Ito's formula for Walsh's Brownian motion and applications

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Statistics and Probability Letters

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Itô’s formula for Walsh’s Brownian motion and applications

Hatem Hajri

^a,∗

, Wajdi Touhami

^b

aMathematics Research Unit, University of Luxembourg, Campus Kirchberg, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Grand-Duchy of Luxembourg, Luxembourg

bFaculté des Sciences de Tunis, Campus Universitaire 2092 - El Manar Tunis, Tunisie

a r t i c l e i n f o

Article history:

Received 15 July 2013

Received in revised form 28 December 2013 Accepted 29 December 2013

Available online 10 January 2014 Keywords:

Walsh’s Brownian motion Itô’s formula

Harmonic functions Stochastic flows

a b s t r a c t

We prove an Itô’s formula for Walsh’s Brownian motion in the plane with angles according to a probability measureµ^on[0,²π[. This extends Freidlin–Sheu formula which corre- sponds to the case whereµhas finite support. We also give some applications.

©2014 Elsevier B.V. All rights reserved.

1. Itô’s formula for Walsh’s Brownian motion

LetE

=

_R²; we will use polar co-ordinates (r

, α

) to denote points inE. We denote byC

(

^E

)

the space of all continuous functions onE. Forf

∈

_C

(

^E

)

, we define

f_α

(

^r

) =

f

(

^r

, α),

^r

>

⁰

, α ∈ [

0

,

²

π [ .

Throughout this paper we fix

µ

a probability measure on

[

0

,

²

π [

, also we define f

(

^r

) =

 2π 0

f

(

^r

, α) µ(

^d

α),

^r

>

⁰

.

Let

(

^Pt⁺

)

t≥0be the semigroup of a reflecting Brownian motion on

[

₀

, ∞[

_{and let}

(

^Pt⁰

)

t≥0be the semigroup of a Brownian motion on

[

0

, ∞[

killed at 0. Then fort

≥

0, defineP_tto act onf

∈

_C0

(

^E

)

as follows:

P_tf

(

⁰

, α) =

P_t⁺f

(

⁰

),

P_tf

(

^r

, α) =

P_t⁺f

(

^r

) +

P_t⁰

(

^fα

−

f

)(

^r

)

forr

>

^{0 and}

α ∈ [

0

,

²

π [

. We recall the following

Theorem 1.1 (Barlow et al., 1989).

(

^Pt

)

t≥0is a Feller semigroup onC0

(

^E

)

, the associated process is by definition the Walsh’s Brownian motion

(

^Zt

)

t≥0(with angles distributed as

µ

^).

We can writeZ_t

= (

^Rt

,

Θt

)

where the radial partR_t

= |

Z_t

|

is a reflected Brownian motion and

(

Θt

)

t≥0is called the angular process ofZ. To well defineΘt, whenZ_t

=

0, we set by conventionΘt

=

0.

∗Corresponding author.

E-mail addresses:[email protected](H. Hajri),[email protected](W. Touhami).

0167-7152/$ – see front matter©2014 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.spl.2013.12.021

Notations:

We denote byE_µ, the state space of Walsh’s Brownian motion, i.e.E_µ

= { (

^r

, α) :

r

≥

0

, α ∈

supp

(µ) }

. We define the tree-metric

ρ

^onR²as follows: forz₁

= (

^r1

, α

1

),

^z2

= (

^r2

, α

2

) ∈

_R²,

ρ(

^z1

,

^z2

) = (

^r1

+

r₂

)

^1{α1̸=α2}

+ |

r₁

−

r₂

|

1{α1=α2}

.

Note that the sample paths of Walsh’s Brownian motion are continuous with respect to the tree-topology, since the process cannot jump from one ray to another one without passing through the origin. These paths are also continuous with respect to the relative topology induced by the Euclidean metric onR².

Now denote byDthe set of all functionsf

:

E_µ

→

_Rsuch that (i) fis continuous for the tree-topology.

(ii) For all

α ∈

supp

(µ)

^,fαisC²on

]

0

, ∞]

andf_α^′

(

⁰

+ ),

^f_α^′′

(

⁰

+ )

exist and are finite.

(iii) For allK

>

^0, sup

0<^r<^K,α∈supp(µ)



|

_fα^′

(

^r

) | + |

_fα^′′

(

^r

) |



< ∞ .

Note that iff

∈

_D, then sup_{α∈supp(µ)}

|

f_α^′

(

⁰

+ ) | < ∞

and in particular

supp(µ)

|

f_α^′

(

⁰

+ ) | µ(

^d

α)

is finite. From now on, we will extend any functiong

:

supp

(µ) →

_Rto

[

0

,

²

π [

by settingg

=

0 outside supp

(µ)

^{. Forf}

∈

_Dandz

= (

^r

, α) ̸=

0, we set f^′

(

^z

) =

f_α^′

(

^r

)

^andf′′

(

^z

) =

f_α^′′

(

^r

)

^.

Now define

B^Z_t

=

R_t

−

R₀

−

L_t

(

^R

)

⁽¹⁾

whereL_t

(

^R

)

stands for the symmetric local time at zero ofR, i.e.L_t

(

^R

) =

lim_ϵ→0 1 2ϵ

t

01{Rs≤ϵ}ds. ThenB^Zis an

(

^Ft^Z

)

t-Brownian motion (Lemma 2.2Barlow et al.(1989)). Our main result is the following

Theorem 1.2. Let

(

^Zt

)

t≥0be a Walsh’s Brownian motion started from z, then for all f

∈

_D, we have f

(

^Zt

) =

f

(

^z

) +

 t

0

1{Zs̸=0}f^′

(

^Zs

)

^dBZs

+

¹ 2

 t

0

1{Zs̸=0}f^′′

(

^Zs

)

^ds

+

 2π 0

f_α^′

(

⁰⁺

) µ(

^d

α)



×

L_t

(

^R

).

When supp

(µ)

is finite this formula is due toFreidlin and Sheu(2000) (see alsoHajri(2011) for another proof based on the skew Brownian motion). Our proof is based on standard approximations and Riemann–Stieltjes integration.

Proof. Note thatH_s

(ω) =

1{Zs(ω)̸=0}f^′

(

^Zs

(ω))

is progressively measurable with respect to the filtration

(

F_t^Z

)

and is locally bounded by the assumption (iii) onf, i.e. a.s.

∀

t

≥

0, sup_s≤t

|

H_s

| < ∞

. Thus the stochastic integral with respect toB^Zis well defined. Again by (iii), the integralt

01{Zs̸=0}f^′′

(

^Zs

)

^dsis well defined.

First, we will consider the casez

=

0. Takef

∈

_D,t

>

^{0 and for}

ε >

^{0, define}

τ

0^ε

=

0 and forn

≥

0

σ

n^ε

=

inf

{

r

≥ τ

n^ε

: |

Z_r

| = ε } , τ

n^ε+1

=

inf

{

r

≥ σ

n^ε

:

Z_r

=

0

} .

Setg

(

^z

) =

f

(

^z

) −

f_α^′

(

⁰

+ ) |

z

|

1{z̸=0}where

α =

arg

(

^z

)

. Note thatg

∈

_Dandg_α^′

(

⁰⁺

) =

0. We will prove our formula first for g. We have

g

(

^Zt

) −

g

(

⁰

) =

lim ε→0

∞



n=0



g

(

^Zσ_n^ε₊₁∧t

) −

g

(

^Zσn^ε∧t

)



=

lim ε→0

A^ε_t

+

D^ε_t where

A^ε_t

=

∞



n=0

g

(

^Zσn^ε+1∧_t

) −

g

(

^Zτn^ε+1∧_t

)

 and

D^ε_t

=

∞



n=0

g

(

^Zτ_n^ε₊₁∧t

) −

g

(

^Zσn^ε∧t

)



=

∞



n=0

g

(

^Zτ_n^ε₊₁∧t

) −

g

(

^Zσn^ε

)

 1{σn^ε≤t}

.

We will first identify lim_ε→0D^ε_t. We need the following

Lemma 1.3. Let h

∈

_D, then for all t

∈ [ σ

n^ε

, τ

n^ε+1

]

we have h

(

^Zt

) =

h

(

^Zσn^ε

) +

 t

σn^ε

1{Zs̸=0}h^′

(

^Zs

)

^dBZs

+

¹ 2

 t

σn^ε

h^′′

(

^Zs

)

^1{Zs̸=0}ds

.

⁽²⁾

Proof. Note thatB^Z_·+σε

n

−

B^Z_σε

n is a Brownian motion independent ofF_σ^Z_nε. Thus by conditioning with respect toF_σ^Z_nε, it will be sufficient to prove that a Walsh’s Brownian motionYstarted from a fixedy

̸=

0 satisfies(2)with

(

^Y

,

^BY

,

⁰

, τ

^′

)

^{in place}

of

(

^Z

,

^BZ

, σ

n^ε

, τ

n^ε+₁

)

^where

τ

^′is the first hitting time of 0 byY. SinceL_t

( |

Y

| ) =

0 for allt

≤ τ

^′, our claim holds by a simple application of Itô’s formula for

|

Y

|

using the functionf_arg(y).

Applying the previous lemma, we get D^ε_t

=

∞



n=0

1{σn^ε≤t}

 t∧τ_n^ε₊₁ σn^ε



g^′

(

^Zs

)

^dBZs

+

¹

2g^′′

(

^Zs

)

^ds



=

∞



n=0

 t∧τ_n^ε₊₁ σn^ε



g^′

(

^Zs

)

^dBZs

+

¹

2g^′′

(

^Zs

)

^ds



=

∞



n=0

 t

0

1_[σε n, τn^ε+1[

(

^s

)



g^′

(

^Zs

)

^dBZs

+

¹

2g^′′

(

^Zs

)

^ds



=

 t

0

∞



n=0

1[σn^ε, τ_n^ε₊₁[

(

^s

)



g^′

(

^Zs

)

^dBZs

+

¹

2g^′′

(

^Zs

)

^ds



where we have used dominated convergence for stochastic integrals (seeRevuz and Yor(1999) page 142) in the last line.

Using dominated convergence again we see that as

ε →

0,D^ε_t converges in probability tot

01{Zs̸=0}

(

^g′

(

^Zs

)

^dBZs

+

¹

2g^′′

(

^Zs

)

^ds

)

^. NowA^ε_t

=

H_t^ε

+

K_t^εwithH_t^ε

=

∞

n=0

g

(

^Zσn^ε+1

) −

g

(

⁰

)

 1_{σε

n+1≤t}and

|

K_t^ε

| ≤ |

g

(

^Zt

) −

g

(

⁰

) |

1{|Zt|≤ε}

→

0 as

ε →

0. Note that

σ

n^εisF^R-measurable,

θ

n

=

arg

(

^Zσn^ε

)

is independent ofF^Rand in particular

σ

n^εand

θ

nare independent. Therefore we obtain

E

[|

H_t^ε

|] ≤

∞



n=0

E

[|

g

(ε, θ

n+1

) −

g

(

⁰

) |

1{σ_n^ε₊₁≤t}

]

≤

∞



n=0

E

[|

g

(ε, θ

n+1

) −

g

(

⁰

) |]

_P

(σ

n^ε+1

≤

t

)

≤

 2π 0

|

_g

(ε, α) −

_g

(

⁰

) |

ε µ(

^d

α) ε

∞



n=0

P

(σ

n^ε+1

≤

t

).

But

ε

∞

n=0P

(σ

n^ε+1

≤

t

) →

_E

[

L_t

(

^R

) ]

as

ε →

0 and for

ε <

^{1, we have}

|

g

(ε, α) −

g

(

⁰

) |

ε ≤

sup

0<r<1,α∈supp(µ)

|

g_α^′

(

^r

) | < ∞

so that by dominated convergence, as

ε →

0,

 2π 0

|

g

(ε, α) −

g

(

⁰

) |

ε µ(

^d

α) →

 2π 0

|

_gα^′

(

⁰⁺

) | µ(

^d

α) =

₀ and finallyH_t^ε

→

0 inL¹. Summarizing, we get

g

(

^Zt

) =

g

(

⁰

) +

 t

0

1{Zs̸=0}g^′

(

^Zs

)

^dBZs

+

¹ 2

 t

0

1{Zs̸=0}g^′′

(

^Zs

)

^ds

.

⁽³⁾

Note that a.s.g

(

^Zt

) =

f

(

^Zt

) −

R_tY_twithY_t

=

f_Θ^′_t

(

⁰

+ )

^1{Zt̸=0}. Set C_t

=

Y_tR_t

−

 t

0

Y_sdR_s

.

We will identifyC_tfollowingProkaj(2009). Put Y_t^ε

=

∞



k=0

Y_t1_[σε k,τk^ε+1[

(

^t

).

ThenY^εis right-continuous and is constant on every interval

[ σ

k^ε

, τ

k^ε+1

[

. SinceRis continuous, thenRis Riemann–Stieltjes integrable with respect toY^εalmost surely and

Y_t^εR_t

−

Y₀^εR₀

−

 t

0

Y_s^εdR_s

=

 t

0

R_sdY_s^ε

.

⁽⁴⁾

As

ε →

0, we haveY_t^ε

→

Y_ta.s. Since

|

Y^ε

| ≤

Mfor someM

>

0, then the dominated convergence for stochastic integrals givest

0Y_s^εdR_s

→

t

0Y_sdR_sin probability. The definition ofY^εentails that

 t

0

R_sdY_s^ε

=

∞



k=0

R_σε kY_σε

k 1_{σε k≤t}

= ε

∞



k=0

Y_σε k1_{σε

k≤t}

.

LetN

(

^t

, ε)

be the number of upcrossings of the interval

[

0

,

^t

]

byR. So

 t

0

R_sdY_s^ε

= ε

N(t,ε)



j=0

f_θ^′_j

(

⁰

+ ).

Since

(

^f_θ^′_j

(

⁰

+ ))

j≥₀is a sequence of independent random variables having the same law

µ

, the Bernoulli law of large numbers yields

 t

0

R_sdY_s^ε

= ε

^N

(

^t

, ε)

N(t,ε)



j=0

f_θ^′

j

(

⁰

+ )

N

(

^t

, ε) →

L_t

(

^R

)

E

[

f_θ^′₀

(

⁰

+ ) ] .

ButE

[

f_θ^′

0

(

⁰

+ ) ] =

2π

0 f_α^′

(

⁰

+ ) µ(

^d

α)

^{and so}t

0R_sdY_s^ε

→ [

2π

0 f_α^′

(

⁰

+ ) µ(

^d

α) ]

L_t

(

^R

)

^as

ε →

0. Letting

ε →

0 in(4), we obtain C_t

=

 2π 0

f_α^′

(

⁰

+ ) µ(

^d

α)

 L_t

(

^R

).

Using the fact that 1{Rs>0}dR_s

=

1{Rs>0}dB^Z_s, it follows that

 t

0

Y_sdR_s

=

 t

0

Y_sdB^Z_s

=

 t

0

1{Zs̸=0}f_Θ^′_s

(

⁰

+ )

^dBZs

.

Now forz

=

0, our formula follows from(3). The casez

̸=

0 holds by discussingt

≤ τ

^andt

> τ

^where

τ

is the first hitting time of 0 byZand using the Itô’s formula satisfied byZ·+τ which is a Walsh’s Brownian motion started from 0.

2. Some applications

2.1. Walsh’s Brownian motion on graphs

The result of this section can be deduced fromHajri and Raimond(inpress) but it has not been announced in that paper.

Our purpose is to give an Itô’s formula for Walsh’s Brownian motion with the more general state space of a graph. This diffusion is defined by means of its infinitesimal generator inFreidlin and Sheu(2000) (see also Chapter 4 inJehring(2009)).

Along an edge it is like a Brownian motion and around a vertex it behaves like a Walsh’s Brownian motion.

LetGbe a metric graph equipped with the shortest path distance, denote byV, the set of its vertices, and by

{

E_i

;

i

∈

I

}

the set of its edges whereIis a finite or countable set. To each edgeE_i, we associate an isometrye_i

:

J_i

→ ¯

E_i, withJ_i

= [

0

,

^Li

]

when L_i

< ∞

andJ_i

= [

0

, ∞ )

^orJi

= ( −∞ ,

⁰

]

whenL_i

= ∞

. WhenL_i

< ∞

, denote

{

g_i

,

^di

} = {

e_i

(

⁰

),

^ei

(

^Li

) }

. WhenL_i

= ∞

, de- note

{

g_i

,

^di

} = {

e_i

(

⁰

), ∞}

whenJ_i

= [

0

, ∞ )

^and

{

g_i

,

^di

} = {∞ ,

^ei

(

⁰

) }

whenJ_i

= ( −∞ ,

⁰

]

. For all

v ∈

V, denoteI_v⁺

= {

i

∈

I

;

g_i

= v }

,I_v⁻

= {

i

∈

I

;

d_i

= v }

andI_v

=

I_v⁺

∪

I_v⁻. We assume that CardI_v

< ∞

for all

v

and that inf_i∈IL_i

>

^{0. To each}

v ∈

V andi

∈

I_v, we associate a parameterp^v_i

∈ [

0

,

¹

]

such that

i∈Ivp^v_i

=

1. Letp

= (

^pi^v

, v ∈

V

,

ⁱ

∈

I_v

)

and denote byD_p^Gthe set of all continuous functionsf

:

G

→

_Rsuch that for alli

∈

I,f

◦

e_iisC²on the interior ofJ_iwith bounded first and second derivatives both extendable by continuity toJ_iand such that for all

v ∈

V



i∈I+ v

p^v_i lim

r→0+

(

^f

◦

e_i

)

^′

(

^r

) =



i∈I− v

p^v_i lim

r→Li−

(

^f

◦

e_i

)

^′

(

^r

)

¹{Li<∞}

+

lim

r→0−

(

^f

◦

e_i

)

^′

(

^r

)

¹{Li=∞}



.

Forf

∈

_Dp^Gandx

=

e_i

(

^r

) ∈

G

\

V, setf^′

(

^x

) = (

^f

◦

e_i

)

^′

(

^r

)

^,f′′

(

^x

) = (

^f

◦

e_i

)

^′′

(

^r

)

and take the following conventions for all

v ∈

V,f^′

(v) =

f^′′

(v) =

0.

LetZbe a Walsh’s Brownian motion onGassociated to the familypand started fromz, i.e. ifE_iis an adjacent edge to

v

^, then from

v

^,Z‘‘jumps’’ toE_iwith probabilityp^v_i. We have the following

Proposition 2.1. There exists a Brownian motion W such that for all f

∈

_Dp^G, we have f

(

^Zt

) =

f

(

^z

) +

 t

0

f^′

(

^Zu

)

^dWu

+

¹ 2

 t

0

f^′′

(

^Zu

)

^du

.

Proof. We takez

= v

a vertex point. We will defineWuntilZhits another vertex

v

^′. The definition ofWshould be clear then using the strong Markov property and the successive hitting times ofVbyZ. Thus we can also assume thatGis a star graph (has only one vertex and eventually an infinite number of edges) which we embed in the complex plane just to

simplify notations. LetG⁺

= ∪

_i∈I+

v E_i

,

^G−

= ∪

_i∈I− v E_iand W_t

=

 t

0

(

¹G+

(

^Zs

) −

1_G−

(

^Zs

))

^dBZs

.

Now our result holds by applyingTheorem 1.2and taking care of the definition of the derivative offin the proposition.

2.2. Harmonic functions on Walsh’s Brownian motion

InJehring(2009), Harmonic functions on Walsh’s Brownian motion are studied. The following lemma (Lemma 3.2 in Jehring(2009)) is the most technical point in proving Theorem 3.2 inJehring(2009). Its proof uses the Excursion theory and non trivial calculations (see Appendix A inJehring(2009)). We will derive it in few lines fromTheorem 1.2.

Lemma 2.2. Let h

:

E

→

_Rbe of the form h

(

^r

, α) =

m_h

(α).

^r

+

h

(

⁰

)

for

(

^r

, α) ∈

E, where the function m_h

: [

0

,

²

π [→

_Ris an integrable function with respect to

µ

and satisfies2π

0 m_h

(α)µ(

^d

α) =

0. Then

{

h

(

^Zt

),

^t

≥

0

}

is a martingale with respect to

(

^FZ

)

tfor every starting point z

∈

E.

Note that whenm_his bounded our formula gives an explicit expression of this martingale:

h

(

^Zt

) =

h

(

^z

) +

 t

0

m_h

(

Θs

)

¹{Zs̸=0}dB^Z_s

.

Proof. ByTheorem 1.2, using the functionh^N

(

^r

, α) = (

^mh

(α) ∧

N

).

^r

+

h

(

⁰

),

^N

≥

0, we see that h^N

(

^Zt

) =

h^N

(

^z

) +

 t

0

(

^mh

(

Θu

) ∧

N

)

^1{Zu̸=0}dB^Z_u

+

 2π 0

(

^mh

(α) ∧

N

)µ(

^d

α) ×

L_t

(

^R

).

Thus for alls

≤

t,

E

[

h^N

(

^Zt

) |

_Fs^Z

] =

h^N

(

^z

) +

 s

0

(

^mh

(

Θu

) ∧

N

)

^1{Zu̸=0}dB^Z_u

+

 2π 0

(

^mh

(α) ∧

N

)µ(

^d

α) ×

E

[

L_t

(

^R

) |

_Fs^Z

]

=

h^N

(

^Zs

) −

 2π 0

(

^mh

(α) ∧

N

)µ(

^d

α) ×

L_s

(

^R

) +

 2π 0

(

^mh

(α) ∧

N

)µ(

^d

α) ×

E

[

L_t

(

^R

) |

_Fs^Z

] .

LettingN

→ ∞

and using dominated convergence (note thatE

[|

m_h

(

Θt

) ||

Z_t

|] < ∞

sincem_h

(

Θt

)

^and

|

Z_t

|

are independent), we getE

[

h

(

^Zt

) |

_Fs^Z

] =

h

(

^Zs

)

^.

2.3. Stochastic flows in the plane

LetZbe a Walsh’s Brownian motion started from 0 and defineW

=

B^Z. Then, we haveR_t

=

W_t

−

min₀≤u≤tW_uby Skorokhod Lemma (seeRevuz and Yor(1999) page 239),

σ (

^R

) = σ (

^W

)

^andΘt is independent ofW for allt

>

^{0. This} situation is close to Tanaka’s equation: ifXsatisfies

X_t

=

 t

0

sgn

(

^Xs

)

^dWs

,

then

σ( |

X

| ) = σ (

^W

)

^{and sgn}

(

^Xt

)

is independent ofWfor allt

>

0. Theorem 2 allows thus to define the analogue of Tanaka’s equation in the plane. This problem has been considered from the stochastic flows view point inHajri(2011) for supp

(µ)

finite.

We say that

(ϕ

s,t

(

^z

))

0≤s≤t,z∈Eis a stochastic flow of mappings (SFM) onEas soon as

ϕ

s,tis measurable with respect to

(

^z

, ω)

^,

ϕ

s,tand

ϕ

0,t−sare equal in law, for any sequence

{[

s_i

,

^ti

] ,

¹

≤

i

≤

n

}

of non-overlapping intervals the mappings

ϕ

si,ti

are independent, and we have the flow property: for alls

≤

t

≤

uandz

∈

E, a.s.

ϕ

s,^u

(

^z

) = ϕ

t,^u

◦ ϕ

s,^t

(

^z

)

^.

Definition 2.3. Let

ϕ

be a SFM andWbe a Brownian motion. We say that

(ϕ,

^W

)

solves Tanaka’s equation with angle

µ

^if for all 0

≤

s

≤

t,z

∈

Eand allf

∈

_Dsuch that2π

0 f_α^′

(

⁰

+ )µ(

^d

α) =

0, a.s.

f

(ϕ

s,t

(

^z

)) =

f

(

^z

) +

 t

s

1{ϕs,^u(z)̸=0}f^′

(ϕ

s,u

(

^z

))

^dWu

+

¹ 2

 t

s

1{ϕs,^u(z)̸=0}f^′′

(ϕ

s,u

(

^z

))

^du

.

⁽⁵⁾ FollowingLe Jan and Raimond(2006), we will construct a SFM

ϕ

solving this equation: by Kolmogorov extension theorem, there exists a probability space

(

Ω

,

^A

,

P

)

on which one can construct a process

(

Θs,t

,

^Ws,t

)

0≤s≤t<∞ taking values in

[

0

,

²

π [×

_Rsuch that (i)–(iv) below are satisfied

(i) W_s,t

:=

_Wt

−

_{Wsfor alls}

≤

_tandWis a Brownian motion.

(ii) For fixeds

<

^t,Θs,tandWare independent and the law ofΘs,tis

µ

^.

(iii) Define min_s,t

=

inf

{

W_u

;

u

∈ [

s

,

^t

]}

. Then for alls

<

^tandu

< v

P

(

Θs,t

=

_Θu,v

|

min_s,t

=

min_u,v

) =

1

.

(iv) For alls

<

^tand

{ (

^si

,

^ti

) ;

1

≤

i

≤

n

}

withs_i

<

^ti, the law ofΘs,tknowing

(

Θsi,ti

)

1≤i≤nandWis given by

µ

on the event min_s,t

̸∈ {

min_si,ti

;

1

≤

i

≤

n

}

and is given by

δ

Θ_si,tion the event

{

min_s,t

=

min_si,ti

}

with 1

≤

i

≤

n.

Note that (i)–(iv) uniquely define the law of

(

Θs1,t1

, . . . ,

Θsn,tn

,

^W

)

^{for alls}i

<

^ti, 1

≤

i

≤

n. This family of laws is consistent by construction.

For 0

≤

s

≤

t,z

∈

E, define

τ

s

(

^z

) =

inf

{

r

≥

s

:

W_s,r

= −|

z

|}

and

ϕ

s,t

(

^z

) =



|

z

| +

W_s,t

,

^arg

(

^z

)



1{t≤τs(z)}

+



W_t

−

min_s,t

,

Θs,t

1{t>τs(z)}

.

FollowingLe Jan and Raimond(2006), one can prove that

ϕ

is a SFM such that

(ϕ,

^W

)

^satisfies(5). It is also possible to extend (5)to stochastic flows of kernels as inLe Jan and Raimond(2006). A stochastic flow of kernels (SFK)K

= (

^Ks,t

;

s

≤

t

)

^{is the} same as a SFM, but the mappings are replaced by kernels, and the flow property being now that for allz

∈

E,s

≤

t

≤

u, a.s.

K_s,u

(

^x

) =

K_s,tK_t,u

(

^x

)

. For example K_s^W_,t

(

^z

) =

E

[ δ

ϕs,t(z)

|

W

] = δ



|z|+Ws,t,^arg(^z)1{t≤τs(z)}

+

 2π 0

δ



Wt−mins,t,α

µ(

^d

α)

^1{t>τs(z)}

is a SFK. By conditioning with respect toWin(5), we see that

(

^KW

,

^W

)

is solution of the following equation.

Definition 2.4. LetKbe a SFK andWbe a Brownian motion. We say that

(

^K

,

^W

)

solves Tanaka’s equation with angle

µ

^if for all 0

≤

s

≤

t,z

∈

Eand allf

∈

_Dsuch that2π

0 f_α^′

(

⁰

+ )µ(

^d

α) =

0, a.s.

K_s,tf

(

^z

) =

f

(

^z

) +

 t

s

K_s,u

(

^f′1R²\{0}

)(

^z

)

^dWu

+

¹ 2

 t

s

K_s,u

(

^f′′1R²\{0}

)(

^z

)

^du

.

⁽⁶⁾

Note that, now we have two solutions to(6):

(δ

ϕ

,

^W

)

^and

(

^KW

,

^W

)

^{. InHajri}(2011), when supp

(µ)

is finite a complete de- scription of the laws of all SFK’sKsolutions of(6)is given under some further assumptions on stochastic flows (see the definitions inLe Jan and Raimond(2004)). Under these assumptions, it should be possible to prove in our setting that

ϕ

is the law-unique SFM solution of(5),K^Wis the unique Wiener (i.e.

σ(

^W

)

-measurable) flow solution of(6)and to give a complete classification of solutions to(6).

Acknowledgements

The research of the first author is supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND).

References

Barlow, M., Pitman, J., Yor, M.,1989. On Walsh’s Brownian motions. Lecture Notes in Math. 1372, 275–293.

Freidlin, M., Sheu, S.,2000. Diffusion processes on graphs: stochastic differential equations, large derivation principle. Probab. Theory Related Fields 116 (2), 181–220.

Hajri, H.,2011. Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 (58), 1563–1599.

Hajri, H., Raimond, O., 2014. Stochastic flows on metric graphs. Electron. J. Probab.http://arxiv.org/abs/1305.0839(in press).

Jehring, K.E., 2009. Harmonic functions on Walsh’s Brownian motion, Ph.D. dissertation, Univ. of California Saint Diego.

Le Jan, Y., Raimond, O.,2004. Flows, coalescence and noise. Ann. Probab. 32 (2), 1247–1315.

Le Jan, Y., Raimond, O.,2006. Flows associated to Tanak’s SDE. ALEA Lat. Am. J. Probab. Math. Stat. 1, 21–34.

Prokaj, V.,2009. Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79, 534–536.

Revuz, D., Yor, M.,1999. Continuous Martingales and Brownian Motion. Springer Verlag, Berlin.