Williams decomposition

We first recall Williams decomposition for the Brownian snake (see [132] for Brownian excursions, [122] for Brownian snake or [1] for general homogeneous branching mechanism without spatial motion).

Under the excursion measures N⁰_x, ˜N_x andN_x, recall thatH_max= sup_[0,σ]H_s. Because of the continuity of H, we can define Tmax = inf{s > 0, Hs = Hmax}. Notice the properties of the Brownian excursions implies that a.e. H_s=H_max only ifs=T_max. We setv_t⁰(x) =N⁰x[H_max> t]

and recall this function does not depend on x. Thus, we shall write v⁰_t for v_t⁰(x). Standard computations give:

v⁰_t = β0

e^β0t−1·

The next result is a straightforward adaptation from Theorem 3.3 of [1] and gives the distribution of (H_max, W_Tmax, R^g_T

max, R^d_Tmax) underN⁰_x.

Proposition 2.4.7 (Williams decomposition underN⁰_x). We have:

(i) The distribution of Hmax underN⁰_x is characterized by:N⁰_x[Hmax> h] =v_h⁰. (ii) Conditionally on {H_max=h0}, WTmax under N⁰_x is distributed as Y_[0,h0) underP˜_x.

(iii) Conditionally on {H_max =h0} and WTmax, R^g_Tmax and R_T^d_max are under N⁰_x independent Poisson point measures on R⁺×Ω¯ with intensity:

1_[0,h

0)(s)ds1_{Hmax(W)<h

0−s} N⁰_W

Tmax(s)[dW].

In other words, for any nonnegative measurable function F, we have N⁰_x

F(H_max, W_Tmax, R_T^g

max, R^d_Tmax)

=− Z ∞

0

∂_hv_h⁰ dhE˜_x

F(h, Y_[0,h),Rˆ^W,(h),g,Rˆ^W,(h),d)

, where under E˜_x and conditionally on Y_[0,h), Rˆ^W,(h),g and Rˆ^W,(h),d are two independent Poisson point measures with intensity νˆ^W,(h)(ds, dW) =1_[0,h)(s)ds1_{{Hmax(W)<h−s}} N⁰_Y

s[dW].

Notice that items (ii) and (iii) in the previous Proposition implies the existence of a measurable family (N^0,(h)_x , h >0) of probabilities on ( ¯Ω,G) such that¯ N^0,(h)_x is the distribution of W (more precisely of (W_Tmax, R^g_T

max, R^d_Tmax)) under N⁰_x conditionally on {H_max=h}.

Remark 2.4.8. In Klebaner & al [74], the Esty time reversal “is obtained by conditioning a [discrete time] Galton Watson process in negative time upon entering state 0 (extinction) at time 0 when starting at state 1 at time −n and letting ntend to infinity”. The authors then observe that in the linear fractional case (modified geometric offspring distribution) the Esty time reversal has the law of the same Galton Watson process conditioned on non-extinction.

Notice that in our continuous setting, the process (H_s,0≤s≤T_max) is under N^0,(h)x a Bessel process up to its first hitting time of h, and thus is reversible: (Hs,0≤s≤Tmax) underN^0,(h)_x is distributed as (h−HTmax−s,0≤s≤Tmax) underN^0,(h)_x . It is also well known (see Corollary 3.1.6 of [37]) that (Hσ−s,0≤s≤σ−T_max) underN^0,(h)_x is distributed as (Hs,0≤s≤T_max) under N^0,(h)_x . We deduce from these two points that (Z_s(1),0≤s≤h) under N^0,(h)_x is distributed as (Zh−s(1),0≤s≤h) underN^0,(h)_x . This result, which holds at fixed h, gives a prelimiting version in continuous time of the Esty time reversal. Passing to the limit as h→ ∞, see Section 2.5.2, we then get the equivalent of the Esty time reversal in a continuous setting.

Before stating Williams decomposition, Theorem 2.4.12, let us prove some properties for the functions v_t(x) = Nx[H_max > t] = Nx[Z_t 6= 0] and ˜v_t(x) = ˜Nx[H_max > t] which will play a significant rôle in the next Section. Recall (2.17) states that

αvt= ˜vt.

Notice also that (2.18) implies that q is bounded from above by (β₀+kβ˜k_∞)/2.

Lemma 2.4.9. Assume (H1)-(H3). We have:

q(x) +v_t⁰≥v˜t(x)≥v⁰_t. (2.46)

2.4 A Williams decomposition 65

where the function Σ defined by:

Σt(x) = 2(v_t⁰+q(x)−˜vt(x)) =∂λψ⁰(v⁰_t)−∂λψ(x,˜ ˜vt(x)) (2.48) satisfies:

0≤Σt(x)≤2q(x)≤β0+kβ˜k_∞. (2.49) Proof. We deduce from item (iii) of Proposition 2.3.7 that, asϕ≥0 (see Lemma 2.3.8),

v˜t(x) = ˜Nx[Zt6= 0] =N⁰_x

where we used (2.22) for the third equality. This proves (2.46).

Using Williams decomposition underN⁰_x, we get:

v˜t(x) =− Using again Williams decomposition under N⁰_x, we have

N^0,(r)_x

We have thanks to item (iii) from Proposition 2.3.7:

This implies (2.47). Notice that, thanks to (2.46),Σ is nonnegative and bounded from above by

2q. ut

Fixh >0. We define the probability measures P^(h) absolutely continuous with respect to P and P on˜ D_h with Radon-Nikodym derivative:

dP^(h)_x|D

Notice this Radon-Nikodym derivative is 1 if the branching mechanism ψ is homogeneous. We deduce from (2.47) and (2.48) that:

dP^(h)_{x |D} In the next Lemma, we give an intrinsic representation of the Radon-Nikodym derivatives (2.52) and (2.53), which does not involve β₀ orv⁰. are nonnegative bounded D_t-martingales respectively under P_x and P˜_x. Furthermore, we have for 0≤t < h:

2.4 A Williams decomposition 67

Notice the limit M_h^(h) of M^(h) and the limit ˜M_h^(h) of ˜M^(h) are respectively given by the right-handside of (2.53) and (2.52).

Remark 2.4.11. Comparing (2.10) and (2.54), we have that P^(h)_x = P^g_x withg(t, x) =∂_hvh−t(x), ifg satisfies the assumptions of Remark 2.3.3.

Proof. First of all, the process ˜M^(h) is clearly D_t-adapted. Using (2.47), we get:

E˜_y

In the homogeneous setting,v⁰ simply solves the ordinary differential equation:

∂hv⁰_h=−ψ⁰(v_h⁰).

Therefore, ˜M^(h) is a D_t-martingale under ˜P_x and the second part of (2.54) is a consequence of (2.52). Then, use (2.14) to get that M^(h) is a D_t-martingale under P_x and the first part of

(2.54). ut

We now give the Williams’ decomposition: the distribution of (H_max, W_Tmax, R_T^g

max, R^d_Tmax) under N_x or equivalently under ˜N_x/α(x). Recall the distribution P^(h)x defined in (2.52) or (2.53).

Theorem 2.4.12 (Williams’ decomposition under N_x). Assume (H1)-(H3). We have:

(i) The distribution of Hmax underN_x is characterized by:N_x[Hmax> h] =vh(x).

(ii) Conditionally on {H_max =h0}, the law of WTmax under N_x is distributed as Y_[0,h0) under P^(h_x ⁰⁾.

(iii) Conditionally on {H_max =h0} and WTmax, R^g_Tmax and R_T^d_max are under N_x independent Poisson point measures on R⁺×Ω¯ with intensity:

1_[0,h0)(s)ds1_{Hmax(W0)<h0−s}α(W_Tmax(s))N_W

Tmax(s)[dW⁰].

In other words, for any nonnegative measurable function F, we have N_x where under E^(h)_x and conditionally onY_[0,h), R^W,(h),g andR^W,(h),d are two independent Poisson point measures with intensity:

ν^W,(h)(ds, dW) =1_[0,h)(s)ds1_{{Hmax(W)<h−s}}α(Y_s)N_Ys[dW]. (2.56) Notice that items (ii) and (iii) in the previous Proposition imply the existence of a measurable familly (N^(h)_x , h >0) of probabilities on ( ¯Ω,G) such that¯ N^(h)_x is the distribution ofW (more precisely of (W_Tmax, R^g_T

max, R^d_Tmax)) under N_x conditionally on {H_max=h}.

Proof. We keep notations introduced in Proposition 2.4.7 and Theorem 2.4.12. We have:

N˜_x

where the first equality comes from (H1) and item (iii) of Proposition 2.3.7; we set f(s, W) = R+∞

0 Zr(W)(ϕ) for the second equality; we use Proposition 2.4.7 for the third equality; we use Lemma 2.4.3 for the fourth with R^W,(h),g andR^W,(h),d which under ˜E^(h)_x and conditionally on Y_[0,h) are two independent Poisson point measures with intensity ν^W,(h); we use (2.51) for the fifth, definition (2.52) of E^(h)_x for the sixth, and (2.47) for the seventh. Then use (2.33) and (2.17)

to conclude. ut

2.4 A Williams decomposition 69

The definition ofN^(h)_x gives in turn sense to the conditional lawN^(h)x =Nx(.|H_max=h) of the (L, β, α) superprocess conditioned to die at time h, for allh >0. The next Corollary is then a

straightforward consequence of Theorem 2.4.12.

Corollary 2.4.13. Assume (H1)-(H3). Let h >0. Letx∈E and Y_[0,h) be distributed according to P^(h)_x . Consider the Poisson point measure N =^P_j∈Jδ_(sj,Z^j₎ on [0, h)×Ω with intensity:

is distributed according to N^(h)x .

We now give the superprocess counterpart of Theorem 2.4.12.

Corollary 2.4.14 (Williams’ decomposition under Pν). Assume (H1)-(H3).

(i) The distribution of Hmax underPν is:Pν(Hmax≤h) = e^−ν(vh).

(ii) Conditionally on{H_max=h₀}, the distribution ofZ underPν is that of the sum ofZ⁰+Z^(h0), where:

(iii) Z^(h0) has distribution:

∂hvh0(x)

ν(∂_hv_h0)ν(dx)N^(h_x0⁰⁾.

(iv) Z⁰ is independent ofZ^(h0) and has distribution Pν(.|H_max < h₀).

Then the measure-valued processZ⁰+Z^(h0) has distribution Pν.

In particular the distribution of Z⁰+Z^(h0) conditionally on h₀ (which is given by (ii)-(iv) from Corollary 2.4.14) is a regular version of the distribution of the (L, β, α) superprocess conditioned to die at a fixed time h0, which we shall writeP^(hν ⁰⁾.

Proof. Let µ be a finite measure on R⁺ and f a nonnegative measurable function defined on R⁺×E. For a measure-valued processA= (At, t≥0) onE, we setA(f µ) =^Rf(t, x)At(dx)µ(dt).

We also write f_s(t, x) =f(s+t, x).

Let Z⁰ andZ^(h0) be defined as in Corollary 2.4.14. In order to characterized the distribution of the process Z⁰+Z^(h0), we shall compute

A=E[e^{−Z0(f µ)−Z(h0)(f µ)}].

We shall use notations from Corollary 2.4.13. We have:

A=−

where we used the definition ofZ⁰ andN for the first equality, and the equalityPν(Hmax< h) =

where we used the definition of G and g for the first and third equalities, Theorem 2.4.12 for the second equality, the master formula for Poisson point measure ^P_i∈Iδ_Zⁱ with intensity ν(dx)Nx[dZ] for the fourth equality (and the obvious notation H_maxⁱ = inf{t≥0;Z_tⁱ = 0}) and Theorem 2.2.3 for the last equality. Thus we get:

E[e^{−Z0(f µ)−Z(h0)(f µ)}] =Eν

he^{−Z(f µ)i}.

This readily implies that the processZ⁰+Z^(h0) is distributed asZ underPν. ut