Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant

Abstract

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green’s functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green’s functions inside the quadrant and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the basic adjoint relationship which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the moment generating function in terms of contour integrals and a conformal mapping.

Appendices

Appendix A. Potential Theory

There have not been many studies to determine explicit expressions for Green’s functions of diffusions. In order to make the article self-contained and give context, in this appendix we illustrate in an informal way the links between partial differential equations and Green’s functions of Markov processes in potential theory.

1.1 A.1. Dirichlet Boundary Condition and Killed Process

Let $\varOmega$ be an open, bounded, smooth subset of $\mathbb {R}^d$ and X an homogeneous diffusion of generator $\mathcal {L}$ starting from x and killed at the boundary $\partial \varOmega$. Assume that X admits a transition density $p_t(x,y)$ and denote by g(x, y) the Green’s function defined by

$$\begin{aligned} g(x,y)= \int _0^{\infty } p_t(x,y) \, \mathrm {d} t. \end{aligned}$$

The forward Kolmogorov equation (or Fokker–Planck equation) with boundary and initial condition says that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}^*_y p_t(x,y) = \partial _t p_t(x,y),\\ p_t(x,\cdot )=0 \text { on } \partial \varOmega ,\\ p_0 (x,\cdot ) = \delta _x. \end{array}\right. } \end{aligned}$$

Integrating this equation in time we can see that Green’s function is a fundamental solution of the dual operator $L^*$ and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}^*_y g (x,\cdot )= -\,\delta _x &{} \quad \text {in } \varOmega ,\\ g (x,\cdot ) = 0 &{} \quad \text {on } \partial \varOmega . \end{array}\right. } \end{aligned}$$

Now, let f be a continuous function on $\overline{\varOmega }$ and $\varphi$ a continuous function on $\partial \varOmega$. If we assume that the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L} u = -\,f &{} \quad \text {in } \varOmega ,\\ u = \varphi &{} \quad \text {on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

admits a unique solution, it is possible to express it in terms of Green’s functions. We have

$$\begin{aligned} u(x)= & {} \mathbb {E}_x \left[ \int _0^\tau f({{X(t)}}) \, \mathrm {d}t\right] + \mathbb {E}_x \left[ \varphi (X_\tau ) \right] \\= & {} \int _\varOmega f(y) g(x,y)\, \mathrm {d}y +\int _{\partial \varOmega } \varphi (y) \partial _{n_y}g(x,y) \, {\mathrm {d}y}, \end{aligned}$$

where $\tau$ is the first exit time of $\varOmega$. (Note that $\partial _{n_y}g$, the inner normal derivative on the boundary of Green’s function, may be interpreted as the density of the distribution of the exit place.) Thanks to Green’s functions it is then possible to solve an interior Poisson’s type equation with Dirichlet boundary conditions which specify the value of u on the boundary and the value of $\mathcal {L}u$ inside $\varOmega$.

1.2 A.2. Neumann Boundary Condition and Reflected Process

Henceforth, let us replace the interior Dirichlet problem by an exterior Neumann boundary problem which specifies the value of the normal derivative of u on the boundary and the value of $\mathcal {L}u$ outside $\varOmega$ in $\varOmega ^c =\mathbb {R}^d {\setminus } \overline{\varOmega }$ :

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L} u = -\,f &{} \quad \text {in } \varOmega ^c,\\ \partial _{n} u = \varphi &{} \quad \text {on } \partial \varOmega . \end{array}\right. } \end{aligned}$$

While the Dirichlet equation was linked to some killed process on the boundary, the Neumann equation is linked to a reflected process. From now, let us denote X the reflected process on $\partial \varOmega$ of generator L inside $\varOmega ^c$. Let us recall that $\varOmega ^c$ is unbounded, we assume that the process is transient and we note g its Green’s function. This time again, g is a fundamental solution of $\mathcal {L}^*$ (with a more complex boundary condition of Robin type linking u and $\partial _n u$). There are some necessary compatibility conditions linking f and $\varphi$ in order for a solution to exist, for example if $\mathcal {L}=\varDelta$ the interior Neumann boundary problem can have a solution only if $\int _\varOmega f =-\,\int _{\partial \varOmega } \varphi$. The solution vanishing at infinity of the Neumann problem, if it exists, is equal to

$$\begin{aligned} u(x)= & {} \mathbb {E}_x \left[ \int _0^\infty f({{X(t)}}) \, \mathrm {d}t\right] + \mathbb {E}_x \left[ \int _0^\infty \varphi ({{X(t)}}) \, \mathrm {d} {{L(t)}}\right] \\= & {} \int _\varOmega f(y) g(x,y)\, \mathrm {d}y +\int _{\partial \varOmega } \varphi (y) h(x,y) \, \mathrm {d}s(y). \end{aligned}$$

We have noted L the local time that the process spends on the boundary $\partial \varOmega$ and h the density of the boundary Green’s measure H which is equal to

$$\begin{aligned} H(x,A)=\mathbb {E}_x \left[ \int _0^\infty \mathbf {1}_A ({{X(t)}}) \, \mathrm {d} {{L(t)}} \right] \quad \text {for } A\subset \partial \varOmega , \end{aligned}$$

and represents the average local time that the process spends on the set A of the boundary. In fact h and the restriction of g to $\partial \varOmega$ are intimately related, for example if $\mathcal {L}=\varDelta$ then $h=g_{\mid _{\partial \varOmega }}$. These formulas present, in an informal way, how to solve a Neumann boundary equation thanks to Green’s functions. The “Appendix C” illustrates this by giving an explicit example in one dimension in (32).

Unfortunately, finding Green’s functions is often a difficult task. Notice that in this paper $\varOmega ^c= \mathbb {R}_+^2$ and $\varOmega$ is therefore neither bounded nor smooth, and the reflection is oblique, rather than normal. This makes our task in this article more complicated.

Appendix B. Carleman Boundary Value Problem

This appendix is a short presentation of the boundary value problems (BVP) theory. It introduces methods and techniques used for the resolution of BVP. The results presented here can be found in the reference books of Litvinchuk [46], Muskhelishvili [48] and Gakhov [32].

1.1 B.1. Sokhotski–Plemelj Formulae

Sokhotski–Plemelj formulas are central in the resolution of Riemann boundary value problems. Let $\mathcal {L}$ a contour (open or closed) smooth and oriented and $f\in \mathbb {H}_\mu (\mathcal {L})$ the set of $\mu$-Hölder continuous functions on $\mathcal {L}$ for $0<\mu \leqslant 1$. A function is sectionally holomorphic if it is holomorphic on the whole complex plane except $\mathcal {L}$ and admits right and left limits on $\mathcal {L}$ (except on its potential ends).

Fig. 8

An oriented smooth open contour $\mathcal {L}$ of ends a and b; right limit $F^-$ and left limit $F^+$ of F on $\mathcal {L}$

Proposition 12

(Sokhotski–Plemelj formulae) The function

$$\begin{aligned} F (z) := \frac{1}{2i\pi }\int _{\mathcal {L}} \frac{f(t)}{t-z} \, \mathrm {d}t, \quad z\notin \mathcal {L} \end{aligned}$$

is sectionally holomorphic. The functions $F^+$ and $F^-$ on $\mathcal {L}$ taking the limit values of F, respectively, on the left and on the right satisfy for $t\in \mathcal {L}$ the formulas

$$\begin{aligned} F^+(t)= \frac{1}{2} f(t) +\frac{1}{2i\pi } \int _{\mathcal {L}} \frac{f(s)}{s-t} \, \mathrm {d}s \quad \text {and} \quad F^-(t)= -\frac{1}{2} f(t) +\frac{1}{2i\pi } \int _{\mathcal {L}} \frac{f(s)}{s-t} \, \mathrm {d}s. \end{aligned}$$

Theses formulas are equivalent to the equations

$$\begin{aligned} F^+ (t)-F^- (t) =f(t) \quad \text {and}\quad F^+(t)+F^-(t)=\frac{1}{i\pi } \int _{\mathcal {L}} \frac{f(s)}{s-t} \, \mathrm {d}s. \end{aligned}$$

These integrals are understood in the sense of the principal value, see [32, Chap. 1, Sect. 12].

Remark 13

(Sectionally holomorphic functions for a given discontinuity) Liouville’s theorem shows that the function F defined above is the unique sectionally holomorphic function ${\varPhi }$ satisfying the equation

$$\begin{aligned} \varPhi ^+ (t)-\varPhi ^- (t) =f(t), \quad \forall t\in \mathcal {L} \end{aligned}$$

and which vanishes at infinity. The solutions of this equation of finite degree at infinity are the functions such that

$$\begin{aligned} \varPhi =F+P \end{aligned}$$

where P is a polynomial.

Remark 14

(Behavior at the ends) It is possible to show that if $\mathcal {L}$ is an oriented open contour from end a to end b, then in the neighborhood of an end c it exists $F_c(z)$, an holomorphic function in the neighborhood of c, such that

$$\begin{aligned} F(z)=\frac{\varepsilon _c}{2i\pi } f(c)\log (z-c) + F_c(z) \quad \text {where}\quad \varepsilon _c= {\left\{ \begin{array}{ll} -\,1 &{}\quad \text {if } c=a,\\ 1 &{} \quad \text {if } c=b. \end{array}\right. } \end{aligned}$$

(27)

1.2 B.2. Riemann Boundary Value Problem

In a standard way, a boundary value problem is composed of a regularity condition on a domain and a boundary condition on that domain.

Definition 15

(Riemann BVP) We say that $\varPhi$ satisfies a Riemann BVP on $\mathcal {L}$ if:

• $\varPhi$ is sectionally holomorphic on $\mathbb {C}{\setminus }\mathcal {L}$ and admits $\varPhi ^+$ as left limit and $\varPhi ^-$ as right limit, $\varPhi$ if of finite degree at infinity;

• $\varPhi$ satisfies the boundary condition

  $$\begin{aligned} \varPhi ^+(t)=G(t)\varPhi ^-(t) +g(t), \quad t\in \mathcal {L} \end{aligned}$$

  where G and g are functions defined on $\mathcal {L}$.

We assume here that G and $g\in \mathbb {H}_\mu (\mathcal {L})$ and that G doesn’t cancel on $\mathcal {L}$. When $g=0$ we talk about a homogeneous Riemann BVP.

1.2.1 B.2.1. Closed Contour

We assume that the contour $\mathcal {L}$ is closed and we denote $\mathcal {L}^+$ the open bounded set of boundary $\mathcal {L}$, and $\mathcal {L}^-$ the complementary of $\mathcal {L}^+ \cup \mathcal {L}$.

Fig. 9

Oriented closed smooth contour $\mathcal {L}$, domains $\mathcal {L}^+$ and $\mathcal {L}^-$ and limit values of $\varPhi$ on the right and on the left

To solve the Riemann BVP we need to introduce the index

$$\begin{aligned} \chi :=\frac{1}{2i\pi } [\log G]_{\mathcal {L}} = \frac{1}{2\pi } [\arg G]_{\mathcal {L}} \end{aligned}$$

which quantifies the variation of the argument of G on the contour $\mathcal {L}$ in the positive direction. Without any loss of generality we assume that 0 is in $\mathcal {L}^+$. It is then possible to define the single-valued function

$$\begin{aligned} \log (t^{-\chi }G(t)), \quad t\in \mathcal {L} \end{aligned}$$

which satisfies the Hölder condition.

Proposition 16

(Solution of homogeneous Riemann BVP on a closed contour) Let us define

$$\begin{aligned} \varGamma (z):= \frac{1}{2i\pi } \int _{\mathcal {L}} \frac{\log (t^{-\chi }G(t))}{t-z} \, \mathrm {d}t, \quad z\notin \mathcal {L} \end{aligned}$$

and

$$\begin{aligned} X(z):= {\left\{ \begin{array}{ll} \exp \varGamma (z), &{}\quad z\in \mathcal {L}^+,\\ z^{-\chi }\exp \varGamma (z),&{} \quad z\in \mathcal {L}^-. \end{array}\right. } \end{aligned}$$

The function X is the fundamental solution of the homogeneous Riemann BVP of Definition 15, i.e., X satisfies the boundary condition $X^+(t)=G(t)X^-(t)$ for $t\in \mathcal {L}$. The function X is of degree $-\chi$ at infinity. If $\varPhi$ is a solution of the homogeneous Riemann BVP, then $\varPhi (z)=X(z)P(z)$ where P is a polynomial.

If we denote k the degree of P, the solution $\varPhi$ is of degree $k-\chi$ at infinity. The fundamental solution X of degree $-\chi$ is then the nonzero homogeneous solution of smallest degree to infinity.

Proof

For $t\in \mathcal {L}$, let us denote $\widetilde{\varGamma } (t)= \frac{1}{2i\pi } \int _{\mathcal {L}} \frac{\log (s^{-\chi }G(s))}{s-t} \, \mathrm {d}s$ where the integral is understood in the sense of principal value. Sokhotski–Plemelj formulas applied at $\varGamma$ show that

$$\begin{aligned} X^+(t)= e^{\widetilde{\varGamma }(t)} \sqrt{t^{-\chi }G(t)} \quad \text {and} \quad X^-(t)=t^{-\chi } e^{\widetilde{\varGamma }(t)} \frac{1}{\sqrt{t^{-\chi }G(t)}} \quad \text {for } t\in \mathcal {L}, \end{aligned}$$

(28)

and then that X is a solution of the homogeneous problem. If $\varPhi$ is a solution of the problem, as $X^\pm (z) \ne 0$ for $z\in \mathcal {L}$ we obtain

$$\begin{aligned} \frac{\varPhi ^+}{X^+}(z)=\frac{\varPhi ^-}{X^-}(z), \quad z\in \mathcal {L}. \end{aligned}$$

By analytic continuation the function $\frac{\varPhi }{X}$ is then holomorphic in the whole complex plane, is of finite degree at infinity and is then a polynomial according to Liouville’s theorem. $\square$

Proposition 17

(Solution of Riemann BVP on a closed contour) We define

$$\begin{aligned} \varphi (z):= \frac{1}{2i\pi } \int _{\mathcal {L}} \frac{g(t)}{X^+ (t)(t-z)} \, \mathrm {d}t, \quad z\notin \mathcal {L}. \end{aligned}$$

The solutions of the Riemann BVP of Definition 15 are the functions such that

$$\begin{aligned} \varPhi (z)=X(z)\varphi (z)+X(z)P_{\chi }(z) \end{aligned}$$

where $P_{\chi }$ is a polynomial of degree $\chi$ for $\chi \geqslant 0$ and $P_{\chi }=0$ for $\chi \leqslant -1$.

Remark 18

(Left limit $X^+$) We have $X^+(t) =(t-b)^{-\chi } e^{\varGamma ^+(t)}$ where $\varGamma ^+(t)$ is the left limit value of $\varGamma$ on $\mathcal {L}$ given by the Sokhotski–Plemelj formulas of Proposition 12, see (28).

Remark 19

(Solubility conditions) For $\chi < -1$ the solutions are holomorphic at infinity (and then bounded) if and only if the following conditions are satisfied:

$$\begin{aligned} \int _{\mathcal {L}} \frac{g(t) t^{k-1}}{X^+(t)} \, \mathrm {d}t=0, \quad k=1, \cdots , -\chi -1. \end{aligned}$$

(29)

Proof

The fundamental solution $X^\pm$ does not cancel on $\mathcal {L}$ and we have the factorization $G=\frac{X^+}{X^-}$. If $\varPhi$ is a solution of the BVP we have

$$\begin{aligned} \frac{\varPhi ^+}{X^+}(t)=\frac{\varPhi ^-}{X^-}(t)+\frac{g}{X^+}(t), \quad t\in \mathcal {L}. \end{aligned}$$

The function $\frac{\varPhi }{X}$ being of finite degree at infinity, Remark 13 gives $\frac{\varPhi }{X}=\varphi +P$. $\square$

1.2.2 B.2.2. Open Contour

We assume that the function $\varPhi$ we are looking for satisfies the Riemann BVP on an open contour oriented from end a to end b and that $\varPhi$ is bounded at the neighborhood of a and b. More generally, one could look for the solutions admitting singularities integrable at the ends. We denote $\delta$, $\varDelta$, $\rho _a$ and $\rho _b$ such that

$$\begin{aligned} G(a)=\rho _a e^{i\delta }, \quad \varDelta = [\arg G]_{\mathcal {L}} \quad \text {et} \quad G(b)=\rho _b e^{i(\delta +\varDelta )} \end{aligned}$$

choosing $-2\pi < \delta \leqslant 0$ and the corresponding determination of the logarithm $\log G$. We define the index

$$\begin{aligned} \chi := \left\lfloor \frac{\delta +\varDelta }{2\pi } \right\rfloor . \end{aligned}$$

Proposition 20

(Solution of Riemann BVP on an open contour) Let us define

$$\begin{aligned} \varGamma (z):= \frac{1}{2i\pi } \int _{\mathcal {L}} \frac{\log (G(t))}{t-z} \, \mathrm {d}t, \quad z\notin \mathcal {L}. \end{aligned}$$

The function

$$\begin{aligned} X(z):= (z-b)^{-\chi } e^{\varGamma (z)} \end{aligned}$$

is a solution of the homogeneous Riemann BVP and is bounded at the ends. This solution is of order $-\chi$ at infinity. If $\varPhi$ is a solution of the homogeneous problem, it may be written as $\varPhi (z)=X(z)P(z)$ where P is a polynomial. We define

$$\begin{aligned} \varphi (z):= \frac{1}{2i\pi } \int _{\mathcal {L}} \frac{g(t)}{X^+ (t)(t-z)} \, \mathrm {d}t, \quad z\notin \mathcal {L}. \end{aligned}$$

The solutions of the Riemann BVP bounded at the ends are the functions

$$\begin{aligned} \varPhi (z)=X(z)\varphi (z)+X(z)P_{\chi }(z) \end{aligned}$$

where $P_{\chi }$ is a polynomial of degree $\chi$ for $\chi \geqslant 0$ and $P_{\chi }=0$ for $\chi \leqslant -1$.

Proof

Due to Remark 14, in the neighborhood of one end c we have

$$\begin{aligned} e^{\varGamma (z)}=(z-c)^{\lambda _c} e^{\varGamma _c(z)} \end{aligned}$$

for $\varGamma _c$ a holomorphic function in the neighborhood of c and

$$\begin{aligned} \lambda _a =-\frac{\delta }{2\pi }+i\frac{\log \rho _a}{2\pi } \quad \text {and} \quad \lambda _b =\frac{\delta +\varDelta }{2\pi }-i\frac{\log \rho _b}{2\pi }. \end{aligned}$$

Since $\delta \leqslant 0$ the function $e^{\varGamma (z)}$ is bounded at a. Furthermore, we notice that the function $X(z)=(z-b)^{-\chi } e^{\varGamma (z)}$ is bounded at b (and at a). The rest of the proof is similar to the closed contour case. $\square$

1.3 B.3. Carleman Boundary Value Problem with Shift

A shift $\alpha (t)$ is a homeomorphism from the contour $\mathcal {L}$ on itself such that its derivative does not cancel and which satisfies Hölder’s condition. Most of the time the condition $\alpha (\alpha (t))=t$ is satisfied and we say that $\alpha$ is a Carleman automorphism of $\mathcal {L}$. In this paper the shift function is the complex conjugation.

Definition 21

(Carleman BVP) The function $\varPhi$ satisfies a Carleman BVP on the closed contour $\mathcal {L}$ (or having its two ends at infinity, as in this paper) if:

• $\varPhi$ is holomorphic on the whole domain $\mathcal {L}^+$ bounded by $\mathcal {L}$ and continuous on $\mathcal {L}$;

• $\varPhi$ satisfies the boundary condition

  $$\begin{aligned} \varPhi (\alpha (t))=G(t)\varPhi (t) +g(t), \quad t\in \mathcal {L}, \end{aligned}$$

  where G and g are two functions defined on $\mathcal {L}$.

We will assume that G and $g\in \mathbb {H}_\mu (\mathcal {L})$ and that G does not cancel on $\mathcal {L}$. When $g=0$ the Riemann BVP is said to be homogeneous.

To solve the Carleman BVP we introduce a conformal glueing function. The following result establishes the existence of such functions.

Proposition 22

(Conformal glueing function) Let $\alpha$ be a Carleman automorphism of the curve $\mathcal {L}$. It exists W, a function

• holomorphic on $\mathcal {L}^+$ deprived of one point where W has a simple pole;

• satisfying the glueing condition

  $$\begin{aligned} W(\alpha (t))=W(t), \quad t\in \mathcal {L}. \end{aligned}$$

  Such a function W establishes a conformal transform (holomorphic bijection) from $\mathcal {L}^+$ to the complex place deprived of a smooth open contour $\mathcal {M}$. This conformal glueing function admits two fixed points A and B of image a and b which are the ends of $\mathcal {M}$.

If we find such a conformal glueing function, we can transform the Carleman BVP into a Riemann BVP. We orient $\mathcal {M}$ from a to b choosing it such that the orientation of $\mathcal {L}$ be conserved by W. We then denote $W^{-1}$ the reciprocal of W and $(W^{-1})^+$ its left limit and $(W^{-1})^-$ its right limit on $\mathcal {M}$. See Fig. 10. For t on the arc $\mathcal {L}$ oriented from B to A, these functions satisfy

$$\begin{aligned} (W^{-1})^+(W(t))=\alpha (t) \quad \text {and}\quad (W^{-1})^-(W(t))=t. \end{aligned}$$

Fig. 10

Conformal glueing function from $\mathcal {L}^+$ to $\mathbb {C}{\setminus } \mathcal {M}$

Let $\varPhi$ be a solution of the Carleman BVP, we define the function $\varPsi$ such that

$$\begin{aligned} \varPsi (W(z)) := \varPhi (z), \quad z\in \mathcal {L}^+. \end{aligned}$$

We then have

$$\begin{aligned} \varPsi (z)= \varPhi (W^{-1}(z)), \quad z\in \mathbb {C}{\setminus } \mathcal {M} \end{aligned}$$

and the limits on the left and on the right of $\varPsi$ on $\mathcal {M}$ are

$$\begin{aligned} \varPsi ^+(t)= \varPhi ((W^{-1})^+(t)) \quad \text {and} \quad \varPsi ^-(t)= \varPhi ((W^{-1})^-(t)), \quad t\in \mathcal {M}. \end{aligned}$$

Let

$$\begin{aligned} H(t)=G((W^{-1})^-(t))\quad \text {and} \quad h(t)= g((W^{-1})^-(t)), \quad t\in \mathcal {M}. \end{aligned}$$

Proposition 23

The function $\varPsi$ satisfies the following Riemann BVP associated to the contour $\mathcal {M}$ and to the functions H and h:

• $\varPsi$ is sectionally holomorphic on $\mathcal {C}{\setminus }\mathcal {M}$;

• $\varPsi$ satisfies the boundary condition

  $$\begin{aligned} \varPsi ^+(t)=H(t)\varPsi ^-(t) +h(t), \quad t\in \mathcal {M}. \end{aligned}$$

Proof

The proof derives from Definition 21, from Proposition 22 and from the above notations. $\square$

As $\varPhi =\varPsi \circ W$, to solve the Carleman BVP of Definition 21, it is enough to determine the conformal glueing function W and to find $\varPsi$ thanks to Section B.2 which explains how to solve the Riemann BVP Proposition 23. Let us define

$$\begin{aligned} X(W(z)):= (W(z)-b)^{-\chi } \exp \left( \frac{1}{2i\pi } \int _{\mathcal {L}_d} \log (G(t))\frac{ W'(t)}{W(t)-W(z)} \, \mathrm {d}t \right) , \quad z\notin \mathcal {L} \end{aligned}$$

and

$$\begin{aligned} \varphi (W(z)):= \frac{-1}{2i\pi } \int _{\mathcal {L}_d} \frac{g(t)}{X^+ (W(t))} \frac{W'(t) }{(W(t)-W(z))} \, \mathrm {d}t, \quad z\notin \mathcal {L}, \end{aligned}$$

where we denote $\mathcal {L}_d=(W^{-1})^- (\mathcal {M})$ (the red curve on the left picture of Fig. 10). We obtain the following proposition.

Proposition 24

(Solution of Carleman BVP) The solutions of the Carleman BVP of Definition 21 are given by

$$\begin{aligned} \varPhi (z)=X(W(z))\varphi (W(z))+X(W(z))P_{\chi }(W(z)) \end{aligned}$$

(30)

where $P_{\chi }$ is a polynomial of degree $\chi$ for $\chi \geqslant 0$ and where $P_{\chi }=0$ for $\chi \leqslant -1$. For $\chi < -1$ the solution to the non-homogeneous problem exists if and only if some solubility conditions of the form (29) are satisfied.

Appendix C. Green’s Functions in Dimension One

This appendix is intended to be an educational approach that illustrates in a simple case the analytical method and the link between Green’s functions and partial differential equations. In this section we study X a Brownian motion (in dimension one) with drift reflected at 0. We are looking for Green’s functions of X. This problem has already been studied in [13]. Here, we solve this question thanks to an analytic study which is much simpler than in dimension two.

Definition 25

(Reflected Brownian motion with drift) We define X, a reflected Brownian motion of variance ${\sigma ^2}$, of drift $\mu$ and starting from $x_0\in \mathbb {R}_+$, as the semi-martingale satisfying the equation

$$\begin{aligned} {{X(t)}}=x_0 + {\sigma } W{(t)}+\mu t + {{L(t)}}, \end{aligned}$$

where L(t) is the (symmetric) local time in 0 of X(t) and $W_t$ is a standard Brownian motion.

Definition 26

(Green measures) Let $A\subset \mathbb {R}$ be a measurable set. Green’s measure of the process X starting from $x_0$ is defined by

$$\begin{aligned} {G({x_0},A)} = \mathbb {E}_{x_0} \left( \int _0^\infty \mathbb {1}_A ({{X(s)}}) \, \mathrm {d} s \right) = \int _0^\infty \mathbb {P}_{x_0} ({{X(s)}} \in A ) \, \mathrm {d} s. \end{aligned}$$

Its density with respect to the Lebesgue measure is denoted $g(x_0,x)$ and is called Green’s function. Green’s function satisfies

$$\begin{aligned} g(x_0,x) = \int _0^\infty p(t,x_0,x) \, \mathrm {d} t, \end{aligned}$$

where $p(t,x_0,x)$ is the transition density of the process X.

If $\mu >0$, the process is transient. In this case, $G({x_0},A)<\infty$ for bounded subset $A\subset \mathbb {R}_+$. Furthermore, notice that if $f:\mathbb {R\rightarrow \mathbb {R}_+}$ is measurable, by Fubini’s theorem we have

$$\begin{aligned} \int _{\mathbb {R}} f(x) \ g(x_0,x) \, \mathrm {d} x =\mathbb {E}_{x_0} \left[ \int _0^{\infty } f({{X(t)}}) \, \mathrm {d} t \right] . \end{aligned}$$

Proposition 27

(Green’s functions and Laplace transform) If $\mu >0$, for all $x\in \mathbb {R}_+$ Green’s function of X is equal to

$$\begin{aligned} g(x_0,x) = \frac{1}{\mu }e^{\frac{2\mu }{{\sigma ^2}} (x-x_0)} \mathbf {1}_{ \{0 \leqslant x < x_0\} } + \frac{1}{\mu } \mathbf {1}_{ \{ x_0 \leqslant x \} } \end{aligned}$$

(31)

and its Laplace transform $\psi ^{x_0}$ is equal to

$$\begin{aligned} \psi ^{x_0} (\theta ) :=\int _0^\infty e^{\theta x} g(x_0,x) \, \mathrm {d}x = - \frac{e^{\theta x_0} +\theta \frac{{\sigma ^2}}{2\mu } e^{-\frac{2\mu }{{\sigma ^2}} x_0}}{\mu \theta +\frac{1}{2}{\sigma ^2} \theta ^2}. \end{aligned}$$

Proof

As in the two dimensional case, we are going to determine the Laplace transform of Green’s function thanks to a functional equation. If f is a function $\mathcal {C}^2$, Itô formula gives

$$\begin{aligned} f({{X(t)}})&= f(x_0) + \int _0^t f' ({{X(s)}}) \, \mathrm {d} {{X(t)}} + \frac{1}{2} \int _0^t f'' ({{X(s)}}) \, \mathrm {d} \langle X \rangle _s\\&= f(x_0) + \int _0^t f' ({{X(s)}}) ( \mathrm {d} W(t) + \mu \mathrm {d} t + \mathrm {d} {{L(t)}} ) + \frac{1}{2} \int _0^t f'' ({{X(s)}}) \sigma \, \mathrm {d} s. \end{aligned}$$

For $f(x)= e^{\theta x}$ and $\theta <0$ we take the expectation of this formula and we obtain

$$\begin{aligned} \mathbb {E}_{x_0} [e^{ \theta {{X(t)}}}]= & {} e^{\theta x_0} + \theta {\mathop {\underbrace{\mathbb {E}_{x_0} \left[ \int _0^t e^{\theta {{X(s)}}} \, \mathrm {d} W(s) \right] }}_{\begin{array}{c} =0 \text { because it is} \\ \text {the expectation of a martingale} \end{array}}}+ \left( \mu \theta +\frac{1}{2}{\sigma ^2} \theta ^2\right) \mathbb {E}_{x_0} \left[ \int _0^t e^{\theta {{X(s)}}} \, \mathrm {d} s \right] \\&+ \theta \mathbb {E}_{x_0} \left[ \int _0^t e^{\theta {{X(s)}}} \, \mathrm {d} {{L(s)}} \right] . \end{aligned}$$

As $\theta <0$ and as ${{X(t)}} \underset{t \rightarrow \infty }{\longrightarrow } \infty$ (as $\mu >0$), we have $\mathop {\lim }\nolimits _{t \rightarrow \infty } \mathbb {E}[e^{ \theta {{X(t)}}}] =0$. Let t tend to infinity. We obtain

$$\begin{aligned} 0&= e^{\theta x_0} + \left( \mu \theta +\frac{1}{2}{\sigma ^2} \theta ^2\right) \mathbb {E}_{x_0} \left[ \int _0^\infty e^{\theta {{X(s)}}} \, \mathrm {d} s \right] + \theta \mathbb {E}_{x_0} \left[ \int _0^\infty e^{\theta {{X(s)}}} \, \mathrm {d} {{L(s)}} \right] \\&= e^{\theta x_0} + \left( \mu \theta +\frac{1}{2}{\sigma ^2} \theta ^2\right) \psi ^{x_0} (\theta ) + \theta \mathbb {E} L({\infty }) \end{aligned}$$

as $e^{\theta {{X(s)}}}=1$ on the support of $\mathrm {d} {{L(s)}}$ which is the set $\{s \geqslant 0: {{X(s)}} =0 \}$. By evaluating at $\theta ^* = -\,2\mu / {\sigma ^2}$ we find that $\mathbb {E} L({\infty }) = - \frac{e^{\theta ^* x_0}}{\theta ^*} = \frac{{\sigma ^2} }{2\mu }e^{-\frac{2\mu }{{\sigma ^2}} x_0}$. We obtain

$$\begin{aligned} \psi ^{x_0} (\theta ) = - \frac{e^{\theta x_0} +\theta \frac{{\sigma ^2}}{2\mu } e^{-\frac{2\mu }{{\sigma ^2}} x_0}}{\mu \theta +\frac{1}{2}{\sigma ^2}\theta ^2}. \end{aligned}$$

Inverting this Laplace transform we find formula (31). $\square$

Remark 28

(Partial differential equation) It is easy to verify that $g(x_0,x)$ satisfies the following partial differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{{\sigma ^2}}{2}\frac{\partial ^2}{\partial x^2} g(x_0,x) -\mu \frac{\partial }{\partial x} g(x_0,x) = -\,\delta _{x_0} (x), &{} \\ {\sigma ^2}\frac{\partial }{\partial x} g(x_0,0) -2\mu g(x_0,0) =0, &{} \end{array}\right. } \end{aligned}$$

(32)

which is similar to Eq. (3) in dimension two.

Appendix D. Generalization to a Non-positive Drift

In this paper, results are obtained for a positive drift: $\mu _1>0$ and $\mu _2>0$. In this appendix, we explain how to generalize these results to transient cases with a non-positive drift, that is when $\mu _1\leqslant 0$ or $\mu _2\leqslant 0$. First of all, in these cases the ellipse $\gamma =0$ is oriented differently, see Fig. 11.

Fig. 11

On the left $\mu _1<0$ and $\mu _2>0$, on the right $\mu _1<0$ and $\mu _2<0$

This leads to another set of convergence for the moment generating function. This is the main difference with the case of a positive drift. Analogously to Proposition 5, we can show that

• when $\mu _1>0$ and $\mu _2\leqslant 0$:

  • $\psi _1(\theta _2)$ is finite on $\{\theta _2\in \mathbb {C} : \mathfrak {R}\theta _2\leqslant \theta _2^{**} \wedge 0 \}$,

  • $\psi _2(\theta _1)$ is finite on $\{\theta _1\in \mathbb {C} : \mathfrak {R}\theta _1 < 0 \}$,

  • $\psi (\theta )$ is finite on $\{\theta \in \mathbb {C}^2 : \mathfrak {R}\theta _1 < 0 \text { and } \mathfrak {R}\theta _2\leqslant \theta _2^{**} \wedge 0 \}$;

• when $\mu _1 \leqslant 0$ and $\mu _2>0$:

  • $\psi _1(\theta _2)$ is finite on $\{\theta _2\in \mathbb {C} : \mathfrak {R}\theta _2<0 \}$,

  • $\psi _2(\theta _1)$ is finite on $\{\theta _1\in \mathbb {C} : \mathfrak {R}\theta _1 \leqslant \theta _1^{*} \wedge 0 \}$,

  • $\psi (\theta )$ is finite on $\{\theta \in \mathbb {C}^2 : \mathfrak {R}\theta _2 < 0 \text { and } \mathfrak {R}\theta _1 \leqslant \theta _1^{*} \wedge 0 \}$;

• when $\mu _1<0$ and $\mu _2<0$:

  • $\psi _1(\theta _2)$ is finite on $\{\theta _2\in \mathbb {C} : \mathfrak {R}\theta _2< 0 \}$,

  • $\psi _2(\theta _1)$ is finite on $\{\theta _1\in \mathbb {C} : \mathfrak {R}\theta _1 < 0 \}$,

  • $\psi (\theta )$ is finite on $\{\theta \in \mathbb {C}^2 : \mathfrak {R}\theta _1< 0 \text { and } \mathfrak {R}\theta _2< 0 \}$.

In these sets the same functional Eq. (10) still holds. As in Lemmas 6 and 7 but with some small technical differences in the proofs, it is then possible to continue the function $\psi _1$. We can therefore establish the same BVP as in Lemma 8. The resolution of this BVP is similar and leads to the same formula as (24). This generalization is the same phenomenon explained in [31, Sect. 3.6].