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Reflected BM in the quarter planeAnalytic approach
Results
Analytic approach for reflected Brownian motionin the quadrant
AofA 2016
Sandro FranceschiJoint work with Irina Kourkova and Kilian Raschel
LPMA, University Pierre et Marie Curie & LMPT, University of Tours
Krakow, Poland, July 4–8, 2016
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Introduction
Random processes in the quarter plane:
Discrete case (random walk) is studied a lot, remarkable exactformulas exist, it is popular in combinatorics.
Continuous case (Brownian motion) serves as anapproximation of large queuing networks.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Introduction
Goals:
Extend to the continuous case the analytic methoddevelopped by Malyshev in the seventies for the discrete case,
Compute explicitely generating function of the stationarydistribution thanks to boundary value problems,
Study the asymptotics of the stationary distributionthanks to saddle point methods.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
1 Reflected BM in the quarter planeReflected Brownian motionStationary distributionGenerating functions
2 Analytic approachFunctional equationRiemann surfaceKey steps
3 ResultsBoundary value problemResolution and explicit expressionAsymptotics
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Plan
1 Reflected BM in the quarter planeReflected Brownian motionStationary distributionGenerating functions
2 Analytic approachFunctional equationRiemann surfaceKey steps
3 ResultsBoundary value problemResolution and explicit expressionAsymptotics
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Reflected brownian motion in R2+
Let(Wt)t∈R+ a planar brownian motion of covariance matrix Σ
µ =
(µ1
µ2
)∈ R2 a drift
R = (R1,R2) =
(r11 r12
r21 r22
)∈ R2×2 a reflection matrix
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Reflected Brownian motion in R2+
Definition
Let us define Bt the reflected Brownian motion in the quadrant as
Bt = B0 + Wt + µt + RLt ∈ R2+
where Lit is a continuous non-decreasing process, that increasesonly when the process touches the boundary. (Lt is a local time)
Theorem (Reiman, Taylor,Williams, 1988 and 1993)
Such a process exists iffr11 > 0, r22 > 0 and eitherr12, r21 > 0 or r11r22− r12r21 > 0.
Process doesn’t exist Process exists
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Reflected Brownian motion in R2+
Definition
Let us define Bt the reflected Brownian motion in the quadrant as
Bt = B0 + Wt + µt + RLt ∈ R2+
where Lit is a continuous non-decreasing process, that increasesonly when the process touches the boundary. (Lt is a local time)
Theorem (Reiman, Taylor,Williams, 1988 and 1993)
Such a process exists iffr11 > 0, r22 > 0 and eitherr12, r21 > 0 or r11r22− r12r21 > 0.
Process doesn’t exist Process exists
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Recurrence criterion
Definition
Bt is said to be recurrent positive if for all neighbourhood of zeroV ⊂ R2
+ we have E[τV ] <∞ where τV = inf{t > 0 : Bt ∈ V }.
Proposition (D. Hobson and L. Rogers, 1993)
The process and its stationary distribution exist and is unique iff:
r11 > 0, r22 > 0, r11r22 − r12r21 > 0,
r22µ1 − r12µ2 < 0, r11µ2 − r21µ1 < 0.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Recurrence criterion
r11 > 0, r22 > 0, r11r22− r12r21 > 0, r22µ1− r12µ2 < 0, r11µ2− r21µ1 < 0.
Recurrentcases
Transientcases
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Stationary distribution and boundaries
Let π be the stationary distribution (or invariant measure) on R2+
of the relecting Brownian motion B.
Thanks to ergodic theorems, the invariant measure of the setA ∈ R2
+ is the average of the time proportion spent in A:
π(A) = limt→∞
E[1
t
∫ t
01A(Bu)du]
What about the boundaries?
We define ν1 a measure on aboundary, such that for A ∈ {0} × R
ν1(A) = Eπ[1
t
∫ t
01A(Bu)dL1
u].
Similarly we define ν2. We notice that ν1 et ν2 are a kind ofstationary distribution (or invariant measure) on the boundaries.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Stationary distribution and boundaries
Let π be the stationary distribution (or invariant measure) on R2+
of the relecting Brownian motion B.
Thanks to ergodic theorems, the invariant measure of the setA ∈ R2
+ is the average of the time proportion spent in A:
π(A) = limt→∞
E[1
t
∫ t
01A(Bu)du]
What about the boundaries? We define ν1 a measure on aboundary, such that for A ∈ {0} × R
ν1(A) = Eπ[1
t
∫ t
01A(Bu)dL1
u].
Similarly we define ν2. We notice that ν1 et ν2 are a kind ofstationary distribution (or invariant measure) on the boundaries.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Generating function of the stationary distribution
In the discrete case the generating functions of the stationarydistribution πi ,j on Z2
+ is the generating series∑
Z2+πi ,jx
iy j .
In the continuous case the generating function of thestationary distribution is the Laplace transform:
φ(θ) = φ(θ1, θ2) =
∫∫R2
+
eθ1x+θ2yπ(x , y)dxdy
On the boundaries we define in an analogous way thefollowing generating functions:
φ2(θ1) =
∫R+
eθ1xν2(x)dx
φ1(θ2) =
∫R+
eθ2yν1(y)dy
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Generating function of the stationary distribution
In the discrete case the generating functions of the stationarydistribution πi ,j on Z2
+ is the generating series∑
Z2+πi ,jx
iy j .
In the continuous case the generating function of thestationary distribution is the Laplace transform:
φ(θ) = φ(θ1, θ2) =
∫∫R2
+
eθ1x+θ2yπ(x , y)dxdy
On the boundaries we define in an analogous way thefollowing generating functions:
φ2(θ1) =
∫R+
eθ1xν2(x)dx
φ1(θ2) =
∫R+
eθ2yν1(y)dy
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Reflected Brownian motionStationary distributionGenerating functions
Generating function of the stationary distribution
In the discrete case the generating functions of the stationarydistribution πi ,j on Z2
+ is the generating series∑
Z2+πi ,jx
iy j .
In the continuous case the generating function of thestationary distribution is the Laplace transform:
φ(θ) = φ(θ1, θ2) =
∫∫R2
+
eθ1x+θ2yπ(x , y)dxdy
On the boundaries we define in an analogous way thefollowing generating functions:
φ2(θ1) =
∫R+
eθ1xν2(x)dx
φ1(θ2) =
∫R+
eθ2yν1(y)dy
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Plan
1 Reflected BM in the quarter planeReflected Brownian motionStationary distributionGenerating functions
2 Analytic approachFunctional equationRiemann surfaceKey steps
3 ResultsBoundary value problemResolution and explicit expressionAsymptotics
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Functional equation
This equation binds the different generating functions.
Theorem
γ(θ)φ(θ) = γ1(θ)φ1(θ2) + γ2(θ)φ2(θ1)
whereγ(θ) = −1
2 (σ11θ21 + σ22θ
22 + 2σ12θ1θ2)− (µ1θ1 + µ2θ2),
γ1(θ) = 〈R1|θ〉 = r11θ1 + r21θ2,
γ2(θ) = 〈R2|θ〉 = r12θ1 + r22θ2.
It is an equation which connects what happens inside the quarterplane and on its boundaries.The function γ is called the kernel.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Proof of the functional equation
Remark: The following relationships characterize the stationarydistribution in different cases:
π(P − I ) = 0 for Markov chains,
πQ = 0 for continuous time Markov chains,∫Gfdπ = 0 for Markov processes where G is the generator.
The counterpart for the reflected Brownian motion in the quadrantis the “basic adjoint relationship”:
∀f ∈ C2b(R2
+)
∫R2
+
Gf (z)π(dz) +∑i=1,2
∫R2
+
Di f (z)νi (dz) = 0
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Proof of the functional equation
Remark: The following relationships characterize the stationarydistribution in different cases:
π(P − I ) = 0 for Markov chains,
πQ = 0 for continuous time Markov chains,∫Gfdπ = 0 for Markov processes where G is the generator.
The counterpart for the reflected Brownian motion in the quadrantis the “basic adjoint relationship”:
∀f ∈ C2b(R2
+)
∫R2
+
Gf (z)π(dz) +∑i=1,2
∫R2
+
Di f (z)νi (dz) = 0
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
where the generator “inside” the quadrant is
Gf (z) =1
2
2∑i ,j=1
σi ,j∂2f
∂z1∂z2(z) +
2∑i=1
µi∂f
∂zi(z)
and for i = 1, 2 the generators on the boundaries are
Di f (x) = 〈R i |∇f 〉.
↪→ We just have to take f = e〈θ|.〉 in the basic adjoint relationshipto obtain the functional equation. Indeed∫
R2+
Ge〈θ|z〉π(dz) +∑i=1,2
∫R2
+
Die〈θ|z〉νi (dz) = 0
givesγ(θ)φ(θ) = γ1(θ)φ1(θ2) + γ2(θ)φ2(θ1).
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
where the generator “inside” the quadrant is
Gf (z) =1
2
2∑i ,j=1
σi ,j∂2f
∂z1∂z2(z) +
2∑i=1
µi∂f
∂zi(z)
and for i = 1, 2 the generators on the boundaries are
Di f (x) = 〈R i |∇f 〉.
↪→ We just have to take f = e〈θ|.〉 in the basic adjoint relationshipto obtain the functional equation. Indeed∫
R2+
Ge〈θ|z〉π(dz) +∑i=1,2
∫R2
+
Die〈θ|z〉νi (dz) = 0
givesγ(θ)φ(θ) = γ1(θ)φ1(θ2) + γ2(θ)φ2(θ1).
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Riemann surface
We introduce the Riemann surface S as the zeros of thekernel γ :
S = {(θ1, θ2) ∈ C2 : γ(θ1, θ2) = 0)}
On the Riemann surface S the first part of the functionalequation disappear and we have 0 = γ1φ1 + γ2φ2.Here S is a sphere. In the discrete case it’s a torus.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Functional equationRiemann surfaceKey steps
Key steps of the analytic method
Find a functional equation
Study the kernel and the Riemann surface (sphere, torus, ...)
Continue meromorphically the generating functions on theRiemann surface
Establish a boundary value problem to find explicit expressions
Study the singularities and use the saddle point method todetermine the asymptotics
Introduce the group of the process, its finitness is linked to thealgebraic nature of the solution
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Plan
1 Reflected BM in the quarter planeReflected Brownian motionStationary distributionGenerating functions
2 Analytic approachFunctional equationRiemann surfaceKey steps
3 ResultsBoundary value problemResolution and explicit expressionAsymptotics
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
What is a BVP with shift?
A boundary value problem with shift is made of two conditions:
a regularity condition on some set
a boundary condition with shift
Example:1 f is meromorphic on the
unit disc D and has onlyone pole of order one in 0
2 f (z) = f (z) for z ∈ U theunit circle
The solution of this BVP with shift is f (z) = z + 1z . It is a gluing
function which glues together the upper and the lower part of U.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
What is a BVP with shift?
A boundary value problem with shift is made of two conditions:
a regularity condition on some set
a boundary condition with shift
Example:1 f is meromorphic on the
unit disc D and has onlyone pole of order one in 0
2 f (z) = f (z) for z ∈ U theunit circle
The solution of this BVP with shift is f (z) = z + 1z . It is a gluing
function which glues together the upper and the lower part of U.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
What is a BVP with shift?
A boundary value problem with shift is made of two conditions:
a regularity condition on some set
a boundary condition with shift
Example:1 f is meromorphic on the
unit disc D and has onlyone pole of order one in 0
2 f (z) = f (z) for z ∈ U theunit circle
The solution of this BVP with shift is f (z) = z + 1z . It is a gluing
function which glues together the upper and the lower part of U.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Boundary of the BVP
R = {θ2 ∈ C : γ(θ1, θ2) = 0 where θ1 ∈ R− and θ2 /∈ R}
The curve R is an hyperbola symetric with respect to thex-axis (and GR is the blue domain).
If θ2 ∈ R and γ(θ1, θ2) = 0 then γ(θ1, θ2) = 0.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Statement of the BVP
Lemma
The function φ1 satisfy the following BVP with shift:
1 φ1 is meromorphic on GR with at most one pole p of order 1,and is bounded at infinity;
2 φ1 is continuous on GR \ {p} and
φ1(θ2) = G (θ2)φ1(θ2), ∀θ2 ∈ R.
where we have defined for θ2 ∈ R
G (θ2) =γ1
γ2(Θ−1 (θ2), θ2)
γ2
γ1(Θ−1 (θ2), θ2).
with the bi-valued algebraic function Θ−1 (θ2) associated to S
defined by γ(Θ−1 (θ2), θ2) = 0.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Statement of the BVP
Lemma
The function φ1 satisfy the following BVP with shift:
1 φ1 is meromorphic on GR with at most one pole p of order 1,and is bounded at infinity;
2 φ1 is continuous on GR \ {p} and
φ1(θ2) = G (θ2)φ1(θ2), ∀θ2 ∈ R.
where we have defined for θ2 ∈ R
G (θ2) =γ1
γ2(Θ−1 (θ2), θ2)
γ2
γ1(Θ−1 (θ2), θ2).
with the bi-valued algebraic function Θ−1 (θ2) associated to S
defined by γ(Θ−1 (θ2), θ2) = 0.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Proof of the BVP
Let θ2 ∈ R and θ1 such that (θ1, θ2) ∈ S and (θ1, θ2) ∈ S.Thus
0 = γ1(θ1, θ2)φ1(θ2) + γ2(θ1, θ2)φ2(θ1)
0 = γ1(θ1, θ2)φ1(θ2) + γ2(θ1, θ2)φ2(θ1)
We deduce that for θ2 ∈ R
⇒ φ1(θ2) =γ1
γ2(θ1, θ2)
γ2
γ1(θ1, θ2)︸ ︷︷ ︸
G(θ2)
φ1(θ2)
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Gluing function
The conformal gluing function w glues together the upper andlower parts of the hyperbola R. It can be expressed in terms of thegeneralized Chebyshev polynomial,
Ta(x) = cos(a arccos(x)) =1
2
{(x +√x2 − 1
)a+(x−√x2 − 1
)a}as follow:
w(θ2) = Tπβ
(−
2θ2 − (θ+2 + θ−2 )
θ+2 − θ
−2
),
where we put
β = arccos− σ12√σ11σ22
.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Resolution of the BVP: case of orthogonal reflexion
In the case of an orthogonal reflexion we canreduce the problem to a more simple BVPwith G = 1. Thanks to an invariant lemmaand to the gluing function we obtain:
Theorem (F. , Raschel, 2016)
Let R be the identity matrix. The Laplace transform φ1 is worth
φ1(θ2) =−µ1w
′(0)
w(θ2)− w(0)θ2,
where w is the gluing function.
In the general case we are able to find explicitly φ1 in termof Cauchy integral and the gluing function.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Resolution of the BVP: case of orthogonal reflexion
In the case of an orthogonal reflexion we canreduce the problem to a more simple BVPwith G = 1. Thanks to an invariant lemmaand to the gluing function we obtain:
Theorem (F. , Raschel, 2016)
Let R be the identity matrix. The Laplace transform φ1 is worth
φ1(θ2) =−µ1w
′(0)
w(θ2)− w(0)θ2,
where w is the gluing function.
In the general case we are able to find explicitly φ1 in termof Cauchy integral and the gluing function.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Asymptotics of the stationary distribution
Theorem (Dai-Miyazawa 2011)
If Dcx is the hatched set
π(Dcx )∼x→∞bxκc e−αcx
where constants αc and κc can be explicitly computed and κctakes one of the values : −3/2, −1/2, 0 ou 1.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Method to determine the asymptotics
We try to make an asymptoticdeveloppement in all directions, that isof π(r cosα, r sinα) when r →∞ andα→ α0.
Method:
Continue meromorphically the generating functions on S
Study the singularities on S
Inverse the Laplace transforms
Use saddle point methods on the Riemann surface S
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Method to determine the asymptotics
We try to make an asymptoticdeveloppement in all directions, that isof π(r cosα, r sinα) when r →∞ andα→ α0.
Method:
Continue meromorphically the generating functions on S
Study the singularities on S
Inverse the Laplace transforms
Use saddle point methods on the Riemann surface S
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Asymptotics in all directions
Theorem (F. , Kourkova, 2016)
Let α0 ∈ (0, π/2). When r →∞ and α→ α0 according to theparameters we have
π(r cosα, r sinα) = (1 + o(1)) ·
C0√re−r〈(cosα,sinα)|θ(α)〉,
C1e−r〈(cosα,sinα)|ηθ∗〉,
C2e−r〈(cosα,sinα)|ζθ∗∗〉,
where C0, C1 and C2 are constants which can be computed infunction of φ1, φ2 and the parameters.
The decay rates ηθ∗ and ζθ∗∗ come from the poles and θ(α) fromthe saddle point.We are able to find the full asymptotic development.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Asymptotics according to the direction
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Thank you for your attention!
J. G. DAI and M. MIYAZAWA - ”Reflecting brownian motion in two dimensions:Exacts asymptotics for the stationary distribution”, Stochastic Systems, 1 (2011), p.146-208.
G. FAYOLLE, R. IASNOGORODSKI and V. MALYSHEV - Random walks inthe quarter-plane, Application of Mathematics (New York), vol. 40, Springer, (1999).
S. FRANCESCHI and k. RASCHEL - ”Tutte-s invariant approach for Brownianmotion reflected in the quadrant”, arXiv: 1602.03054 (2016)
S. FRANCESCHI and I. KURKOVA - ”Asymptotic expansion of stationarydistribution for reflected brownian motion in the quadrant via analytic approach”arXiv:1604.02918 (2016)
I. KURKOVA and V. MALYSHEV - ”Martin boundary and elliptic curves”,Markov Process and Related Fields, 4 (1998), p. 203-272.
R. J. WILLIAMS - ”Semimartingale reflecting Brownian motions in the orthant.”,
Stochastic Networks , 13 (1995).
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Kernel
The kernel γ can be written asγ(θ1, θ2) = a(θ1)θ2
2 + b(θ1)θ2 + c(θ1). The two branches aregiven by
Θ±2 (θ1) =−b(θ1)±
√d(θ1)
2a(θ1),
where d(θ1) = b2(θ1)− 4a(θ1)c(θ1) is the discriminant.
The polynomial d has two roots, called θ±1 which are thebranching points of Θ2.
We notice that d is negative on (−∞, θ−1 ) ∪ (θ+1 ,∞).
Branches Θ±2 take complex conjugate values on this set.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
Reflected BM in the quarter planeAnalytic approach
Results
Boundary value problemResolution and explicit expressionAsymptotics
Group of the process
It is the group 〈ζ, η〉 generated by ζ and η, given by
ζ(θ1, θ2) =
(θ1,
c(θ1)
a(θ1)
1
θ2
), η(θ1, θ2) =
(c(θ2)
a(θ2)
1
θ1, θ2
).
By construction if γ(θ1, θ2) = 0 thenγ(ζ(θ1, θ2)) = γ(η(θ1, θ2)) = 0. They are automorphism of S.We have
ζ(θ1,Θ+2 (θ1)) = ζ(θ1,Θ
−2 (θ1))
The algebricity of generating functions depends of thefiniteness of the group.
Sandro Franceschi Analytic approach for reflected Brownian motion in the quadrant
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