t-Martin Boundary of Killed Random Walks in the Quadrant

Abstract

We compute the t-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete t-harmonic functions. Our approach is uniform in t, and shows that there are three regimes for the Martin boundary.

Appendix: Explicit Expressions for w

It turns out that a suitable expression for w has been found in [6, Sect. 5.5.2]. Let us briefly explain why such a function appears in [6]. The main goal of Fayolle et al. [6] is to develop a theory for solving a functional equation satisfied by the stationary probabilities generating function of reflected random walks in the quarter plane. The functional equation in [6] is closed to ours [compare [6, Eq. (1.3.6)] with (6)]. Roughly speaking, the general solution of Fayolle et al. [6] can be expressed as

$$\displaystyle{ \int f(y) \frac{w'(y)} {w(x) - w(y)}\text{d}y }$$

for some function f, with the same function w as ours. Our situation is therefore simpler, since we can express the solutions directly in terms of w, without any integral.

In this section we simply state the expression of w, and we refer to [6, Chap. 5] or to [12, Sect. 3] for the details. The expression is

$$\displaystyle{ w(x) = \frac{u(x_{0})} {u(x) - u(x_{0})} - \frac{u(x_{0})} {u(0) - u(x_{0})} }$$

(19)

(the second term $\frac{u(x_{0})} {u(0)-u(x_{0})}$ in (19) is to ensure that w(0) = 0), where the function u is different in the two cases t ∈ (t ₀, ∞) and t = t ₀.

1.1 Case t ∈ (t ₀, ∞)

In that case, u can be expressed in terms of Weiertrass elliptic functions, with the formula

$$\displaystyle{ u(x) = \wp _{1,3}(s^{-1}(x) -\omega _{ 2}/2), }$$

(20)

where

• ℘ _1, 3 is the Weierstrass elliptic function associated with the periods ω ₁ and ω ₃ defined in (21), i.e.,

  $$\displaystyle{ \wp _{1,3}(\omega ) = \frac{1} {\omega ^{2}} +\sum _{n_{1},n_{3}\in \mathbf{Z}}\left \{ \frac{1} {(\omega -n_{1}\omega _{1} - n_{3}\omega _{3})^{2}} - \frac{1} {(n_{1}\omega _{1} + n_{3}\omega _{3})^{2}}\right \}, }$$

• ω ₁ and ω ₂ are defined as below [with δ as in (8)]:

  $$\displaystyle{ \omega _{1} = i\int _{x_{1}}^{x_{2} } \frac{\text{d}x} {\sqrt{-\delta (x)}},\qquad \omega _{2} =\int _{ x_{2}}^{x_{3} } \frac{\text{d}x} {\sqrt{\delta (x)}},\qquad \omega _{3} =\int _{ X(y_{1})}^{x_{1} } \frac{\text{d}x} {\sqrt{\delta (x)}}, }$$

  (21)

• s(ω) = g ⁻¹(℘ _1, 2(ω)), where ℘ _1, 2 is the Weierstrass elliptic function associated with the periods ω ₁ and ω ₂, and g ⁻¹ is the reciprocal function of

  $$\displaystyle{ g(x) = \left \{\begin{array}{lll} \frac{\delta ''(x_{4})} {6} + \frac{\delta '(x_{4})} {x - x_{4}} & \text{if}&x_{4}\neq \infty, \\ \frac{\delta ''(0)} {6} + \frac{\delta '''(0)x} {6} &\text{if}&x_{4} = \infty,\end{array} \right. }$$

  (22)

• x ₀ ∈ (X(y ₁), x ₂)∖{0} is arbitrary.

1.2 Case t = t ₀

We have

$$\displaystyle{ u(x) = \left ( \frac{\pi } {\omega _{3}}\right )^{2}\left \{\sin \left (\frac{\pi } {\theta }\left [\arcsin \left ( \frac{1} {\sqrt{\frac{1} {3} -\frac{2g(x)} {\delta ''(1)}} }\right ) - \frac{\pi } {2}\right ]\right )^{-2} -\frac{1} {3}\right \}, }$$

(23)

with g as in (22) and

$$\displaystyle{ \theta =\arccos \left (-\frac{\sum _{-1\leq i,j\leq 1}ijp_{i,j}x_{2}^{i}y_{2}^{\,j}} {2\sqrt{\alpha (x_{2 } )\tilde{\alpha }(y_{2 } )}} \right ). }$$

1.3 Remarks

It can be shown that:

• The expressions given in (20) and (23) are a priori complicated, but it may happen that for some { p _k, ℓ }, they become much simpler. If p _0, 1 + p _1, 0 + p _0, −1 + p _−1, 0 = 1 for instance, the function u is rational. More generally, if ω ₂∕ω ₃ ∈ Q, then u is an algebraic function. See [12, Proposition 15 and Remark 16] for further remarks on u.

• The function u is continuous w.r.t. the eigenvalue t ∈ [t ₀, ∞), see [5, Sect. 2.2].

• At t = t ₀ we have x ₂ = x ₃ and y ₂ = y ₃ (in fact t ₀ = inf{t > 0: x ₂ = x ₃} = inf{t > 0: y ₂ = y ₃}).