Martin boundary of a space-time Brownian motion with drift killed at the boundary of a moving cone
Sandro Franceschi


An obstruction theory for strictly commutative algebras in positive characteristic
    Oisín Flynn-Connolly
    
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We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+γ t) killed at the moving boundary of the cone {(t,x): 0⩽ x ⩽ t }. This article determines the parabolic Martin boundary and all harmonic functions associated with this process. To that end, the asymptotics of Green's functions are determined along all directions. We also find the exit probabilities at the edges, the probability of remaining in the cone forever and the laws of the exit point and exit time. From this, we derive an explicit formula for the transition kernel of the process.
These results arise from two different methods initially introduced to study random walks. An analytical approach, developed in the 1970s by Malyshev and based on the steepest descent method on a Riemann surface, is used to determine the asymptotics of the Green's functions. A recursive compensation approach, inspired by the method developed in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic functions.





§ INTRODUCTION




  
Main process and overview of the results


Let W(t) be a standard Brownian motion, γ∈ (0,1) a drift, (t_0,y_0) a starting point such that 0<t_0<y_0.
We define the space-time Brownian motion by

    B(t):=(t_0+t,y_0+W(t)+γ t).

Let the cone

    C:= { (t',y): 0<y<t' }

which defines a two-sided moving boundary,
and T the first exit time of the cone

    T:=inf{t⩾ 0 : B(t)∉ C } .

We also define the exit times on each edge of the cone

    T_1:=inf{ t⩾ 0 : B(t)=(t_0+t,t_0+t) ,  t>0 }  and  
    T_2:=inf{ t⩾ 0 : B(t)=(t_0+t,0) ,  t>0 }

and we have T=T_1 ∧ T_2.

The purpose of the present work is to determine the parabolic Martin boundary associated with the space-time Brownian motion killed at the boundary of the cone C (Theorem <ref>). This result derives from the asymptotics of Green's functions (Theorem <ref>).
An explicit expression is given for all the associated harmonic functions (Theorem <ref>), that is the positive functions u satisfying 

    (∂^2_y +γ∂_y +∂_t)u(t,y)=0
       for all  (t,y)∈ C,
    
    
    u(t,y)=0    for all  (t,y)∈∂ C.

We find the persistence probability ℙ^α(T=∞) (Proposition <ref>) and the probabilities of exit on an edge ℙ^α(T_1<T_2) and ℙ^α(T_2<T_1) (Proposition <ref>) for the process conditioned to drift in the direction α via Doob's h-transform.
We also compute the laws of T, T_1, T_2 and of the law of the exit point of the process B(T) when T<∞ (Theorem <ref>). Finally, this gives a new original approach to obtain the transition kernel of the process killed at time T (Corollary <ref>), which we note


    p^k,C_(t_0,y_0)(t,y) dy :=ℙ_(t_0,y_0)(B(t)=(t_0+t,dy), T>t).




















  
Proof strategy and structure of the article
Section <ref> sets out some fundamental analytical preliminaries for our study. For practical reasons we first transform via a simple linear application, the space-time Brownian motion in the cone C into a degenerate Brownian motion in the quadrant ℝ_+^2.
Then, we find a kernel functional equation (Proposition <ref>) connecting the Laplace transform of the Green's functions of this new process and the Laplace transform of the exit densities on the axes.
Section <ref> is devoted to the asymptotics of Green's functions.
In this section, the key element of the proofs is the application of the steepest descent method on a Riemann surface generated by the kernel. 

Section <ref>

gives a probabilist interpretation of all the harmonic functions in the cone in terms of the persistence probabilities of some conditioned processes via Doob's h-transform. Then, the full Martin boundary is determined.

Section <ref> 
is based on a recursive compensation approach used to determine explicit expressions for the harmonic functions. The minimal Martin boundary is shown to be homeomorphic to a portion of a certain parabola defined by the kernel. Persistence and exit probabilities are computed.


Finally, Section <ref> is dedicated to the inversion of the Laplace transforms to obtain the law of the exit point and the transition kernel of the process.



  
Related literature

The literature on Brownian motion and moving boundaries is very rich, from the 1960s to the present day. The study of the crossing probability of the boundary is at the heart of many of these articles. The boundary can be one-sided or two-sided, depending on the problems considered. We cannot claim to be exhaustive here and will simply cite a few articles related to our problem. In a seminal paper published in 1960 <cit.>, Anderson considers a Brownian motion between two non-parallel straight lines and determines the probability of crossing one of the two straight lines before the other for a finite or infinite time horizon.
In 1964 <cit.>, see <cit.> for an English version, Skorokhod founds
the probability density of the escape location of a Lévy process that does not cross 

two parallel straight lines, which provides the law of the process killed at the boundary.
In 1967 <cit.> Sheep considered for the first time in the continuous setting a square-root boundary. This type of boundary will subsequently be the subject of numerous studies <cit.>. In the 1970s until recently Novikov solved many interesting related issues <cit.>. 

There are also links between moving boundary results and Brownian motion in Weil chambers and reflection groups. For example, Biane, Bougerol, and O’Connell <cit.> studied the probability that a Brownian motion with drift stays forever in a Weyl chamber, also called the persistence probability. Recently, Defosseux <cit.> studied space-time Brownian motion in an affine Weyl chamber. Some of the results of this last paper are found again in the present article using other methods.

Since the seminal work of Martin <cit.>, many books <cit.> and articles have studied Brownian motion and random walks in cones, the asymptotics of their Green's functions, their Martin boundary and the associated harmonic functions. In the discrete setting, examples include
the work of Malyshev, Kourkova et al. on the Martin boundary of random walks in a quarter plane
<cit.> using a saddle point method on a Riemann surface generated by the kernel of a functional equation. This powerful technique used in the present paper has recently been developed with success in the continuous setting for reflected Brownian motion in wedges or half-planes <cit.>.

In the higher dimensional case, the famous article of Ney and Spitzer
<cit.> computes the asymptotics of the Green's function of random walks with drift in ℤ^d and shows that the Martin boundary is homeomorphic to the unit sphere.
Many other interesting articles by Ignatiouk et al. deals with the Martin boundary of random walks in half-spaces and orthants <cit.>. Denisov and Wachtel study the tail asymptotics for the exit time of a multidimensional random walk in a cone <cit.>. Duraj et al. determine the asymptotic of the Green function for random walks without drift confined to multidimensional convex cones <cit.>.
Garbit and Raschel study the survival probability of multidimensional random walks in pyramids <cit.>.
We also mention the article of Hoang et al. on the construction of harmonic functions in wedges using Boundary value problems <cit.>.

In the continuous setting, several papers study Brownian motion in cones in higher dimensions and we mention a few of them here. De Blassie determines the distributions of first exit times of Brownian motion in cones and computes asymptotics <cit.>. Bañuelos and Smits study the asymptotic behavior of Brownian motion in cones and express their results with infinite series involving its transition density <cit.>. Garbit and Raschel study the asymptotics of the tail distribution of the first exit time of Brownian motion with drift for a large class of cones
<cit.>.


Finally, we mention the compensation approach developed by Adan, Wessels and Zijm <cit.> which is an inspiration for this paper. The recent article of Hoang et al. <cit.> uses this method to compute discrete harmonic functions in the quadrant. The paper of Ichiba et al. <cit.> also uses this approach to compute the invariant measure of a degenerate competing three-particle system.












§ ANALYTICAL PRELIMINARIES





  
Killed degenerate Brownian motion 
in the quadrant
A simple linear transformation maps the space-time Brownian motion B(t) defined in (<ref>) in the cone C onto a degenerate Brownian motion with drift Z(t) in the quadrant ℝ_+^2.

To that purpose, we define the linear transform ℓ by

    ℓ(t',y):=(t'-y,y) 
      and  ℓ^-1(x,y):=(x+y,y).

Thanks to ℓ we perform the change of variable

    t'=t_0+t=x+y .

We denote B(t)=(t_0+t,y_0+W(t)+γ t)=(t',Y(t)) the coordinates of the space-time Brownian motion. We define the drifted degenerate Brownian motion by 

    Z(t)=(X(t),Y(t))    :=ℓ (B(t))
    
        = (t_0-y_0 - W(t)+(1-γ )t , y_0+W(t)+γ t)
    
        =z_0+𝐖(t)+μ t

where 𝐖(t):=(-W(t),W(t)) is a degenerate Brownian motion in ℝ^2 of covariance matrix 
[  1 -1; -1  1 ]
and we denote μ:=(1-γ,γ)∈ℝ_+^2 the positive drift and z_0=(x_0,y_0):=(t_0-y_0,y_0)∈ℝ_+^2 the starting point.

We have ℓ(C)=(ℝ_+^*)^2 where C is defined in (<ref>). The stopping time T defined in (<ref>) is equal to

    T=inf{ t⩾ 0 : Z(t) ∉ (ℝ_+^*)^2 }.

It is the first exit time of the quadrant for the process Z(t). The stopping times on each of the boundaries defined in (<ref>) are equal to

    T_1=inf{ t⩾ 0 : Z(t) ∈{0}×ℝ_+ }  and  
    T_2=inf{ t⩾ 0 : Z(t) ∈ℝ_+×{0}}

and we still have T=T_1∧ T_2.









  
Green's functions and exit densities

For a set S⊂ (ℝ_+^*)^2 we define the Green's measure of the process killed at the boundary by

    G_z_0(S):=∫_0^∞ℙ(Z(t)∈ S ,t<T)dt.

Let us remark that for S⊂ (ℝ_+^*)^2 we have ℙ(Z(t)∈ S ,t<T)=ℙ(Z(t∧ T) ∈ S).
We assume that this measure has a density denoted

    g_z_0(z) dz :=G_z_0(dz).

Let us recall that in (<ref>) we noted p^k,C_(t_0,y_0)(t,y) the transition kernel of the space time Brownian motion killed in the cone C. It is important to remark that, up to the following change of variables, the Green's function of the process Z killed at the boundary of the quadrant is equal to the transition kernel of B the space time Brownian motion killed at the boundary of C :

    p^k,C_(t_0,y_0)(t,y)=g_z_0(z)
       where  z_0=(t_0-y_0,y_0)  and  z=(t_0+t-y,y).

One may consult the famous Doob's book  <cit.> which explains how Brownian Motion transition kernel may be seen as Green functions. 

The exit measures on the boundaries are defined for S⊂ℝ_+^2 by

    A_1(S)    :=ℙ(Z(T_1)∈ S , T_1<T_2)=𝔼(1_Z(T_1)∈ S1_T_1<T_2),
    
    
    A_2(S)    :=ℙ(Z(T_2)∈ S , T_2<T_1)
    =𝔼(1_Z(T_2)∈ S1_T_2<T_1),

and the support of the measure A_1 (resp. A_2) lies on {0}×ℝ_+ (resp. ℝ_+×{0}).
We denote their densities f_1 and f_2 defined as follows

    f_1(y) dy δ_0(dx) 
    :=A_1(dx,dy),
      
    f_2(x) dx δ_0(dy) 
    :=A_2(dx,dy),

where δ_0 is the Dirac measure in 0.
One may notice that (t_0+T)1_T<∞=(X(T)+Y(T))1_T<∞= Y(T_1) 1_T_1<T_2 + X(T_2) 1_T_2<T_1 and
we deduce that the cumulative distribution function of T is equal to F_T(t):=ℙ(T<t)=A_1({0}× [0,t_0+t])+A_2( [0,t_0+t]×{0}) and then f_T the density of T is equal to 

    f_T(t)=f_1(t_0+t)+f_2(t_0+t).

Note that to lighten the notation we have not noted the dependence at z_0 for A_1, A_2, f_1, f_2 and f.





  
Functional equation
Let us define the Laplace transform of the Green's function for (p,q)∈ℂ^2 by

    L(p,q):=∬_ℝ_+^2 g_z_0(x,y)e^px+qydxdy=𝔼(∫_0^∞ e^(p,q)· Z_t1_t<Tdt ) .

Remembering that Z=(X,Y) the Laplace transform of the exit densities are defined by

    L_1(q):=∫_0^∞ f_1(y) e^q ydy=
    𝔼(e^q Y(T_1)1_T_1<T_2) ,
      
    L_2(p):=∫_0^∞ f_2(x) e^p xdx
    =𝔼(e^p X(T_2)1_T_2<T_1) .

Once again, we omit to note the dependence at z_0 for L, L_1 and L_2. One may remark that L(0,0)=𝔼[T]=∞ since ℙ(T=∞)>0 and

    L_1(0)=  ℙ(T_1<T_2) ,
      
    L_2(0)= ℙ(T_2<T_1),
      
    L_1(0)+L_2(0)=ℙ(T<∞).


For (p,q)∈ℂ^2 such that p <0 and q<0 the Laplace transforms converge and we have

    K(p,q)L(p,q)=L_1(q)+ L_2(p)-e^px_0+qy_0

where the kernel K is defined by

    K(p,q):=1/2(p-q)^2+(1-γ) p+γ q.




We denote 

    𝒢:=1/2(∂/∂ x-∂/∂ y)^2+(1-γ) ∂/∂ x+γ∂/∂ y
 
the infinitesimal generator of Z. We apply Dynkin's formula, for f twice differentiable we obtain

    𝔼 f(Z(t∧ T))=f(z_0)+ 𝔼∫_0^t ∧ T𝒢f(Z(s))ds

which can be rewritten

    𝔼 (f(Z(t∧ T_1))1_T_1<T_2)+𝔼 (f(Z(t∧ T_2))1_T_2<T_1)=f(z_0)+ 𝔼∫_0^t𝒢f(Z(s)) 1_s<Tds .

By taking f(x,y)=e^px+qy we obtain 

    𝔼(e^q Y(t∧ T_1)1_T_1<T_2)+𝔼(e^p X(t∧ T_2)1_T_2<T_1)=e^px_0+qy_0+ K(p,q) 𝔼(∫_0^t e^(p,q)· Z_s1_s<Tds ) .

The two expectations of the left-hand side are bounded by 1 since p <0 and q<0 and then the expectation of the right-hand side is also finite.
By making t tend towards infinity 





and by definition of L, L_1 and L_2 we obtain the functional equation (<ref>).




  
Study of the kernel and analytic continuation

The kernel K(p,q) defined in (<ref>)
leads us to introduce four algebraic functions P^+, P^-, Q^+ and Q^- satisfying

    K(P^±(q),q)=K(p,Q^±(p))=0,

analytic on ℂ∖ [(1-γ)^2/2,∞) for P^±, on ℂ∖ [γ^2/2,∞) for Q^±, and defined by 

    P^±(q):=γ-1+q±√((1-γ)^2-2q)  and  
    Q^±(p):=-γ+p±√(γ^2-2p).

Thanks to the functional equation (<ref>) we can now continue meromorphically L_1 to the domain {q∈ℂ: q < (1-γ)^2 /2} by the formula

    L_1(q)= e^P^-(q) x_0 +q y_0 - L_2(P^-(q)).

We just have to verify that P^-(q) <0 when q < (1-γ)^2 /2 and to remember that L_2 is analytic in the complex half-plane with negative real part.
In the same way, we extend L_2 analytically to the domain {p∈ℂ: p < γ^2 /2}.




  
PDE and derivative of the Green's functions

Let z=(x,y)∈ℝ^2.
Let us recall that we defined in (<ref>)

the infinitesimal generator 𝒢. We now define its dual 𝒢^* by

    𝒢^*:=1/2(∂/∂ x-∂/∂ y)^2-(1-γ) ∂/∂ x-γ∂/∂ y .

The Green's function g_z_0 satisfies the following classical parabolic partial differential equation <cit.>, 
denoting δ_z_0 the Dirac measure in z_0∈ℝ^2 we have

    𝒢^* g_z_0=-δ_z_0   on  (ℝ_+^*)^2,
    
    
    g_z_0=0    on ℝ_+×{0}∪{0}×ℝ_+.




We have the following links between the Green's density and the exit densities:

    f_1(y)= 1/2∂ g_z_0/∂ x (0,y) ,
      
    f_2(x)=1/2∂ g_z_0/∂ y (x,0).





A weak formulation of the partial differential equation (<ref>) applied to an exponential test function leads to a functional equation linking the Laplace transforms of the Green's function and their derivatives. 
To do this, it is enough to integrate the partial differential equation (<ref>) and to make successive integration by parts on the integral 

    ∬_ℝ_+^2𝒢^*g(z_0,z)e^z·(p,q)dz = -e^z_0· (p,q).

Then, defining L_1 (q):=1/2∫_0^∞∂ g/∂ x (0,y) e^q ydy and symmetrically L_2 (p) we obtain

    K(p,q)L(p,q)=L_1(q)+ L_2(p)-e^px_0+qy_0.

Comparing this equation with the functional equation (<ref>), we obtain by identification that L_1 =L_1 and L_2 =L_2 and then we can conclude.




A function u(t,y) satisfies (<ref>) if and only if h(x,y):=u(x+y,y) satisfies 𝒢h=0 on (ℝ_+^*)^2 and h=0 on the boundaries ℝ_+×{0}∪{0}×ℝ_+. We say that h (resp. u) is harmonic for the killed degenerate Brownian motion Z (resp. is harmonic for the killed space-time Brownian motion B).




§ ASYMPTOTICS VIA MALYSHEV'S ANALYTICAL APPROACH





 §.§ Asymptotics of Green's functions


By using the functional equation (<ref>), inverting the Laplace transform and applying the steepest descent method, it is possible to compute the asymptotics of Green's function.


Let α∈ (0,π/2) and z=(x,y)∈ℝ_+^2. When z→∞ and y/x →tanα we have

    g_z_0(z)∼h^α(z_0)/√(|z|)e^-z· (p(α),q(α))/√(2π (cosα+sinα))

where the so-called saddle point (p(α),q(α)) is defined by

    (p(α),q(α))   :=argmax{pcosα+qsinα :(p,q)∈ℝ^2, K(p,q)=0 }
       = ( γ^2 /2 - 1/2(1+1/tanα)^2 , (1-γ)^2/ 2 - 1/2(1+tanα)^2)


and h^α is a harmonic function for the process (i..e 𝒢 h^α=0 on ℝ_+^2 and h^α=0 on ℝ×{0}∪{0}×ℝ) defined for α∈(0,π/2) by

    h^α(z_0):= e^p(α) x_0+q(α) y_0-L_1(q(α))-L_2(p(α)) .




We also have the asymptotics of the exit densities on the boundaries

    f_1(y)y→∞∼h^π/2(z_0)/y^3/2e^-y q(π/2)/√(2π)  and  
    f_2(x)x→∞∼h^0(z_0)/x^3/2e^-x p(0)/√(2π)

where we define

    h^π/2(z_0):=x_0 e^x_0 p(π/2) +y_0 q(π/2)-L_2'(p(π/2))
      and  
    h^0(z_0):=y_0 e^x_0 p(0) +y_0 q(0) - L_1'(q(0))
    .







The proof is done below in Section <ref>.




It should be noted that for α>0 we have h^α→ 0 when α→ 0 (it derives from Proposition <ref> below). That is why we have defined h^0 differently. We can interpret h^0 
as the q-derivative of h^α when α>0 evaluated in q(0),
i.e. h^0= . ∂ h^α/∂ q|_q(0). 

A similar remark holds for h^π /2.











 §.§ Proof of Theorem <ref>



The proof of  Theorem <ref> follows the same steps as in <cit.> and <cit.>. We will first show asymptotics for the Green functions g_z_0 (<ref>)  and then asymptotics for the exit densities f_1 and f_2 (<ref>).



  
From double to simple integral via the residue theorem
The Laplace inversion formula, see <cit.> and
<cit.>, implies that for ϵ>0 we have 





    g_z_0(z)= 1/(2iπ)^2∫_-ϵ-i∞^-ϵ+i∞∫_-ϵ-i∞^-ϵ+i∞ L(p,q) e^-px-qydpdq.

We are now going to use the functional equation (<ref>) and the residue theorem to transform the double integral into a sum of simple integrals.

The Green's function satisfies g_z_0(z)
= I_1+I_2+I_3 where

    I_1
    :=
    -1/2iπ∫_-ϵ-i∞^-ϵ+i∞L_1(q) /√((1-γ)^2-2q) e^-P^+(q)x-qydq,
    
    
    I_2:=
    -1/2iπ∫_-ϵ-i∞^-ϵ+i∞L_2(p) /√(γ^2 -2p) e^-px-Q^+(p)ydp,
    
    
    I_3:=
    1/2iπ∫_-ϵ-i∞^-ϵ+i∞e^p x_0+Q^+(p)y_0/√(γ^2 -2p) e^-px-Q^+(p)ydp.




The following equalities are explained below.

    g(z_0,z)    = 1/(2iπ)^2∫_-ϵ-i∞^-ϵ+i∞∫_-ϵ-i∞^-ϵ+i∞L_1(q)+L_2(p)-e^px_0+qy_0/K(p,q)
    e^-px-qydpdq
    
       =
    1/2iπ∫_-ϵ-i∞^-ϵ+i∞L_1(q)  /1/2(P^+(q)-P^-(q)) e^-P^+(q)x-qydq
    +
    1/2iπ∫_-ϵ-i∞^-ϵ+i∞L_2(p) /1/2(Q^+(p)-Q^-(p)) e^-px-Q^+(p)ydp
    
       -
    1/2iπ∫_-ϵ-i∞^-ϵ+i∞-e^p x_0+Q^+(p)y_0/1/2(Q^+(p)-Q^-(p)) e^-px-Q^+(p)ydp
    
       =I_1+I_2+I_3

The first equality above comes from the functional equation (<ref>). The second equality above comes from the residue theorem applied to an integration contour 𝒞_R represented in Figure <ref> to obtain I_1 (and a similar contour to obtain I_2 and I_3). In a classic way, the residue theorem implies that
1/2iπ∫_𝒞_RL_1(q)/K(p,q)
e^-px-qydp = L_1(q)/1/2(P^+(q)-P^-(q)) e^-P^+(q)x-qy since P^+(q) is the only pole of the integrand inside 𝒞_R.
Furthermore, the integral on the red half circle and the blue segments tends to 0 when R →∞ and we obtain that -∫_-ϵ-i∞^-ϵ+i∞L_1(q)/K(p,q)
e^-px-qydp =lim_R→∞∫_𝒞_RL_1(q)/K(p,q)
e^-px-qydp taking into account the orientation of the contour. Details are omitted and one can refer to <cit.> or <cit.> which perform similar calculations. 
The third equality just comes from the definition of P^± and Q^± in (<ref>).







  
Steepest descent method
We are going to denote  by (r,α)∈ℝ_+× [0,π/2] the polar coordinate of the point z=(x,y)∈ℝ_+^2, i.e. (x,y)=(rcosα,r sinα) and (r,α)=(√(x^2+y^2), arctan (y/x)).
Let us define the functions

    F(p,α):= p cosα +Q^+(p) sinα  and  
    G(q,α):= P^+(q) cosα +q sinα.

The point (p(α),q(α)) defined in (<ref>) is a saddle point for these functions. Indeed, 
for α∈ (0,π/2) 



the first derivatives (according to p for F and q for G) satisfy

    F'_p(p(α),α)=G'_q(q(α),α)=0,

and the second derivatives satisfy

    F”_p(p(α),α)
    =-(cosα+sinα)^3/sin^2α
    < 0,    
    G”_q(q(α),α)=-(cosα+sinα)^3/cos^2α< 0.


 

When r→∞ and α→α_0 the following asymptotics holds:

    I_1  = - L_1(q(α_0))
    e^-z· (p(α_0),q(α_0))/√(2π r (cosα_0+sinα_0) )
    +o( 1/√(r) e^-z· (p(α_0),q(α_0))) .

Similar asymptotics hold for I_2 and I_3. 



The integral I_1 is typical to apply the classical saddle point method. A famous reference is Fedoryuk's book
<cit.>.



Starting from (<ref>) we then obtain

    I_1
       =-1/2iπ∫_-ϵ-i∞^-ϵ+i∞L_1(q) /√((1-γ)^2-2q) e^-rG(q,α)dq
    
       r→∞α→α_0=-1/2iπL_1(q(α_0)) /√((1-γ)^2-2q(α_0))
    i√(-2π/r G”_q (q(α_0)))
    e^-r G(q(α_0),α_0)
    +o( 1/√(r) e^-r G(q(α_0),α_0)) .

Let us give some details about the application of this method.
To obtain the asymptotics we shift the integration contour onto the steepest descent line which passes through the saddle point q(α). This is possible since the integrand is a meromorphic function and we do not cross the only pole since q(α)<(1-γ)^2/2 for α∈(0,π/2), see (<ref>). Then, the asymptotics derives from the classical Laplace's method, see for example <cit.>. To handle the convergence in α→α_0 we just apply a simple parameter-dependent Morse lemma, see <cit.> which details this technique.
To conclude, a straightforward calculation using (<ref>) and (<ref>) shows that

    1/2π1 /√((1-γ)^2-2q(α))√(-2π/ G”_q (q(α)))
    =
    1/√(2π  (cosα+sinα) )
 
and it just remains to remark that r G(q(α),α)=z· (p(α),q(α)).



The main asymptotics (<ref>) of  Theorem <ref> directly derives from Lemmas <ref> and <ref>. One will compute more explicitly h^α(z_0)=e^p(α) x_0+q(α) y_0-L_1(q(α))-L_2(p(α)) in the following section which will show that this function is non-zero for α∈(0,π/2).




















  
Asymptotics on the boundaries via Tauberian lemmas


When q→ q^+ := q(π/2)=(1-γ)^2/2 we have

    L_1(q)=L_1(q^+)-
    (x_0 e^P^-(q^+) x_0 +q^+ y_0 -L_2' (P^-(q^+))) √(2(q^+-q)) +  o(√(q^+-q))

and a symmetrical result holds for L_2.



Evaluating 
(<ref>) at (P^-(q),q) and making a Taylor series expansion when q→ q^+ we obtain 

    L_1(q)    = e^P^-(q) x_0 +q y_0 -L_2(P^-(q))
    
       = 
    L_1(q^+)+
    (x_0 e^P^-(q^+) x_0 +q^+ y_0 -L_2' (P^-(q^+))) (P^-(q)) - P^-(q^+)) +  o(P^-(q)) - P^-(q^+))

and it just remains to notice that
P^-(q)) - P^-(q^+)= - √(2(q^+-q)).


 Lemma <ref> and classical Tauberian inversion 
lemmas

<cit.>,
<cit.>,
imply that




    f_1(y)y→∞∼ (x_0 e^P^-(q^+) x_0 +q^+ y_0 -L_2' (P^-(q^+)))-√(2)/Γ(-1/2) y^-3/2e^-q^+ y

and we deduce the asymptotics (<ref>) of Theorem <ref> noticing that Γ(-1/2)=-2√(π).




§ MARTIN BOUNDARY AND PERSISTENCE PROBABILITIES





 §.§ Doob's h-transform and persistence


In this short section, we give a probabilist interpretation of the harmonic functions h^α, found in Theorem <ref>, in term of the persistence probabilities of the process conditioned to drift in a given direction.

For α∈[0,π/2] we consider the probability ℙ^α defined by the Doob's h-transform for the harmonic function e^xp(α)+y q(α) associated to the process Z(t)=z_0+𝐖(t)+μ t.
We have

    ℙ^α (Z(t) ∈ S)=𝔼 ( e^(p(α),q(α)) · (𝐖(t)+μ t)1_Z(t) ∈ S)

for a set S⊂ℝ^2. 
Under ℙ^α the process Z(t) is a Markov process of transition kernel 

    P_t(z_0,z)=e^(x-x_0)p(α)+(y-y_0) q(α)P_t(z_0,z)

where P_t is the transition kernel of Z(t) under ℙ. Under ℙ^α the process Z(t) is conditioned to drift in the direction α (in the sense of the Doob's h-transform).

We introduce the corresponding modified Green's measure, for S⊂ℝ_+^2 we set

    G^α_z_0(S):=∫_0^∞ℙ^α(Z(t)∈ S ,t<T)dt

and its density

    g^α_z_0 (z):=e^(x-x_0)p(α)+(y-y_0) q(α) g_z_0(z).

We define L^α the Laplace transform relative to these modified Green's functions and we have

    L^α (p,q)=L(p+p(α),q+q(α)) e^-x_0 p(α)-y_0 q(α).

On the same way, we introduce the modified exit measures

    A_1^α(S ) :=ℙ^α(Z(T_1)∈ S , T_1<T_2)
      and  
    A_2^α(S) :=ℙ^α(Z(T_2)∈ S , T_2<T_1)

and their densities 

    f_1^α(y):=e^-x_0 p(α)+(y-y_0) q(α) f_1(y)
      and  
    f_2^α (x):= e^(x-x_0)p(α)-y_0 q(α) f_2(x).

We denote their Laplace transforms L_1^α and L_2^α and we have

    L_1^α (q)=L_1(q+q(α)) e^-x_0 p(α)-y_0 q(α),
      
    L_2^α (p)=L_2(p+p(α)) e^-x_0 p(α)-y_0 q(α)
    .





One may remark that

    L_1^α(0)=  ℙ^α(T_1<T_2) ,
      
    L_2^α(0)= ℙ^α(T_2<T_1),
      
    L_1^α(0)+L_2^α(0)=ℙ^α(T<∞).





Let z_0=(x_0,y_0), for α∈ (0,π) we have

    h^α (z_0)= e^x_0 p(α)+y_0 q(α)ℙ^α(T=∞)
    =e^x_0 p(α)+y_0 q(α) - 𝔼 (e^X_T p(α)+Y_T q(α)1_T<∞)

and 

    h^0(z_0)=y_0e^x_0 p(0) +y_0 q(0)- 𝔼(Y_T_1 e^q(0) Y_T_11_T_1<T_2),
      
    h^π/2(z_0)=x_0e^x_0 p(π/2) +y_0 q(π/2)- 𝔼(X_T_2 e^p(π/2) X_T_21_T_2<T_1)
    .

When x_0→∞ and y_0→∞
we obtain

    h^α(z_0) ∼ e^x_0 p(α)+y_0 q(α),
      
    h^0(z_0) ∼ y_0e^x_0 p(0) +y_0 q(0)  and  
    h^π/2(z_0) ∼ x_0 e^x_0 p(π/2) .




By Theorem <ref>, we have

    h^α(z_0)   = e^p(α) x_0+q(α) y_0-L_1(q(α))-L_2(p(α))
    
       = e^p(α) x_0+q(α)y_0(1-L_1^α(0)-L_2^α(0))
    
       = e^p(α) x_0+q(α)y_0(1-ℙ^α(T_1<T_2)-ℙ^α(T_2<T_1))
    
       = e^p(α) x_0+q(α)y_0ℙ^α (T=∞) 
    
       = e^p(α) x_0+q(α)y_0-e^p(α) x_0+q(α)y_0ℙ^α (T<∞) 
    
       =
    e^x_0 p(α)+y_0 q(α) - 𝔼 (e^X_T p(α)+Y_T q(α)1_T<∞) 
    ,

and also

    h^0(z_0)    =y_0 e^x_0 p(0) +y_0 q(0)-  L_1'(q(0))
    
       =
    y_0 e^x_0 p(0) +y_0 q(0)- 
    𝔼(Y_T_1 e^q(0) Y_T_11_T_1<T_2) .

We now assume that x_0→∞ and y_0→∞. Noticing that ℙ^α(T=∞) → 1 we obtain the asymptotics for h^α when α∈(0,π/2). Remembering that p(0)>0, q(0)<0 and that Y_T_1>y_0, we see that 𝔼(Y_T_1 e^q(0) Y_T_11_T_1<T_2)=o(y_0 e^x_0 p(0) +y_0 q(0)) and the asymptotics of h^0 follows. A symmetrical proof holds for h^π/2.

These asymptotics will be useful in the proof of Theorem <ref> to show that the Martin boundary is minimal. 



 §.§ Martin boundary


We refer to the classic book by Doob <cit.> which gives a comprehensive presentation of Martin boundary theory.

For α∈[0,π/2], the function z_0↦ h^α(z_0) is harmonic for the process Z, i.e. 𝒢 h^α =0 inside the quarter plane and h^α =0 on its boundaries. The Martin boundary of the process Z killed outside of the quarter plane is denoted ∂_M^ℝ_+^2 Z and is homeomorphic to [0,π/2] through the map Φ given by


    [         Φ : [0,π/2]                   ⟶         ∂_M^ℝ_+^2 Z;                   α                   ⟼ h^α (·)/h^α (1,1) .;                     ]




We define the Martin's kernel for the arbitrary point (1,1)∈( ℝ_+^*)^2 by

    k(z_0,z):=g_z_0(z)/g_(1,1)(z).

By the Martin's boundary theory in a parabolic context <cit.> we have to show that for α∈ [0,π/2], z=(x,y), z→∞ and y/x →tanα we have

    k(z_0,z)→ k^α(z_0):= h^α (z_0)/h^α (1,1).

For α∈(0,π/2) it directly derives from the asymptotics of the Green's function g_z_0 (<ref>) found in Theorem <ref>. For α=0 or π/2 it is enough to use formula (<ref>) of Theorem <ref> and Proposition <ref> and to apply l'Hôpital's rule. The continuity of Φ is clear on (0,π/2). The continuity at 0 derives from Remark <ref>, noticing that p'(0)=0, with a simple Taylor approximation we obtain that h^α=α q'(0)h^0 +o(α) when α→ 0 and then k^α→ k^0 when α→ 0. The same reasoning holds for the continuity at π/2.
 
The injectivity of Φ follows from the asymptotics of h^α when x_0→∞ and y_0→∞ which are all different for different values of α, see (<ref>) of Proposition <ref>.
Since [0,π/2] is compact and Φ is a continuous bijection, Φ is an homeomorphism.

In Theorem <ref>, we will deduce from the asymptotics (<ref>) that the Martin boundary is minimal. 



§ HARMONIC FUNCTIONS VIA THE COMPENSATION APPROACH





 §.§ Minimal harmonic functions

 


The harmonic function h^α(z_0) defined in (<ref>) has been expressed in terms of the Laplace transforms L_1 and L_2 and then interpreted in (<ref>) as a persistence probability of a conditioned process. But this is not a very tractable expression since these Laplace transforms and these probabilities are not known explicitly. 

In this section, we compute explicitly all the harmonic functions h^α(z_0) thanks to a recursive compensation approach.  


For z=(x,y)∈ℝ_+^2, we define the functions h^(p_0,q_0) by

    h^(p_0,q_0)(x,y)
    :=
    ∑_n∈ℤ (-1)^n e^x p_n +y q_n

where (p_n,q_n) is a recursive sequence of points on the parabola 𝒫:={(p,q)∈ℝ_+^2 : K(p,q)=0 } starting from (p_0,q_0)∈𝒜:={(p(α),q(α)): α∈(0,π/2) } the red arc of the parabola, see Figures <ref> and <ref>, and defined by

    p_2n:=p_0+2n(p_0-q_0)-2n(n+γ)
    
    
    q_2n:=q_0+2n(p_0-q_0)-2n(n+γ-1)
      and  
    
    
    
    
    (p_2n+1,q_2n+1)=(p_2n,q_2n+2).
 
Then, the functions h^(p_0,q_0) are positive and harmonic for the process Z killed at the boundary of the quadrant, that is

    𝒢h^(p_0,q_0)  =0 
       on  (ℝ_+^*)^2
    
    h^(p_0,q_0) =0    on ℝ×{0}∪{0}×ℝ.




This sum is typical of the compensation approach. First, we may notice that the convergence of the sum is trivial since the sequence (p_n,q_n) tends to minus infinity quadratically. Let us notice that a function f(z)=e^z· (p,q) satisfies 𝒢f=0 if and only if K(p,q)=0, i.e. (p,q)∈𝒫. It is easy to verify that for all n∈ℤ we have (p_n,q_n)∈𝒫, see Figure <ref>. Therefore, we deduce that 𝒢h^(p_0,q_0)  =0 for z=(x,y)∈ℝ_+^2. Furthermore, using (p_2n+1,q_2n+1)=(p_2n,q_2n+2) we have the following recursive compensations

    h^(p_0,q_0)(x,y) =        4=0   when  x=0    4=0   when  x=0 
      ⋯ +     e^p_-2x+q_-2y   -    e^p_-1x+q_-1y   +   e^p_0x+q_0y   -   e^p_1x+q_1y   +     e^p_2x+q_2y   + ⋯
         3=0   when  y=0       3=0   when  y=0

and we deduce that h^(p_0,q_0) satisfies the boundary conditions of (<ref>). 

It only remains to show the positivity. First, there exists t>0 large enough such that when x+y>t we have h^(p_0,q_0)(z) >0, indeed

    h^(p_0,q_0)(x,y) =  
     e^p_0x+q_0y -e^p_1x+q_1y - e^p_-1x+q_-1y^⩾ 0   when  x+y  large enough
        + ∑_ℕ^*e^p_2nx+q_2ny - e^p_2n+1x+q_2n+1y^⩾ 0  always
       +  ∑_ℕ^*e^p_-2nx+q_-2ny - e^p_-2n-1x+q_-2n-1y .^⩾ 0  always

For all starting point z_0=(x_0,y_0)∈ℝ_+^2 of the process Z=(X,Y), remembering that X(t)+Y(t)=t_0+t and since h^(p_0,q_0) is harmonic, we deduce that

    h^(p_0,q_0) (x_0,y_0)=𝔼[h^(p_0,q_0)(Z(t))]
    ⩾ 0 .







It remains to define the two harmonic functions associated with the extremal points of the arc ∂𝒜= {(p(0),q(0)),(p(π/2),q(π/2)) }. Indeed, when (p_0,q_0)→(p(0),q(0)) or 
(p(π/2),q(π/2)) we have h^(p_0,q_0)→ 0. That is why we have to define h^(p(0),q(0)) and h^(p(π/2),q(π/2)) differently. 

For z=(x,y)∈ℝ_+^2, we define the function h^(p(0),q(0)) by

    h^(p(0),q(0))(x,y)
    :=
    ∑_n∈ℤ ( -2n x +(1-2n)y) e^xp_2n +y q_2n

where (p_2n,q_2n) is a recursive sequence of points on the parabola 𝒫 starting from (p_0,q_0)=(p(0),q(0))=(γ^2/2,-γ+γ^2/2)∈∂𝒜, and defined by (<ref>), which means in this case

    p_2n:=γ^2/2 -2n^2,
    
    
    q_2n:=γ^2/2 -γ -2n(n-1).
 
Then, the function h^(p_0,q_0) is positive and harmonic for the process Z killed at the boundary of the quadrant, that is satisfying (<ref>). A symmetric expression holds for h^(p(π/2),q(π/2)).



First, we again notice that the convergence of the sum is trivial since the sequence (p_2n,q_2n) tends to minus infinity quadratically. 
Here we cannot compensate in the same way that in the previous proposition because we would obtain the null function since (p_2n,q_2n)=(p_-2n+1,q_-2n+1). Therefore, we compensate using linear polynomials in front of the exponential terms.
With a straightforward computation we notice that a function f(x,y)=(λ_1x+λ_2y)e^xp+yq satisfies 𝒢f=0 if and only if K(p,q)=0 and λ_1((1-γ)+p-q)+λ_2(γ+q-p)=0. It is easy to verify that 
K(p_2n,q_2n)=0 and -2n((1-γ)+p_2n-q_2n)+(1-2n)(γ+q_2n-p_2n)=0.

Therefore we deduce that 𝒢h^(p_0,q_0)  =0 for z=(x,y)∈ℝ_+^2. Furthermore, using the fact that (p_2n,q_2n)=(p_-2n,q_-2n+2) we have the following recursive compensations, we obtain that h^(p(0),q(0))(x,y) is equal to

    1=0  when  y=0   4=0   when  y=0    4=0   when  y=0 
          y e^p_0x+q_0y   - (2x+y)    e^p_2x+q_2y   +   (2x+3y)e^p_-2x-q_-2y   -(4x+3y)   e^p_4x+q_4y   +(4x+5y)     e^p_-4x+q_-4y   + ⋯
         3=0   when  x=0       3=0   when  x=0

and we deduce that h^(p(0),q(0)) satisfies the boundary conditions of (<ref>). Positivity is proved in the same way as in the proof of Proposition <ref>. It is easy to see that the function h^(p(0),q(0)) is positive for x+y large enough and we conclude in the same way thanks to the harmonic character of the function.


The following proposition shows that the compensation approach actually
allows to construct all positive minimal harmonic functions.

This implies a correspondence between minimal positive harmonic functions and the arc of parabola 𝒜.

For all α∈[0,π/2] we have

    h^(p(α),q(α)) = h^α

where (p(α),q(α)) is the saddle point defined in (<ref>). The Martin boundary given in Theorem <ref> is minimal. It is homeomorphic to the arc of parabola 𝒜 through the map 



    [                                       𝒜                                       ⟶                             ∂_M^ℝ_+^2 Z;                             (p(α),q(α))                                       ⟼ h^(p(α),q(α))(z_0)/h^(p(α),q(α))(1,1) .;                                         ]



 See Figures <ref> and <ref>.


 Recall that the Martin boundary ∂_M^ℝ_+^2 Z is determined in Theorem <ref> and is homeomorphic to [0,π/2]. In Propositions <ref> and <ref> we have seen that the functions h^(p(α),q(α)) are harmonic for α∈[0,π/2]. By Martin boundary theory <cit.>, 
we have

    h^(p(α),q(α)) (z_0) =∫_0^π/2h^β (z_0)/h^β (1,1)dm_α(β)
 
for some measure m_α on [0,π/2]. Let z_0=(x_0,y_0) and α∈(0,π/2), when x_0→∞ and y_0→∞ we have h^(p(α),q(α)) (z_0)∼ e^x_0 p(α)+ y_0 q(α) by (<ref>) and h^β (z_0) ∼ e^x_0 p(β)+y_0 q(β) by Proposition <ref>. For α=0, we have h^(p(0),q(0)) (z_0)∼ y_0 e^x_0 p(0)+ y_0 q(0) by (<ref>) and h^β (z_0) ∼ y_0 e^x_0 p(β)+y_0 q(β) and a symetrical result holds for α=π/2. Then, from these asymptotics and (<ref>) we can deduce that m_α =h^α (1,1) δ_α (where δ_α is the dirac measure at α) and therefore h^(p(α),q(α)) = h^α. Indeed, it is not possible to create an exponential asymptotic of a certain parameter by averaging exponential asymptotics of other parameters.


We deduce that these harmonic functions are minimal since there is a unique measure on the Martin boundary which satisfies (<ref>). The Martin boundary is then equal to the minimal Martin boundary and is homeomorphic to the arc of parabola 𝒜 through the composition between the map Φ given in Theorem <ref> and the homeomorphism [0,π/2]→𝒜 :α↦ (p(α),q(α)).


The same kind of phenomenon appears for non-singular random walks <cit.> and singular one <cit.>.



 §.§ Persistence and exit probabilities


From (<ref>), (<ref>) and (<ref>) we find explicitly the persistence probability of the conditioned process: ℙ^α(T=∞)=e^-x_0 p(α)-y_0 q(α)  h^α(z_0). The next remark focuses on the non-conditioned process. 



Let α_γ∈ (0,π/2) such that tanα_γ=γ/(1-γ), we have (p(α_γ),q(α_γ))=(0,0) and then ℙ=ℙ^α_γ. Then, the probability 


that the process stays in the cone forever is equal to

    ℙ_(t_0,y_0)(∀ t>0 : X_t∈ C)=ℙ(T=∞)=h^(0,0)(z_0)= h^α_γ (z_0) .

For z_0=(x_0,y_0)=(t_0-y_0,y_0) and remembering equation (<ref>) we obtain

    ℙ(T=∞)
       =∑_n∈ℤ (-1)^n e^p_n x_0 + q_n y_0
    =
    ∑_n∈ℤ e^p_2n x_0 + q_2n y_0 - ∑_n∈ℤ e^p_2n+1^p_2n x_0 + q_2n+1^q_2n+2 y_0
       =
    ∑_n∈ℤ 2 sinh( q_2n - q_2n+2/2 y_0 ) e^p_2n x_0 + q_2n+ q_2n+2/2 y_0 
       = 2 ∑_n∈ℤsinh((2n+γ)y_0) e^-2n(n+γ)t_0-γ y_0.






We can also express this quantity in another way to match Defosseux's result <cit.>. By noting that p_-2n x_0 + q_-2n y_0=-2n^2t_0+2nγ t_0-2ny_0 and p_2n x_0 +q_2n+2 y_0=-2n^2t_0-2nγ t_0-2γ y_0-2ny_0 
and with a simple change of variable in the sum we obtain

    ℙ(T=∞)
       =
    ∑_n∈ℤ e^p_-2n x_0 + q_-2n y_0 - ∑_n∈ℤ e^p_2n x_0 +q_2n+2 y_0
       =  2 ∑_n∈ℤsinh(γ (y_0+2nt_0)) e^-2(n y_0+n^2 t_0)-γ y_0.




We are now interested in ℙ^α(T_1<T_2) and ℙ^α(T_2<T_1) which are the exit probabilities on each side of the cone of the conditioned process. 

For α∈ [0,π] we have

    L_1(q(α))
    = e^p(α) x_0+q(α)y_0ℙ^α(T_1<T_2)
    =∑_n∈ -ℕ^* (-1)^n+1 e^p_n x_0 + q_n y_0

and 

    L_2(p(α))
    = e^p(α) x_0+q(α)y_0ℙ^α(T_2<T_1)
    =∑_n∈ +ℕ^* (-1)^n+1 e^p_n x_0 + q_n y_0

where (p_0,q_0)=(p(α),q(α)) and the sequence (p_n,q_n)_n∈ℤ is defined in (<ref>).


 The proof is again based on a compensation method.
Let us define

    L_1^α (z_0):=∑_n∈ -ℕ^* (-1)^n+1 e^p_n x_0 + q_n y_0.
 
Let us verify that the function L_1^α is harmonic and satisfies the following conditions:

    (𝒢L_1^α) (z_0)  =0 
       for  z_0∈(ℝ_+^*)^2,
    
    L_1^α (z_0) =0    for  z_0=(x_0,0)∈ℝ×{0} ,
    
    L_1^α (z_0) =e^ q(α)y_0   for  z_0=(0,y_0)∈{0}×ℝ.

The first condition is trivial by the construction of the sequel (p_n,q_n)∈𝒫, see Proposition <ref>. The boundary conditions can be seen by summing in packets of two (taking into account the first term which cannot be compensated on one side) as follows

    L_1^α (z_0)=     1=e^ q(α)y_0 when  x_0=0    4=0   when  x_0=0    4=0   when  x_0=0 
         e^p_-1x_0+q_-1y_0   -    e^p_-2x_0+q_-2y_0   +   e^p_-3x_0-q_-3y_0   -   e^p_-4x_0+q_-4y_0   +     e^p_-5x_0+q_-5y_0   + ⋯
         3=0   when  y_0=0       3=0   when  y_0=0 .

With a classical martingale argument we remark that the function z_0↦ L_1(q(α)) seen as a function of z_0 is also harmonic, i.e. 𝒢_z_0  L_1(q(α))=0. This function also satisfies the same boundary conditions, to verify it, it is enough to remark that via the Doob's h-transform we have

    L_1(q(α))
    = e^p(α) x_0+q(α)y_0 L_1^α(0)
    = e^p(α) x_0+q(α)y_0ℙ_z_0^α(T_1<T_2)

and of course ℙ^α(T_1<T_2)=0 if z_0∈ℝ×{0} and ℙ^α(T_1<T_2)=1 if z_0∈{0}×ℝ.
The functions L_1^α and L_1(q(α)) seen as a function of z_0 both satisfy (<ref>) and tend to 0 at infinity. We deduce by the maximum principle that these functions are equals, i.e. L_1(q(α))=L_1^α. 



For α_γ∈ (0,π/2) such that tanα_γ=γ/(1-γ) we have (p(α_γ),q(α_γ))=(0,0) and we find that the probability ℙ(T_1<T_2) 

that the process exit of the cone by the vertical axis is equal to

    ℙ(T_2<T_1)=L_2(0)
       =∑_n∈ℕ^* (-1)^n+1 e^p_n x_0 + q_n y_0
    =
    ∑_n∈ℕ e^p_2n+1^p_2n x_0 + q_2n+1^q_2n+2 y_0 - ∑_n∈ℕ e^p_2n+2 x_0 + q_2n+2 y_0
       =
    ∑_n∈ℕ 2 sinh( p_2n - p_2n+2/2 x_0 ) e^p_2n+ p_2n+2/2 x_0 + q_2n+2 y_0
       = 2 ∑_n∈ℕsinh((2n+1+γ)x_0) e^-(2n+2)(n+γ)t_0 -(γ+1) x_0

where (p_0,q_0)=(0,0).
Similarly, we obtain

    ℙ(T_1<T_2)= 2 ∑_n∈ℕsinh((2n+2-γ)y_0) e^-(2n+2)(n+1-γ)t_0 -(2-γ) y_0.

This result matches with Anderson's result <cit.>.



The Laplace transform L_1 can be extended analytically on ℂ∖ [q^+,∞) by the formula

    L_1(q)  
    
    
    
    
    
    
    
    
    
    
    
    
       =
    ∑_n∈ -ℕ^* 2 sinh( (2n-p_0-q_0+γ+1) x_0 ) e^ (p_0+(2n+1)(p_0-q_0)-2n(n+γ+1)-γ-1) x_0 + (q+(2n+2)(P^+(q)-q)-(2n+2)(n+γ)) y_0
       =
    ∑_n∈ -ℕ^*  
     e^(p_0 +2n(p_0-q_0)-2n(n+γ) ) x_0 + (q_0 +(2n+2)(p_0-q_0)-(2n+2)(n+γ)) y_0
          - e^(p_0 +(2n+2)(p_0-q_0)-(2n+2)(n+1+γ) ) x_0 + (q_0 +(2n+2)(p_0-q_0)-(2n+2)(n+γ)) y_0

where (p_0,q_0)=(P^+(q),q) and

    p_2n=P^+(q)+2n(P^+(q)-q)-2n(n+γ)
    
    
    q_2n=q+2n(P^+(q)-q)-2n(n+γ-1)
      and  
    
    
    
    
    (p_2n+1,q_2n+1)=(p_2n,q_2n+2).




The result directly derives from Proposition <ref>, the fact that (p(α),q(α))=(P^+(q(α)),q(α)) and the analyticity of P^+ on ℂ∖ [q^+,∞). Taking (p_0,q_0)=(P^+(q),q) we obtain that

    L_1(q)    =∑_n∈ -ℕ^* (-1)^n+1 e^p_n x_0 + q_n y_0
    
    
    
    =
    ∑_n∈ -ℕ^*  
    (e^p_2n x_0 
    - e^p_2n+2 x_0 )e^ q_2n+2 y_0
       =
    ∑_n∈ -ℕ^* 2 sinh( p_2n - p_2n+2/2 x_0 ) e^p_2n+ p_2n+2/2 x_0 + q_2n+2 y_0

and we conclude noticing that (<ref>) implies (<ref>).



















§ EXIT DENSITIES AND TRANSITION KERNEL





This last section aims at computing the inverse Laplace transforms of L_1 and L_2 to obtain an explicit expression for the exit densities f_1 and f_2. Finally, these expressions are used to calculate the kernel p_(t_0,y_0)^k,C(t,y).




 Let z_0=(x_0,y_0) the starting point and t_0=x_0+y_0. Exit densities are given by the following formulas:

    f_1(y)
    =
    e^-t_0 (1-γ)-y_0 γ/√(2π)e^-1/2 (1-γ)^2(y-t_0) /√((y-t_0)^3)∑_n∈ℤ
    ((2n+1)t_0-y_0) e^-1/2((2n+1)t_0-y_0)^2/y-t_0 +(2n+1)y_0-2n(n+1)t_0


    f_2(x)
    =
    e^-t_0 γ-x_0(1- γ)/√(2π)e^-1/2γ^2(x-t_0) /√((x-t_0)^3)∑_n∈ℤ
    ((2n+1)t_0-x_0) e^-1/2((2n+1)t_0-x_0)^2/x-t_0 +(2n+1)x_0-2n(n+1)t_0










We seek to inverse the Laplace transform of Corollary <ref>. We have

    L_1(q)    = 
    ∑_n∈ℕ  
    (e^p_-2n-2 x_0 +q_-2n y_0 
    - e^p_-2n-2 x_0 +q_-2n-2 y_0 )
    
       = ∑_n∈ℕ
    e^-(P^+(q)-q)((2n+1)t_0-y_0)+q t_0- (2n+2)(n+1-γ) t_0 + 2(n+1-γ)y_0
          -
    e^-(P^+(q)-q)((2n+1)t_0+y_0)+q t_0- (2n+2)(n+1-γ) t_0 - (2n+2)y_0
       = ∑_n∈ℕ
    e^-((2n+1)t_0-y_0)√((1-γ)^2-2q)+q t_0- t_0(2n(n+1)+1-γ)+y_0(2n+1-γ)
          -
    e^-((2n+1)t_0+y_0)√((1-γ)^2-2q)+q t_0- t_0(2n(n+1)+1-γ)-y_0(2n+1+γ)

We denote ℒ^-1_q the inverse Laplace operator related to the q-variable.
For a>0, b>0 and y>t_0 we have

    ℒ^-1_q (e^-b√(a-2q)+q t_0)(y)= b/√(2π)1/√((y-t_0)^3)e^-1/2 (a(y-t_0) +b^2/y-t_0).

Taking the inverse Laplace transform of the sum term by term, it reads

    f_1(y)    = 
    e^-t_0 (1-γ)-y_0 γ/√(2π)1/√((y-t_0)^3)
       ∑_n∈ℕ
    ((2n+1)t_0-y_0) e^-1/2 ((1-γ)^2(y-t_0) +((2n+1)t_0-y_0)^2/y-t_0)
    e^- 2n(n+1)t_0+(2n+1)y_0
          - 
    ((2n+1)t_0+y_0) e^-1/2 ((1-γ)^2(y-t_0) +((2n+1)t_0+y_0)^2/y-t_0)
    e^- 2n(n+1)t_0-(2n+1)y_0

which coincides with the stated result summing over ℤ. The formula for f_2(x) is obtained replacing 1-γ by γ and y by x.


With some tedious computations, one may verify that this result matches with Anderson's result <cit.> by deriving its formula (4.32).

The previous result gives an explicit expression for the density of T since we have seen in (<ref>) that f_T(t)=f_1(t_0+t)+f_2(t_0+t). The persistence probability of the process after a time t is given by 

    ℙ(T>t    ) =ℙ(T=∞)+∫_t^∞ f_T(s)ds
    
       t→∞=
    h^α_γ(z_0) + 
    (  h^0(z_0)/√(2π)e^-γ^2/2(t+t_0)/t^3/2 +  h^π/2(z_0)/√(2π)e^-(1-γ)^2/2(t+t_0)/t^3/2)(1+o(1))

where the asymptotics derives from (<ref>) and α_γ is defined in Remark <ref>.




We now use the previous result about the exit densities 
to obtain an 
explicit formula for the Green's function of the process Z, and then for the transition density of the space-time Brownian motion B, see (<ref>).


For (t_0+t,y)∈ C, the transition density defined in (<ref>) is equal to

    p_(t_0,y_0)^k,C(t,y) =1/√(2π t) e^-(y-y_0-γ t)^2/2t- 𝔼_(t_0,y_0)( 1/√(2π (t-T)) e^-(y-Y_T-γ (t-T))^2/2(t-T)1_T⩽ t)
    
       =
    1/√(2π t)∑_n∈ℤ(
    
    e^-(y-2n t_0-y_0-γ t)^2/2t+γ (2n t_0 +y_0)
    -
    
    e^-(y+2n t_0+y_0-γ t)^2/2t - γ (2n t_0 +y_0)
    )
     e^-γ y_0 -2n^2t_0 -2ny_0




Let us define p_t^γ(y):=1/√(2π t) e^-(y-γ t)^2/2t the transition kernel of the free Brownian motion with drift γ starting from 0 and recall that we note p^k,C_(t_0,y_0)(t,y) dy 
=ℙ_(t_0,y_0)(B(t)=(t_0+t,dy), T>t) the transition kernel of the killed space-time Brownian motion in C. 
A direct calculus based on the Markov property gives

    p^k,C_(t_0,y_0)(t,y) dy 
    
    
       =
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy))-
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy), T⩽ t)
    
       =
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy))
    -
    ∫_0^t
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy) | T=T_2=u)f_2(t_0+u)du
    
       -
    ∫_0^t
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy) | T=T_1=v)f_1(t_0+v)dv
    
       =
    ℙ_(t_0,y_0)(B(t)=(t_0+t,dy))-
    ∫_0^t ℙ_(t_0+u,0)(B(t)=(t_0+t,dy))f_2(t_0+u)du
    
       -
    ∫_0^t ℙ_(t_0+v,t_0+v)(B(t)=(t_0+t,dy))f_1(t_0+v)dv
    
       = ( p_t^γ(y-y_0)
    -
    ∫_0^t p_t-u^γ(y)f_2(t_0+u)du_=:I_2(t)
    -
    ∫_0^t p_t-v^γ(y-t_0-v)f_1(t_0+v)dv _=:I_1(t)) dy

and equality (<ref>) follows.
We defined in the previous formula the integrals I_1(t) and I_2(t) and we are going to compute these integrals by computing their Laplace transforms. We denote ℒ_t the Laplace transform operator related to the t variable and  ⋆ the convolution operation. We have

    ℒ_t(I_2(t))(p)    =
    
    
    
    
    ℒ_t ( 
     ( p_·^γ(y)  ⋆ f_2(t_0+·)) (t) ) (p)
    
       = 
     ℒ_t ( 
     p_t^γ(y) ) (p)  ×ℒ_t ( f_2(t_0+t)  ) (p)

One may compute that

    ℒ_t (   
     p_t^γ(y)
     ) (p)
    =1/√(2(p+γ^2/2)) e^γ y-y√(2(p+γ^2/2)).

Recalling that Theorem <ref> gives

    f_2(t_0+u)
    =
    e^-t_0 γ-x_0(1- γ)/√(2π)e^-1/2γ^2u /u^3/2∑_n∈ℤ
    ((2n+1)t_0-x_0) e^-1/2((2n+1)t_0-x_0)^2/u +(2n+1)x_0-2n(n+1)t_0

we compute the following Laplace transform (we note sgn the sign function)

    ℒ_t ( 
    ((2n+1)t_0-x_0)/√(2π)t^3/2 e^-1/2γ^2t -1/2((2n+1)t_0-x_0)^2/t) (p)
    = sgn(n) e^-|(2n+1)t_0-x_0|√(2(p+γ^2/2))

and we obtain

    ℒ_t ( 
     f_2(t_0+t)  ) (p)=
     e^-t_0 γ-x_0(1- γ)∑_n∈ℤ
    sgn(n)
     e^-sgn(n)((2n+1)t_0-x_0)√(2(p+γ^2/2)) +(2n+1)x_0-2n(n+1)t_0.

Using (<ref>), (<ref>) and (<ref>) we obtain the Laplace transform of I_2. By inverting this Laplace transform we compute

    I_2(t)    =  e^-t_0 γ-x_0(1- γ)∑_n∈ℤ sgn(n)
    p_t^γ (y+sgn(n)((2n+1)t_0-x_0))
     e^-sgn(n)γ((2n+1)t_0-x_0) +(2n+1)x_0-2n(n+1)t_0
       =
     e^-γ y_0 ∑_n∈ℤ sgn(n)
    p_t^γ (y+sgn(n)(2n t_0+y_0))
     e^-2n^2t_0 -2ny_0 - sgn(n) γ (2n t_0 +y_0)  
       =
     e^-γ y_0 ∑_n∈ℕ 
    p_t^γ (y+2n t_0+y_0)
     e^-2n^2t_0 -2ny_0 - γ (2n t_0 +y_0)  
        - 
     e^-γ y_0 ∑_n∈ -ℕ^* 
    p_t^γ (y-2n t_0-y_0)
     e^-2n^2t_0 -2ny_0 + γ (2n t_0 +y_0)

In the same way, we show that for x+y=t_0+t we have

    I_1(t)
       =
     e^-(1-γ) x_0 ∑_n∈ℕ 
    p_t^1-γ (x+2n t_0+x_0)
     e^-2n^2t_0 -2nx_0 - (1-γ) (2n t_0 +x_0)  
        - 
     e^-(1-γ) x_0 ∑_n∈ -ℕ^* 
    p_t^1-γ (x-2n t_0-x_0)
     e^-2n^2t_0 -2nx_0 + (1-γ) (2n t_0 +x_0)  
       =
     e^-γ y_0 ∑_n∈ℕ 
    p_t^γ (y-2(n+1) t_0+y_0)
     e^-2(n+1)^2t_0 +2(n+1)y_0 +γ (2(n+1) t_0 -y_0)  
        - 
     e^-γ y_0 ∑_n∈ -ℕ^* 
    p_t^γ (y+2n t_0-y_0)
     e^-2n^2t_0 +2ny_0 - γ (2n t_0 -y_0)  
       =
     e^-γ y_0 ∑_n∈ -ℕ^* 
    p_t^γ (y+2n t_0+y_0)
     e^-2n^2t_0 -2ny_0 - γ (2n t_0 +y_0)  
        - 
     e^-γ y_0 ∑_n∈ℕ^* 
    p_t^γ (y-2n t_0-y_0)
     e^-2n^2t_0 -2ny_0 + γ (2n t_0 +y_0)

Since p^k,C_(t_0,y_0)(t,y)=p_t^γ(y-y_0)-I_1(t)-I_2(t) we obtain formula (<ref>) which concludes the proof.

We find again the result obtained by Defosseux <cit.>. 


The approach developed in this article which combines the steepest descent method and a recursive compensation method allowed us to determine the Martin boundary and all the harmonic functions associated with the space-time Brownian motion. By using these two methods together, we also developed a new and original approach which enabled us to recover some interesting results.



 §.§ Acknowledgments
 
This project has received funding from Agence Nationale de la Recherche, ANR JCJC programme under the Grant Agreement RESYST ANR-22-CE40-0002.
The author would like to thank Manon Defosseux, Kilian Raschel and Pierre Tarrago for interesting discussions related to space-time Brownian motion, the compensation approach and Martin's boundary theory.


apalike





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