Stochastic flows related to Walsh Brownian motion

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 16 (2011), Paper no. 58, pages 1563–1599.

Journal URL

http://www.math.washington.edu/~ejpecp/

Stochastic flows related to Walsh Brownian motion

Hatem Hajri

Département de Mathématiques Université Paris-Sud 11, 91405 Orsay, France

Hatem.Hajri@math.u-psud.fr

Abstract

We define an equation on a simple graph which is an extension of Tanaka’s equation and the skew Brownian motion equation. We then apply the theory of transition kernels developed by Le Jan and Raimond and show that all the solutions can be classified by probability measures.

Key words:Stochastic flows of kernels, Skew Brownian motion, Walsh Brownian motion.

AMS 2000 Subject Classification:Primary 60H25; Secondary: 60J60.

Submitted to EJP on January 17, 2011, final version accepted July 18, 2011.

1 Introduction and main results.

In [10], [11] Le Jan and Raimond have extended the classical theory of stochastic flows to in- clude flows of probability kernels. Using the Wiener chaos decomposition, it was shown that non Lipschitzian stochastic differential equations have a unique Wiener measurable solution given by random kernels. Later, the theory was applied in[12]to the study of Tanaka’s equation:

ϕ_s,t(x) =x+ Z t

s

sgn(ϕ_s,u(x))W(du), s≤t,x ∈R, (1) where sgn(x) =1_{x>0}−1_{x≤0},W_t =W_0,t1_{t>0}−W_t,01_{t≤0} and(W_s,t,s≤t)is a real white noise (see Definition 1.10[11]) on a probability space(Ω,A,P). IfK is a stochastic flow of kernels (see Definition 3 below) andW is a real white noise, then by definition,(K,W)is a solution of Tanaka’s SDE if for alls≤t,x∈R, f ∈C_b²(R)(f isC² onRand f^′,f^′′are bounded)

K_s,tf(x) = f(x) + Z t

s

K_s,u(f^′sgn)(x)W(du) + 1 2

Z t s

K_s,uf^′′(x)du a.s. (2) It has been proved[12]that each solution flow of (2) can be characterized by a probability measure on[0, 1]which entirely determines its law. Define

τ_s(x) =inf{r ≥s:W_s,r=−|x|},s,x ∈R. Then, the uniqueF^W adapted solution (Wiener flow) of (2) is given by

K_s,t^W(x) =δ_x+sgn(x)Ws,t1_{t≤τs(x)}+1 2(δ_W⁺

s,t+δ_−W⁺

s,t)1_{t>τ

s(x)}, W_s,t⁺ :=W_s,t− inf

u∈[s,t]W_s,u. Among solutions of (2), there is only one flow of mappings (see Definition 4 below) which has been already studied in[18].

We now fixα∈]0, 1[and consider the following SDE having a less obvious extension to kernels:

X_t^s,x =x+W_s,t+ (2α−1)˜L_s,t^x , t≥s,x ∈R, (3) where

˜L_s,t^x = lim

ǫ→0⁺

1 2ǫ

Z t s

1_{|X^s,x

u |≤ǫ}du (The symmetric local time).

Equation (3) was introduced in[8]. For a fixed initial condition, it has a pathwise unique solution which is distributed as the skew Brownian motion (SBM) with parameter α (SBM(α)). It was shown in[1]that whenα6= ¹

2, flows associated to (3) are coalescing and a deeper study of (3) was provided later in[3]and[4]. Now, consider the following generalization of (1):

X_s,t(x) =x+ Z t

s

sgn(X_s,u(x))W(du) + (2α−1)˜L_s,t^x (X), s≤t,x ∈R, (4) where

˜L_s,t^x (X) = lim

ǫ→0⁺

1 2ǫ

Z t s

1_{|X

s,u(x)|≤ǫ}du.

Each solution of (4) is distributed as the SBM(α). By Tanaka’s formula for symmetric local time ([15]page 234),

|X_s,t(x)|=|x|+ Z t

s

g

sgn(X_s,u(x))d X_s,u(x) +˜L_s,t^x (X), wheregsgn(x) =1_{x>0}−1_{x<0}. By combining the last identity with (4), we have

|X_s,t(x)|=|x|+W_s,t+˜L_s,t^x (X). (5) The uniqueness of solutions of the Skorokhod equation ([15]page 239) entails that

|X_s,t(x)|=|x|+W_s,t− min

s≤u≤t[(|x|+W_s,u)∧0]. (6)

Clearly (5) and (6) imply thatσ(|X_s,u(x)|;s≤u≤ t) =σ(W_s,u;s≤u≤t)which is strictly smaller thanσ(X_s,u(x);s≤u≤ t)and so X_s,·(x) cannot be a strong solution of (4). For these reasons, we call (4) Tanaka’s SDE related toSBM(α).

From now on, for any metric spaceE,C(E)(respectivelyC_b(E)) will denote the space of all contin- uous (respectively bounded continuous)R-valued functions on E. Let

• C²_b(R^∗) ={f ∈C(R): f is twice derivable onR^∗,f^′,f^′′∈C_b(R^∗), f_|]0,+^′ _∞[,f_|^′′_]0,+∞[ (resp. f_|^′_]−∞,0[,f_|]^′′_−∞,0[)have right(resp. left)limit in 0}.

• D_α={f ∈C_b²(R^∗):αf^′(0+) = (1−α)f^′(0−)}.

For f ∈D_α, we set by convention f^′(0) = f^′(0−),f^′′(0) = f^′′(0−). By Itô-Tanaka’s formula ([13]

page 432) or Freidlin-Sheu formula (see Lemma 2.3[5]or Theorem 3 in Section 2) and Proposition 3 below, both extensions to kernels of (3) and (4) may be defined by

K_s,tf(x) =f(x) + Z t

s

K_s,u(ǫf^′)(x)W(du) + 1 2

Z t s

K_s,uf^′′(x)du, f ∈D_α, (7) whereǫ(x) =1 (respectivelyǫ(x) =sgn(x)) in the first (respectively second) case, but due to the pathwise uniqueness of (3), the unique solution of (7) whenǫ(x) =1, isK_s,t(x) =δ_X^s,x

t (this can be justified by the weak domination relation, see (24)). Our aim now is to define an extention of (7) related to Walsh Brownian motion in general. The latter process was introduced in[17]and will be recalled in the coming section. We begin by defining our graph.

Definition 1. (Graph G)

Fix N ≥1andα₁,· · ·,α_N>0such that XN i=1

α_i=1.

In the sequel G will denote the graph below (Figure 1) consisting of N half lines(D_i)_1≤i≤N emanating from 0. Let ~e_i be a vector of modulus 1 such that D_i = {h~e_i,h > 0} and define for all function

f :G−→Rand i∈[1,N], the mappings :

f_i : R_{+ −→} R h 7−→ f(h~e_i)

Define the following distance on G:

d(h~e_i,h^′~e_j) = (

h+h^′ if i6= j,(h,h^′)∈R²₊,

|h−h^′| if i= j,(h,h^′)∈R²₊. For x∈G, we will use the simplified notation|x|:=d(x, 0).

We equip G with its Borelσ-fieldB(G)and use the notation G^∗=G\ {0}. Now, define

•C_b²(G^∗) ={f ∈C(G):∀i∈[1,N],f_i is twice derivable onR^∗₊,f_i^′,f_i^′′∈C_b(R^∗₊)and both have finite limits at0+}.

•D(α₁,· · ·,α_N) ={f ∈C_b²(G^∗): XN i=1

α_if_i^′(0+) =0}.

For all x ∈G, we define~e(x) =~e_i if x ∈D_i,x 6=0(convention~e(0) =~e_N). For f ∈C_b²(G^∗), x 6=0, let f^′(x) be the derivative of f at x relatively to~e(x) (= f_i^′(|x|) if x ∈ D_i) and f^′′(x) = (f^′)^′(x) (= f_i^′′(|x|)if x ∈ D_i). We use the conventions f^′(0) = f_N^′(0+),f^′′(0) = f_N^′′(0+). Now, associate to each ray D_i a signǫ_i∈ {−1, 1}and then define

ǫ(x) =

(ǫ_i if x∈D_i,x 6=0 ǫ_N if x=0

To simplify, we suppose thatǫ₁=· · ·=ǫ_p=1, ǫ_p+1=· · ·=ǫ_N=−1for some p≤N . Set G⁺= [

1≤i≤p

D_i, G⁻= [

p+1≤i≤N

D_i.Then G=G⁺[ G⁻.

G

O ~ei

(Di,α_i) G⁺

G−

ǫ(x) =−1 ǫ(x) =1

Figure 1: GraphG.

We also putα⁺=1−α⁻:=P_p

i=1α_i.

Remark 1. Our graph can be simply defined as N pieces ofR₊ in which the N origins are identified.

The values of the~e_i will not have any effect in the sequel.

Definition 2. (Equation (E)).

On a probability space(Ω,A,P), let W be a real white noise and K be a stochastic flow of kernels on G (a precise definition will be given in Section 2). We say that(K,W)solves(E)if for all s ≤ t,f ∈ D(α₁,· · ·,α_N),x∈G,

K_s,tf(x) = f(x) + Z t

s

K_s,u(ǫf^′)(x)W(du) +1 2

Z t s

K_s,uf^′′(x)du a.s.

If K=δ_ϕ is a solution of(E), we simply say that(ϕ,W)solves(E).

Remarks 1. (1) If(K,W)solves(E), thenσ(W)⊂σ(K)(see Corollary 2) below. So, one can simply say that K solves(E).

(2) The case N =2,p= 2,ǫ₁ =ǫ₂ =1(Figure 2) corresponds to Tanaka’s SDE related to SBM and includes in particular the usual Tanaka’s SDE [12]. In fact, let (K^R,W) be a solution of (7) with α=α₁,ǫ(y) =sgn(y)and defineψ(y) =|y|(~e₁1_y≥0+~e₂1_y<0),y ∈R. For all x∈G, define K_s,t^G(x) = ψ(K_s,t^R(y))with y =ψ⁻¹(x). Let f ∈ D(α₁,α₂),x ∈ G and g be defined onR by g(z) = f(ψ(z)) (g ∈ D_α1). Since K^R satisfies(7) in (g,ψ⁻¹(x)) (g is the test function and ψ⁻¹(x) is the starting point), it easily comes that K^G satisfies(E)in(f,x). Similarly, if K^Gsolves(E), then K^Rsolves (7).

O

+ +

Figure 2: Tanaka’s SDE.

(3) As in (2), the case N=2,p=1,ǫ₁=1,ǫ₂=−1(Figure 3) corresponds to (3).

O +

−

Figure 3: SBM equation.

In this paper, we classify all solutions of(E) by means of probability measures. We now state the first

Theorem 1. Let W be a real white noise and X^s,x_t be the flow associated to (3) with α=α⁺. Define Z_s,t(x) =X_t^s,ǫ(x)|x|,s≤t,x ∈G and

K_s,t^W(x) = δ_{x+~e(x)ǫ(x)Ws,t}1_{t≤τs,x} +

Xp i=1

α_i

α⁺δ_~ei|Zs,t(x)|1_{Zs,t(x)>0}+ XN i=p+1

α_i

α⁻δ_~ei|Zs,t(x)|1_{{Zs,t(x)≤0}}

1_{t>τs,x},

where τ_s,x = inf{r ≥ s : x +~e(x)ǫ(x)W_s,r = 0}. Then, K^W is the unique Wiener solution of (E).

This means that K^W solves(E) and if K is another Wiener solution of(E), then for all s≤ t,x ∈G, K_s,t^W(x) =K_s,t(x)a.s.

The proof of this theorem follows [10] (see also [14] for more details) with some modifications adapted to our case. We will use Freidlin-Sheu formula for Walsh Brownian motion to check that K^W solves(E). Unicity will be justified by means of the Wiener chaos decomposition (Proposition 8). Besides the Wiener flow, there are also other weak solutions associated to(E) which are fully described by the following

Theorem 2. (1) Define

∆_k=

u= (u₁,· · ·,u_k)∈[0, 1]^k: Xk

i=1

u_i=1 , k≥1.

Supposeα⁺6=¹

2.

(a) Let m⁺and m⁻be two probability measures respectively on∆_pand∆_N−p satisfying : (+)

Z

∆_p

u_im⁺(du) = α_i

α⁺,∀1≤i≤p, (-)

Z

∆_N−p

u_jm⁻(du) =α_j+p

α⁻ , ∀1≤ j≤N−p.

Then, to(m⁺,m⁻)is associated a stochastic flow of kernels K^m+,m− solution of(E).

• To (δ₍^α1

α+,···,_α^α+^p),δ₍^αp+1

α− ,···,^αN

α−))is associated a Wiener solution K^W.

• To ( Xp

i=1

α_i

α⁺δ0,..,0,1,0,..,0, XN i=p+1

α_i

α⁻δ0,..,0,1,0,..,0) is associated a coalescing stochastic flow of map- pingsϕ.

(b) For all stochastic flow of kernels K solution of(E)there exists a unique pair of measures(m⁺,m⁻) satisfying conditions(+)and(−)such that K^{l aw}= K^m+,m−.

(2) Ifα⁺= ¹

2,N>2, there is just one solution of(E)which is a Wiener solution.

Remarks 2. (1) If α⁺ = 1, solutions of (E) are characterized by a unique measure m⁺ satisfying condition(+)instead of a pair(m⁺,m⁻)and a similar remark applies ifα⁻=1.

(2) The caseα⁺= ¹

2,N=2does not appear in the last theorem since it corresponds to d X_t=W(d t).

This paper follows ideas of[12]in a more general context and is organized as follows. In Section 2, we remind basic definitions of stochastic flows and Walsh Brownian motion. In Section 3, we use a

“specific”SBM(α⁺)flow introduced by Burdzy-Kaspi and excursion theory to construct all solutions of(E). Unicity of solutions is proved in Section 4. Section 5 is an appendix devoted to Freidlin-Sheu formula stated in[5]for a general class of diffusion processes defined by means of their generators.

Here we first establish this formula using simple arguments and then deduce the characterization of Walsh Brownian motion by means of its generator (Proposition 3).

2 Stochastic flows and Walsh Brownian motion.

LetP(G)be the space of probability measures on G and(f_n)_n∈Nbe a sequence of functions dense in{f ∈C₀(G),||f||∞≤1}withC₀(G)being the space of continuous functions onGwhich vanish at infinity. We equipP(G) with the distanced(µ,ν) = (P

n2⁻ⁿ(R

f_ndµ−R

f_ndν)²)¹² for allµ andν inP(G). Thus,P(G)is a locally compact separable metric space. Let us recall that a kernelK on G is a measurable mapping fromGintoP(G). We denote byEthe space of all kernels onG and we equipEwith theσ-fieldE generated by the mappingsK7−→µK,µ∈ P(G), withµKthe probability measure defined asµK(A) =R

GK(x,A)µ(d x)for everyµ∈ P(G). Let us recall some fundamental definitions from[11].

2.1 Stochastic flows.

Let(Ω,A,P)be a probability space.

Definition 3. (Stochastic flow of kernels.) A family of(E,E)-valued random variables(K_s,t)_s≤tis called a stochastic flow of kernels on G if,∀s6t the mapping

K_s,t : (G×Ω,B(G)⊗ A) −→ (P(G),B(P(G))) (x,ω) 7−→ K_s,t(x,ω) is measurable and if it satisfies the following properties:

1. ∀s<t<u,x∈G a.s. ∀f ∈C₀(G),K_s,uf(x) =K_s,t(K_t,uf)(x)(flow property).

2. ∀s6t the law of K_s,t only depends on t−s.

3. For all t₁<t₂<· · ·<t_n, the family{K_ti,ti+1, 1≤i≤n−1}is independent.

4. ∀t≥0,x ∈G,f ∈C₀(G), lim

y→xE[(K_0,tf(x)−K_0,tf(y))²] =0.

5. ∀t≥0,f ∈C₀(G), lim

x→+∞E[(K_0,tf(x))²] =0.

6. ∀x∈G,f ∈C₀(G), lim

t→0+E[(K_0,tf(x)−f(x))²] =0.

Definition 4. (Stochastic flow of mappings.) A family(ϕ_s,t)_s≤t of measurable mappings from G into G is called a stochastic flow of mappings on G if K_s,t(x):=δ_ϕs,t(x)is a stochastic flow of kernels on G.

Remark 2. Let K be a stochastic flow of kernels on G and set P_tⁿ=E[K_0,t^⊗n],n≥1. Then,(Pⁿ)_n≥1 is a compatible family of Feller semigroups acting respectively on C₀(Gⁿ)(see Proposition 2.2[11]).

2.2 Walsh Brownian motion.

Recall that for all f ∈C₀(G), f_iis defined onR₊. From now on,we extend this definition onRby setting f_i=0on]− ∞, 0[. We will introduce Walsh Brownian motionW(α₁,· · ·,α_N), by giving its transition density as defined in[2]. OnC₀(G), consider

P_tf(h~e_j) =2 XN

i=1

α_ip_tf_i(−h) +p_tf_j(h)−p_tf_j(−h),h>0, P_tf(0) =2 XN

i=1

α_ip_tf_i(0).

where(p_t)_t>0is the heat kernel of the standard one dimensional Brownian motion. Then(P_t)_t≥0 is a Feller semigroup onC₀(G). A strong Markov process Z with state spaceG and semigroup P_t, and such thatZis càdlàg is by definition the Walsh Brownian motionW(α₁,· · ·,α_N)onG.

2.2.1 Construction by flipping Brownian excursions.

For alln≥0, letD_n= {₂^kn, k∈N_} andD=∪n∈ND_n. For 0≤u< v, definen(u,v) = inf{n∈N: D_n∩]u,v[6=;}and f(u,v) =infD_n(u,v)∩]u,v[.

LetB be a standard Brownian motion defined on(Ω,A,P)and(~γ_r,r ∈D) be a sequence of inde- pendent random variables with the same law

XN i=1

α_iδ_~e

i which is also independent ofB. We define B⁺_t =B_t− min

u∈[0,t]B_u, g_t=sup{r ≤t :B_r⁺=0}, d_t=inf{r≥t :B⁺_r =0},

and finallyZ_t=~γ_rB_t⁺,r= f(g_t,d_t)ifB⁺_t >0, Z_t=0 if B⁺_t =0. Then we have the following Proposition 1. (Z_t,t≥0)is an W(α₁,· · ·,α_N)on G started at 0.

Proof. We use these notations min_s,t= min

u∈[s,t]B_u, ~e_0,t=~e(Z_t),Fs=σ(~e_0,u,B_u; 0≤u≤s).

Fix 0≤s<t and denote byE_s,t={min_0,s=min_0,t}(={g_t≤s}a.s.). Leth:G−→Rbe a bounded measurable function. Then

E[h(Z_t)|Fs] =E[h(Z_t)1_E

s,t|Fs] +E[h(Z_t)1_E^c

s,t|Fs], with

E[h(Z_t)1_Ec

s,t|Fs] = XN

i=1

E[h_i(B⁺_t)1_{g

t>s,~e_0,t=~e_i}|Fs] = XN

i=1

α_iE[h_i(B⁺_t )1_{g

t>s}|Fs].

IfB_s,r=B_r−B_s, then the density of( min

r∈[s,t]B_s,r,B_s,t)with respect to the Lebesgue measure is given by

g(x,y) = 2 p2π(t−s)³

(−2x+y)exp(−(−2x+y)²

2(t−s) )1_{y>x,x<0} ([7]page 28).

Since(B_s,r,r≥s)is independent ofFs, we get

E[h_i(B⁺_t)1_{gt>s}|Fs] = E[h_i(B_s,t− min

r∈[s,t]B_s,r)1_{min

r∈[0,s]Bs,r>min

r∈[s,t]Bs,r}|Fs]

= Z

R

1_{−B⁺

s>x}( Z

R

h_i(y−x)g(x,y)d y)d x

= 2

Z

R₊

h_i(u)p_t−s(B_s⁺,−u)du (u= y−x)

and soE[h(Z_t)1_E^c

s,t|Fs] =2 XN

i=1

α_ip_t−sh_i(−B⁺_s ). On the other hand E[h(Z_t)1_E

s,t|Fs] =E[h(~e_0,s(B_t−min_0,s))1_E

s,t∩(B_t>min0,s)|Fs]

=E[h(~e_0,s(B_t−min_0,s))1_{B

t>min0,s}|Fs]−E[h(~e_0,s(B_t−min_0,s))1_Ec

s,t∩(B_t>min0,s)|Fs].

Obviously on{~e_0,s=~e_k}, we have E[h(~e_0,s(B_t−min_0,s))1_{B

t>min0,s}|Fs] = E[h_k(B_s,t+B⁺_s )1_{B

s,t+B⁺_s>0}|Fs]

= p_t−sh_k(B_s⁺) and

E[h(~e_0,s(B_t−min_0,s))1_Ec

s,t∩(B_t>min0,s)|Fs] =E[h_k(B_s,t+B_s⁺)1_{−B⁺

s>min

r∈[s,t]Bs,r,B_s,t+B⁺_s>0}|Fs]

= Z

R

h_k(y+B_s⁺)1_{y+B⁺

s>0}

‚Z

R

1_{−B⁺

s>x}g(x,y)d x

Œ

d y=p_t−sh_k(−B_s⁺).

As a result,E[h(Z_t)|Fs] =P_t−sh(Z_s)wherePis the semigroup ofW(α₁,· · ·,α_N).

Proposition 2. Let M = (M_n)_n≥0be a Markov chain on G started at0with stochastic matrix Q given by

Q(0,~e_i) =α_i, Q(n~e_i,(n+1)~e_i) =Q(n~e_i,(n−1)~e_i) = 1

2 ∀i∈[1,N], n≥1. (8) Then, for all0≤t₁<· · ·<t_p, we have

(1 2ⁿM_⌊2²ⁿ_t

1⌋,· · ·, 1 2ⁿM_⌊2²ⁿ_t

p⌋)−−−−→_n ^law

→+∞ (Z_t

1,· · ·,Z_t

p), where Z is an W(α₁,· · ·,α_N)started at0.

Proof. LetBbe a standard Brownian motion and define for alln≥1 : T₀ⁿ(B) =T₀ⁿ(|B|) =0 and for k≥0

T_k+1ⁿ (B) = inf{r ≥T_kⁿ(B),|B_r−B_Tⁿ

k|= 1

2ⁿ}, T_k+1ⁿ (|B|) = inf{r ≥T_kⁿ(|B|),||B_r| − |B_Tⁿ

k||= 1

2ⁿ}.

Then, clearly T_kⁿ(B) = T_kⁿ(|B|)and so(T_kⁿ(|B|))_k≥0 ^{l aw}= (T_kⁿ(B))_k≥0. It is known ([6]page 31) that

n→lim+∞Tⁿ

⌊2²ⁿt⌋(B) = t a.s. uniformly on compact sets. Then, the result holds also for Tⁿ

⌊2²ⁿt⌋(|B|).

Now, letZ be theW(α₁,· · ·,α_N)started at 0 constructed in the beginning of this section from the reflected Brownian motionB⁺. LetT_kⁿ=T_kⁿ(B⁺)(defined analogously to T_kⁿ(|B|)) andZ_kⁿ=2ⁿZ_Tⁿ

k. Then obviously(Z_kⁿ,k≥0)^{l aw}= M for alln≥0. Since a.s. t −→Z_t is continuous, it comes that a.s.

∀t ≥0, lim_n→+∞ ¹

2ⁿZⁿ

⌊2²ⁿt⌋=Z_t. 2.2.2 Freidlin-Sheu formula.

Theorem 3. [5]Let(Z_t)_t≥0 be a W(α₁,· · ·,α_N)on G started at z and let X_t=|Z_t|. Then (i) (X_t)_t≥0is a reflecting Brownian motion started at|z|.

(ii) B_t=X_t−˜L_t(X)− |z|is a standard Brownian motion where

˜L_t(X) = lim

ǫ→0⁺

1 2ǫ

Z _t

0

1_{|Xu|≤ǫ}du.

(iii) ∀f ∈C²_b(G^∗),

f(Z_t) = f(z) + Z t

0

f^′(Z_s)d B_s+1 2

Z t 0

f^′′(Z_s)ds+ ( XN

i=1

α_if_i^′(0+))˜L_t(X). (9) Remarks 3. (1) By taking f(z) =|z|and applying Skorokhod lemma, we find the following analogous of (6),

|Z_t|=|z|+B_t− min

s≤u≤t[(|z|+B_u)∧0].

From this observation, when ǫ_i = 1 for all i ∈ [1,N], we call (E), Tanaka’s SDE related to W(α₁,· · ·,α_N).

(2) For N ≥3, the filtration(F_t^Z) has the martingale representation property with respect to B[2], but there is no Brownian motion W such thatF_t^Z =F_t^W [16].

Using this theorem, we obtain the following characterization of W(α₁,· · ·,α_N) by means of its semigroup.

Proposition 3. Let

•D(α₁,· · ·,α_N) ={f ∈C_b²(G^∗): XN i=1

α_if_i^′(0+) =0}.

•Q= (Q_t)_t≥0be a Feller semigroup satisfying Q_tf(x) = f(x) + 1

2 Z t

0

Q_uf^′′(x)du ∀f ∈D(α₁,· · ·,α_N).

Then, Q is the semigroup of W(α₁,· · ·,α_N).

Proof. Denote by P the semigroup ofW(α₁,· · ·,α_N),A^′ andD(A^′)being respectively its generator and its domain onC₀(G). If

D^′(α₁,· · ·,α_N) ={f ∈C₀(G)\

D(α₁,· · ·,α_N),f^′′∈C₀(G)}, (10) then it is enough to prove these statements:

(i)∀t>0, P_t(C₀(G))⊂D^′(α₁,· · ·,α_N).

(ii)D^′(α₁,· · ·,α_N)⊂D(A^′) and A^′f(x) =¹

2f^′′(x)on D^′(α₁,· · ·,α_N).

(iii)D^′(α₁,· · ·,α_N)is dense inC₀(G)for||.||∞.

(iv) IfRandR^′are respectively the resolvents ofQandP, then R_λ=R^′_λ ∀ λ >0 on D^′(α₁,· · ·,α_N).

The proof of (i) is based on elementary calculations using dominated convergence, (ii) comes from (9), (iii) is a consequence of (i) and the Feller property of P (approximate f by P¹

n

f). To prove (iv), letAbe the generator ofQ and fix f ∈ D^′(α₁,· · ·,α_N). Then, R_λf is the unique element of D(A)such that(λI−A)(R_λf) = f. We haveR^′_λf ∈D^′(α₁,· · ·,α_N)by (i),D^′(α₁,· · ·,α_N)⊂D(A)by hypothesis. HenceR^′_λf ∈D(A)and sinceA=A^′on D^′(α₁,· · ·,α_N), we deduce thatR_λf =R^′_λf.

3 Construction of flows associated to ( E).

In this section, we prove(a)of Theorem 2 and show thatK^W given in Theorem 1 solves(E).

3.1 Flow of Burdzy-Kaspi associated to SBM.

3.1.1 Definition.

We are looking for flows associated to the SDE (3). The flow associated toSBM(1)which solves(3) is the reflected Brownian motion above 0 given by

Y_s,t(x) = (x+W_s,t)1_{t≤τ

s,x}+ (W_s,t− inf

u∈[τs,x,t]W_s,u)1_{t>τ

s,x}, where

τ_s,x =inf{r≥s:x+W_s,r=0}. (11)

and a similar expression holds for the SBM(0) which is the reflected Brownian motion below 0.

These flows satisfy all properties of the SBM(α),α ∈]0, 1[ we will mention below such that the

“strong”flow property (Proposition 4) and the strong comparison principle (12). Whenα∈]0, 1[, we follow Burdzy-Kaspi[4]. In the sequel, we will be interested inSBM(α⁺)and so we suppose in this paragraph thatα⁺∈ {/ 0, 1}.

With probability 1, for all rationalssandxsimultaneously, equation(3)has a unique strong solution withα=α⁺. Define

Y_s,t(x) =infX_t^u,y

u,y∈Q

u<s,x<Xu,y s

, L_s,t(x) = lim

ǫ→0⁺

1 2ǫ

Z t s

1_{|Y

s,u(x)|≤ǫ}du.

Then, it is easy to see that a.s.

Y_s,t(x)≤Y_s,t(y) ∀s≤t,x ≤y. (12) This implies that x7−→Y_s,t(x)is increasing and càdlàg for alls≤t a.s.

According to [4] (Proposition 1.1), t 7−→ Y_s,t(x) is Hölder continuous for all s,x a.s. and with probability equal to 1:∀s,x∈R,Y_s,·(x)satisfies (3). We first check thatY is a flow of mappings and we start by the following flow property:

Proposition 4. ∀t≥s a.s.

Y_s,u(x) =Y_t,u(Y_s,t(x)) ∀u≥t,x ∈R.

Proof. It is known, since pathwise uniqueness holds for the SDE (3), that for a fixeds≤t≤u,x ∈R, we haveY_s,u(x) =Y_t,u(Y_s,t(x)) a.s. ([9]page 161). Now, using the regularity of the flow, the result extends clearly as desired.

To conclude thatY is a stochastic flow of mappings, it remains to show the following Lemma 1. ∀t≥s, x∈R, f ∈C₀(R)

y→xlimE[(f(Y_s,t(x))− f(Y_s,t(y)))²] =0.

Proof. We takes=0. For g∈C₀(R²), set

P_t⁽²⁾g(x) =E[g(Y_0,t(x₁),Y_0,t(x₂))], x = (x₁,x₂).

Ifǫ >0, f_ǫ(x,y) =1_{{|x−y|≥ǫ}}, then by Theorem 10 in[13], P_t⁽²⁾f_ǫ(x,y)−−−→_y

→x 0.

For all f ∈C₀(R), we have

E[(f(Y_0,t(x))−f(Y_0,t(y)))²] =P_t⁽²⁾f^⊗2(x,x) +P_t⁽²⁾f^⊗2(y,y)−2P_t⁽²⁾f^⊗2(x,y).

To conclude the lemma, we need only to check that

ylim→xP_t⁽²⁾f(y) =P_t⁽²⁾f(x), ∀x ∈R², f ∈C₀(R²).

Let f = f₁⊗ f₂with f_i ∈C₀(R), x = (x₁,x₂),y = (y₁,y₂)∈R². Then

|P_t⁽²⁾f(y)−P_t⁽²⁾f(x)| ≤M X2 k=1

P_t⁽²⁾(|1⊗f_k−f_k⊗1|)(y_k,x_k),

where M >0 is a constant. For allα >0,∃ǫ >0,|u−v|< ǫ⇒ ∀1≤k≤2 : |f_k(u)−f_k(v)|< α.

As a result

|P_t⁽²⁾f(y)−P_t⁽²⁾f(x)| ≤2Mα+2M X2 k=1

||f_k||_∞P_t⁽²⁾f_ǫ(x_k,y_k),

and we arrive at lim sup_y→x|P_t⁽²⁾f(y) − P_t⁽²⁾f(x)| ≤ 2Mα for all α > 0 which means that lim_y→x P_t⁽²⁾f(y) =P_t⁽²⁾f(x). Now this easily extends by a density argument for all f ∈C₀(R²).

In the coming section, we present some properties related to the coalescence ofY we will require in Section 3.2 to construct solutions of(E).

3.1.2 Coalescence of the Burdzy-Kaspi flow.

In this section, we suppose ¹₂ < α⁺<1. The analysis of the case 0< α⁺< ¹₂ requires an application of symmetry. Define

T_x,y =inf{r≥0, Y_0,r(x) =Y_0,r(y)}, x,y ∈R.

By the fundamental result of[1], T_x,y <∞ a.s. for all x,y ∈R. Due to the local time, coalescence always occurs in 0; Y_0,r(x) =Y_0,r(y) =0 if r =T_x,y. Recall the definition ofτ_s,x from (11). Then T_x,y >sup(τ_0,x,τ_0,y)a.s. ([1]page 203). Set

L_t^x =x+ (2α⁺−1)L_0,t(x), U(x,y) =inf{z≥ y :L_t^x =L_t^y =zfor some t≥0},y≥x. According to[3](Theorem 1.1), there existsλ >0 such that

∀u≥ y >0, P(U(0,y)≤u) = (1− y u)^λ.

Thus for a fixed 0< γ <1, we get lim_y→0+P(U(0,y)≤ y^γ) =lim_y→0+(1−y^1−γ)^λ=1.

From Theorem 1.1[3], we haveU(x,y)−x ^{l aw}= U(0,y−x)for all 0< x< y and so

ylim→x+P(U(x,y)−x ≤(y−x)^γ) =1, ∀x ≥0. (13)

Lemma 2. For all x∈R, we havelim_y→xT_x,y =τ_0,x in probability.

Proof. In this proof we denoteY_0,t(0)simply by Y_t. We first establish the result for x =0. For all t>0, we have

P(t≤T_0,y)≤P(L_0,t(0)≤L_0,T

0,y(0)) =P(L⁰_t ≤U(0,y))

since(2α⁺−1)L_0,T0,y(0) =U(0,y). The right-hand side converges to 0 as y→0+by (13). On the other hand, by the strong Markov property at timeτ_0,y fory <0,

G_t(y):=P(t≤T_0,y) =P(t≤τ_0,y) +E[1_{t>τ

0,y}G_t−τ0,y(Y_τ

y)].

For allε >0, E[1_{t>τ

0,y}G_t−τ0,y(Y_τ

0,y)] =E[1_{t−τ

0,y>ε}G_t−τ0,y(Y_τ

0,y)] +E[1_{0<t−τ

0,y≤ε}G_t−τ0,y(Y_τ

0,y)]

≤E[G_ε(Y_τ0,y)] +P(0<t−τ_0,y ≤ε).

From previous observations, we have Y_τ

0,y > 0 a.s. for all y < 0 and consequently Y_τ

0,y −→ 0+

as y →0−. Since lim_z→0+G_ε(z) =0, by letting y →0− and using dominated convergence, then ε→0, we get lim sup

y→0−

G_t(y) =0 as desired for x =0. Now, the lemma easily holds after remarking that

T_x,y−τ_0,x ^{l aw}= T_0,y−x if 0≤x< y and T_x,y−τ_0,x ^{l aw}= T_0,x−y if x < y≤0.

Fors6t,x∈R, define

g_s,t(x) =sup{u∈[s,t]:Y_s,u(x) =0} (sup(;) =−∞). (14) We use Lemma 2 to prove