λkwlwm

F(e_l, e_m), e_k

H

= λ⁻_k^ε C₀λ^δ_lλ^δ_m

A^α−1+εF

A^δ−αe_l, A^δ−αe_m , e_k

H, and, by condition (2.9), it follows thatq_k=

l,mq_k,l,m∈ [0,1), providedC₀is large enough, andq_k→0 ask→ ∞.

Thus the crucial condition (2.3) is fulfilled.

2.5.2. A specific example

Conditions (2.8) and (2.9) are quite easy to check in concrete examples. For example, for Navier–Stokes equations (in both dimensions 2 and 3), condition (2.8) follows from well-known results concerning the growth of eigenvalues for the Laplacian, while condition (2.9) follows from Lemma 2.1 of Temam [20].

The conditions of the previous section are by no means optimal and can be easily improved once the non-linearity is explicitly given. We can rely on cancellations yieldingqk,l,m=0 for many indices. Consider for example the 1D Burgers equation with Dirichlet boundary condition on[0, π]. Hereek=sin(kx),λk=k²andF(u, v)=¹₂(u·∂xv+

∂_xu·v). It is easy to verify that F(el, em)=

l+m

4 el+m−|l−m| 4 e_|l−m|

so that we can directly compute qk=cw_k

k _∞

l=1

1 w_lw_k+l +2

k−1

l=1

1 w_lw_k−l

,

up to some normalisation constantc >0, so that forwk=C0λ^α/2_k =C0k^α, withα >1 andC0large, we have, with computations similar to those of Lemma 2.3,qk∈ [0,1)andqk→0 ask→ ∞. Again (2.3) is fulfilled.

3. The branching particle representation formula 3.1. Existence and uniqueness

Denote bya_∞=sup_k∈Zd|a_k|the norm of bounded families(a_k)_k∈Zd of elements ofC^r, with|a_k| = a_k·a^∗_k the norm inC^r. We start by giving a short theorem, whose proof is purely deterministic, which ensures that there is a unique local solution to (2.1). The proof is a rather standard application of the Banach fixed point theorem, and the result will be improved in Section 4 using probabilistic tools.

Theorem 3.1(Unique local existence). Assume that χ (0)∈^∞

C^r

, γ∈L^∞

[0, T], ^∞ C^r

for allT >0.

Then there exists a timeT₀>0, depending only onχ (0),γ, and the constants appearing in the equation, such that the mild formulation(2.4)has a unique solutionχ∈L^∞_loc([0, T0), (C^r)^Zd).

Moreover, we have eitherT₀= ∞orχ (t )_∞→ ∞ast→T₀. Finally, if the functionst →γ_k(t )areC^k, then t →χ_k(t )areC^k+1in time and solve Eq.(2.1).

Proof. LetBbe a ball of radiusR >0 centred at the constant function with valueχ (0), in the space L^∞

[0, t], ^∞ C^r

.

Forχ∈BdefineF (χ )by the right-hand side of (2.4). Then forR0=R+ χ (0)∞, F (χ )k(t )−χk(0)χ (0)

∞+CRpkR0+CBqkR₀²+γ (t )

∞

1−e^−λkt∗ .

If we chooseR >χ (0)_∞+supγ (t )_∞andt_∗small enough we see thatF mapsBinto itself. Here we have used assumption (2.3) to control the large|k|s. Moreover, ifχ¹andχ²are inB, then fortt_∗,

F χ¹

−F χ²

k(t )(C_Rp_k+2CBR₀q_k)

1−e^−λkt∗ sup

tt_∗

χ¹(t )−χ²(t )

∞.

HenceF is a strict contraction inL^∞([0, t], ^∞(C^r))if we chooset_∗small enough. Here we need again, for large|k|, the assumption (2.3).

The assertion for the timeT₀follows in a standard manner by gluing together local solutions. The continuity of t→χ_k(t )follows from the mild form (2.4). It is even differentiable with bounded derivative. The C^k-regularity follows by differentiating (2.4) and the higher regularity follows from differentiating (2.1). 2

3.2. The branching tree

We now give a construction of the branching process that will be used to represent the solutions of (2.1). For the general theory of branching processes see [1,8]. We will label particles of the process with labels taken from the set I=_∞

n=0{0,1,2}ⁿ(cf. Fig. 1). The history of a particleα=(α₁, . . . , α_n)can be read off by interpretingα_j=0 as a regeneration, andα_j =1 (or 2) as being child 1 (or 2) in a binary branching event at generationj (binary branching events will correspond to quadratic terms in the underlyingPDEwhereas regenerations, which may also be thought of a branching events with a single offspring, will correspond to lower order linear terms).

Forα∈ {0,1,2}ⁿ we write |α| =n which we call the length of the label. We writeα= ∅for the single label of length zero. Whenα=(α₁, . . . , α_n)we write α|j for the label α|j =(α₁, . . . , α_j)of its ancestor at generation j∈ {0,1, . . . , n−1}(and setα|0= ∅). Fori∈ {0,1,2}we write(i, α)for the label(i, α₁, . . . , α_n)and(α, i)for the label(α₁, . . . , α_n, i)(or(i, α)=(α, i)=(i)ifα= ∅). We construct the branching particle systems on a probability space equipped with the following independent families of I.I.D. variables:(E_α)_α∈Iexponential mean one variables (that will control the overall rates of branching and regenerating);(U_α)_α∈Iuniform[0,1]variables (that will control

Fig. 1. A tree with branches, deaths (") and regenerations (!).

Fig. 2. The construction of the tree. At each event timeτ, there is a random selection between eitherdeath,regenerateorbranchof two new particles to stateslandm(depending on the statekof the parent particle).

whether a particle regenerates, branches or dies) and((Y_α⁽¹⁾(k), Yα⁽²⁾(k))_α∈I,k∈Zd random variables with distribution P[Yα⁽¹⁾(k)=l, Yα⁽²⁾(k)=m] =qk,l,m(which will control the positions of the two offspring of a particle that branches).

We now define a system(Kα, τ_α^B, τ_α^D)_α∈Iof particle positions, birth and death times, inductively over the length n= |α|of the labels. Fixk∈Z^dand setK_∅=k,τ_∅^B=0 andτ_∅^D=λ⁻_k¹E_∅. Assume that the positions, birth and death times have been defined for|α|n. Then, forαof lengthn+1, define birth and death times

τ_α^B=τ_α^D_|n, τ_α^D=τ_α^B+λ⁻¹

Kα|nE_α, and the particle positions

Kα=

⎧⎪

⎨

⎪⎩

Kα|n, α_n+1=0, Y_α⁽¹⁾(Kα|n), α_n+1=1, Y_α⁽²⁾(Kα|n), α_n+1=2.

This defines a complete tree of all possible branching and regenerating particles rooted atk. In the desired evolution the particles will choose whether to regenerate, branch or die according to the probabilitiespk,qk,dk.

We now define indicator variables(I_α)_α∈Ito decide whether a particular branch has survived. DefineI_∅=1 and, forαof lengthn+1,

I_α=

⎧⎨

⎩

1 ifα_n+1=0 andU_α∈ [0, p_Kα|n], 1 ifα_n+1∈ {1,2}andU_α∈ [1−q_K

α|n,1], 0 otherwise.

Fix an isolated cemetery stateΔand define, for|α| =n, Kα=

Kα if _n

j=1I_α|j=1, Δ otherwise.

The collectionTk=(Kα, τ_α^B, τ_α^D)_α∈Inow defines our branching tree rooted atk. It lives in the space defined by T =

Z^d∪ {Δ}

× [0,∞)× [0,∞)_I . We denote the law ofTkonT byP_k.

The descendants of any one particle in the tree form a new tree. To make this precise we define shift maps π_i:T →T, fori=0,1,2 as follows (cf. Fig. 2). ForT =(kα, s_α, t_α)_α∈I∈T we define a new treeπ_i(T)by

π_i(T)=(k(i,α), s_(i,α)−t_∅, t_(i,α)−t_∅)_α∈I.

The treeπi(T)is meant to be the tree of descendants of the particle labelled (i), with their birth and death times shifted so that particle(i)is born at timet=0. The construction of the branching particle system from I.I.D. families implies the following lemma.

Lemma 3.2.LetTk=(Kα, τ_α^B, τ_α^D)_α∈Ihave lawP_k. Then

(1) conditional on{τ_∅^D∈ds,K(0)=k}the treeπ0(Tk)has the law^Pk;

(2) conditional on{τ_∅^D∈ds, K(1)=m,K(2)=l}the treesπ1(Tk)and π2(Tk)are independent and have laws^Pm

andP_l.

We want to ensure that the tree has only finitely many branches before timet. DefineN_[0,t]:T →NbyN_[0,t](T)=

|{α∈I: sαt}|, that is the cardinality of the set of particles born before timet.

Lemma 3.3.UnderP_kthe variablesN_[0,t]are almost surely finite for allt0.

Proof. LetP_k(t )=^Pk[N_[0,t]<∞]. By conditioning on the values ofτ_∅^D,K(0),K(1),K(2)and using Lemma 3.2,

Pk(t )=e^−λkt+ t 0

λke^{−λk(t−s)}

pkPk(s)+

l,m∈Z^d

qk,l,mPl(s)Pm(s)+dk ds.

Hence,(P_k(t ):k∈Z^d, t0)is a bounded, real-valued solution to Eq. (2.1) with forcingγ≡1 and bilinear operators Bk,l,m(χ , χ)=χ χ. By Theorem 3.1, there is only one solution, namelyPk(t )=1 for allk, t. 2

A simple criterion that ensures that the branching process becomes extinct with probability one, that isKα=Δfor all large|α|, is that

qkdk and pk<1 for allk∈Z^d. (3.1)

This is easy to fulfil by modifying (enlarging) the constantsC_BandC_Rin (2.1). Indeed the number of particles alive at timetis an integer valued process whose successive values, under the condition (3.1), form a sub-critical branching process. Therefore it eventually reaches zero. The number of valuesk∈Z^dtaken by particles before this extinction is almost surely finite. The conditions thatλk>0 andpk<1 ensure that the extinction time for the branching particle system is almost surely finite. Note that, as explained in Remark 2.2, we can always choose the system (2.1) in such a way that (3.1) holds.

3.3. The evaluation along the tree

We now fix a forcing functionγ and an initial conditionχ (0). We wish to define evaluation mapsR_t:T →C^r fort0, which will depend onγandχ (0). These will satisfy a recursive property that allows them to be calculated backwards along the tree.

For the sake of simplicity, we introduce the following abbreviations: given a branching treeT =(kα, s_α, t_α)_α∈I and a particle labelledα∈I, withkα=Δ, we say that the particle has a

death: ifk(α,0)=k(α,1)=k(α,2)=Δ,

regeneration: ifk(α,0)=Δandk(α,1),k(α,2)=Δ, branch: ifk(α,0)=Δandk(α,1),k(α,2)=Δ.

Under each probabilityP_k, every particleαfor whichKα=Δmust do exactly one of the above three possibilities.

Lemma 3.4. There exists a family of maps Rt:T →C^r, for t0, satisfying, when N_[0,t](T) <∞, the implicit formula

R_t(T)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

χk_∅(0), t_∅t,

γk_∅(t−t_∅), t_∅< t,death at∅, C_RR_t−t∅

π₀(T)

, t_∅< t,regen. at∅,

C_BB_{k∅,k(1),k(2)} R_t−t∅

π₁(T) , R_t−t∅

π₂(T)

, t_∅< t,branch at∅.

(3.2)

Proof. Informally, since the tree is finite whenN_[0,t]<∞the value ofRt(T)can be calculated backwards along the tree, starting at times=t and working back to times=0: evaluate the initial conditionχ (0)at any particles that are alive at timet, evaluate the forcing functionγ (s)at any particle that dies at times < t, and apply the bilinear operators at the times of branching events.

For a careful proof one can define a sequence of approximationsR⁽_n,t^apprx)in the following way:

R⁽_1,t^apprx)(T)=

χ_k∅(0) ift_∅t,

γ_k∅(t−t_∅) ift_∅< t,death at∅,

1 otherwise,

andR_n^(apprx_+1,t⁾(T)is given by

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

χk_∅(0), t_∅t,

γk_∅(t−t_∅), t_∅< t,death at∅,

C_RR_n,t^(apprx_−t⁾

∅

π₀(T)

, t_∅< t,regener. at∅,

C_BB_{k∅,k(1),k(2)} R_n,t^(apprx_−t⁾

∅

π₁(T)

, R⁽_n,t^apprx_−t⁾

∅

π₂(T)

, t_∅< t,branch at∅.

IfN_[0,t]<∞then only finitely many iterations are needed andRt(T)=R⁽_n,t^apprx)(T)for all largen. 2

In some cases the evaluation can be written more explicitly. LetF (t )(respectivelyB(t )) be the number of particles that have regenerated (respectively branched) before timet. LetD(t )be the set of labels of particles that have died strictly before timet.

Consider the special case wherer=1 and that all the bilinear formsB_k,l,m coincide with the usual product inC.

Then the evaluation is given, almost surely underP_k, by R_t(T)=C_B^{B(t )}C_R^{F (t )}

α∈D(t )

γ_kα(t−t_α)

α:t∈[sα,tα)

χ_kα(0). (3.3)

In the general case, we can only verify, underP_k, that R_t(T)C^{B(t )}_B C_R^{F (t )}

α∈D(t )

γ_kα(t−t_α)

α:t∈[s_α,t_α)

χ_kα(0), (3.4)

and that equality holds in (3.4) if|B_k,l,m(χ , χ)| = |χ||χ|for allk,l,mandχ , χ. 3.4. The representation formula

Consider an initial conditionχ (0)∈^∞(C^r), and a forcingγ∈L^∞([0, T], ^∞(C^r)). The representation formula for solutions of (2.1), when the expectation exists, is given by

χ_k(t )=^Ek[Rt], k∈Z^d. (3.5)

Theorem 3.5.Suppose that there existsC=C(γ , χ (0), T ) <∞so that E_k|R_t|C for allk∈Z^dand allt∈ [0, T].

Thenχdefined in(3.5)is the uniqueL^∞([0, T], ^∞(C^r))solution of problem(2.1)for the dataγ,χ (0).