國立臺灣大學 理學院 應用數學科學研究所 碩士論文
Institute of Applied Mathematical Sciences College of Science
National Taiwan University Master Thesis
複合更新理論應用在散射邊界上
An Application of Compound Renewal Theory on a Diffuse Boundary
蘇于瑞 Su, Yu-Jui
指導教授：陳逸昆 博士, 黃建豪 博士
Advisors: Chen, I-Kun , Ph.D. & Huang, Chien-Hao, Ph.D.
中華民國 109 年 7 月 July, 2020
Acknowledgements
I would like to express my deep gratitude to Professor Chen, I-Kun and Profes- sor Huang, Chien-Hao, my advisors, for their guidance, encouragement, consistent patience and useful recommendations. During these time, I think I’ve become a more genuine person.
I would also like to express my great appreciation to Professor Liu, Tai-Ping for his suggestion on the research topic.
Finally, I wish to thank my parents for their support through out my life.
Abstract
A transport equation with a diffuse boundary condition is studied for the propagation of a gas near vacuum. Our work reaches large time asymptotic results. Also, some stochastic processes concerning the structure of the differ- ential equation and the boundary condition were studied. The characteristic method leads to the explicit solution of the transport equation, which is a renewal function. The local limit theorem is our main tool studying this func- tion. When turning to the specific renewal function in our problem, more structures were revealed and the space-time relationship is given by extra properties of the second rate function.
Keywords: Diffuse Boundary Condition; Random Walks; Renewal The- ory; Regenerative Processes; Compound Renewal Processes; Large Deviations;
Local Limit Theorem; Second Rate Function
摘要
研究的是具有散射邊界條件的漂移方程，此模型描述在上半空間裡一團接近
真空的氣體。我們的工作得到了當時間趨近無窮的漸近結果。此外，研究了一些
與微分方程的結構和邊界條件有關的隨機過程。漂移方程的顯式解是一個更新函
數。局部極限定理是我們研究此函數的主要工具。我們問題中它有更多結構，因
此，這個漸進結果在不同的時空關係下由第二速率函數的額外屬性所決定。
關鍵字: 散射邊界條件;隨機漫步；更新理論；再生過程；複合更新過程；大偏
差理論；局部極限定理；第二速率函數。
Contents
List of Figures i
1 Introduction 1
1.1 Method of Characteristics, the Renewal Measure . . . 3 1.2 Numerical Computation . . . 6
2 Local Limit Theorem 9
2.1 Local Limit Theorem for Sums of Independent Random Vectors . . . 9 2.2 Local Limit Theorem for the Renewal Measure . . . 11
3 The Second Rate Function 13
3.1 Normal and Moderate Large Deviations . . . 15
4 Stochastic Processes 16
References 20
List of Figures
1 Contour of ρ(x, t) for 0 < t < 12 and −3 < x < 3. Started quite concentrated at the origin, it gradually spreads out as t grows. . . 7 2 Contour of Qn∗(x, t) for n = 1, 2, ..., 8, 0 < t < 20 and −5 < x < 5.
The center of Qn∗(x, t) shifts with a unit pace as n grows; meanwhile, their shapes gradually deform to a normal distribution. . . 7 3 The flow rateR∞
−∞ρ(x, t)dx and fτn∗(t), n = 1, 2, 3, ..., for t < 20. The flow rate R ρ(x, t)dx converges to a constant 1/Eτ = 2/√
π ≈ 1.128.
The peak point of fτn∗(t) is reached when t = nEτ . . . 8
1 Introduction
In the work of Liu and Yu (see [5]), a Boltzmann equation under a constant gravi- tational field on the upper half space is considered, for (x, t, v) ∈ H+× R+× R3,
∂tf + v · ∇xf − g · ∇vf = Q(f ) (1.1)
with Maxwell diffuse boundary condition, for x ∈ {z = 0}1
f (x, t, v)|v3>0 = ρ(x, t)p2π/θM(v)|v3>0, ρ(x, t) = ρ(x, y, t) := R
v3<0−v3f (x, t, v)dv,
where
x = (x, y, z) ∈ H+, g = (0, 0, g), g > 0, H+ := {x ∈ R3|z ≥ 0}, R+ := {t ≥ 0},
M (v) := e−|v|2/(2θ) (2πθ)3/2 .
Fix any time t, the quantity f (·, t, ·) is a probability density which gives the proba- bility per unit phase-space volume if it’s normalized.2 This equation models the dif- fusion of gas near vacuum that being reflected off a flat surface with the Maxwellian distribution and being pulled back to the surface by the gravitational force. The quantity ρ is the flux that goes into the boundary {z = 0}. Under the boundary condition, this inward flux equals to the flux that goes outward from the boundary.
The following theorem gives an upper bound estimate of the gas density f show- ing that the propagation of the gas is similar to a two-dimensional heat flow in the central (|x| = o(t)) part.
1We abuse the notation that when x ∈ {z = 0}, x denotes either (x, y, 0) or (x, y) according to the context.
2This is achieved when the initial data f (x, 0, v) is normalized by the fact that the solution f has a fixed total amount ||f ||t:=R f (x, t, v)dxdv, which is constant for any t ∈ R+.
Theorem 1.1 (LY[5]-B). There exists 0 > 0 and C0 > 0 such that for all 0 < < 0, if 0 ≤ f (x, 0, v) ≤ e−
√
x2+y2e−gz/θM (v), then the solution satisfies
0 ≤ f (x, t, v) ≤ C0
(e−gz/θM (v))3/4
1 1+te−
x2+y2 C0(1+t) + e−
√
x2+y2+t C0
p
e−gz/θM (v)
(1.2)
This theorem is obtained by first treating the collision term Q(f ) as a perturba- tion since the case of interest is the one near vacuum; thus, with Q(f ) omitted, a free transport equation
∂tf + v · ∇xf − g · ∇vf = 0 (1.3)
is studied first, which then leads to the following theorem. This theorem has a slightly stronger estimate than Theorem 1.1.
Theorem 1.2 (LY[5]-A). There exists C > 0 such that if |f (x, 0, v)| ≤ e−
√
x2+y2 e−gz/θM (v)β
for a given β ∈ (0, 1), then the solution ˜f of the transport equation satisfies
| ˜f (x, t, v)| = O(1) 1
1 + te−C(1+t)x2+y2 + e−
√
x2+y2+t C
e−gz/θM (v)β
.
Instead of upper bounds of f described in (1.2), our goal is to go further into its large time asymptotics. However, our current result is only on the transport part.
We leave the non-linear collision part for future research.
In the case of a free transport equation, the behaviour of f in H+× R+× R3 is completely determined by the boundary flux ρ and its initial data; thus, the problem is reduced to study ρ(x, t) in R2×R+. (By characteristic method, the result for f can be easily recovered from ρ.) The above Theorem 1.2 is obtained by estimates of upper bounds of ρ(x, t) in various regions of (x, t) (see [5, p.195]). Thus, asymptotics of f are reached when we have the corresponding asymptotics of ρ.
Theorem 1.3 (Main Result). Given a radial space-time ratio α = |x|/t, suppose the initial data satisfies f+(x, t) = O(e−λ(α)·(x,t)).3 Then
ρ(x, t) = ||f ||ψ1(D0(1, α))C(α)
t e−tD(1,α)(1 + o(1)) (1.4)
as t → ∞, where the remainder o(1) is uniform for all sufficiently small compact set K such that α ∈ K. The functions C, D are defined in Section 2.2 and λ is an increasing function defined in Section 3.
1.1 Method of Characteristics, the Renewal Measure
The transport equation can be solved explicitly by characteristic method. A char- acteristic curve (x(t), v(t)) satisfies dtdf (x(t), t, v(t)) = 0; thus, given t > t0,
v(t) =x0(t) = v(t0) − g(t − t0),
x(t) =x(t0) + v(t0) · (t − t0) − g(t − t0)2
2 .
Since f is invariant along a characteristic curve, for each point (x, v) ∈ H+ × R3 the value of f can be shifted along a characteristic curve to its boundary of domain, which are either {t = 0} and {z = 0}. In this way, separating the boundary flux ρ into two parts according to these two types of characteristic curves, we find that ρ satisfies a renewal equation. For x ∈ {z = 0},
3The initial flux f+ is defined in (1.7); ψ1 is the Laplace transform of f+ being the only term concerning the initial data in (1.4).
ρ(x, t) = Z
v3<0
−v3f (x, t, v)dv
= Z
v3<−gt/2
−v3f (x, t, v)dv + Z
−gt/2<v3<0
−v3f (x, t, v)dv
= Z
v3<−gt/2
−v3f (x − v1t, y − v2t, −v3t − g
2t2, 0, v + gt)dv +
Z
−gt/2<v3<0
−v3f x + (2v1v3/g, 2v2v3/g, 0), t + 2v3/g, v − 2(0, 0, v3)dv
= Z
v3<−gt/2
−v3f (x − v1t, y − v2t, −v3t − g
2t2, 0, v + gt)dv +
Z
−gt/2<v3<0
−v3ρ
x + 2v1v3
g , y +2v2v3
g , t + 2v3 g
r 2π
θ M
v − 2(0, 0, v3) dv
= f+(x, t) + Q(x, t) ∗ ρ(x, t), (1.5)
where the convolution ”∗” is taken on both x and t and 4
Q(x, t) = Q(x, y, t) :=
g2
8πθ2texp (−x2θt2+y22 −g8θ2t2), t > 0
0 t ≤ 0
, (1.6)
f+(x, t) :=
Z
v3<−gt/2
−v3f (x − v1t, y − v2t, −v3t − g
2t2, 0, v + gt)dv. (1.7)
If we interpret f+ as the flux caused by particles that never been reflected, then moving forward in time along its characteristic curve, any particle with finite initial speed will eventually reach the ”ground” {z = 0}. Thus, if f is integrable,
||f+|| :=
Z ∞ 0
Z
R2
f+dxdt = Z
H+×R3
f (x, 0, v)dxdv
= Z
H+×R3
f (x, t, v)dxdv =: ||f ||.
The renewal equation (1.5) can be solved uniquely (a proof can be found in e.g.
4The function Q(x, t) defined in [5] has a typo.
[6, p. 115]) with solution given by
ρ(x, t) =R ∗ f+(x, t), (1.8)
R(x, t) :=
∞
X
n=0
Qn∗(x, t), (1.9)
where Q0∗ is defined to be the Dirac delta function. R(x, t) is the renewal function associated with Q.
Being non-negative andR Qdxdt = 1, it can be treated as a probability density of a random vector ξ. Let ξi, i = 1, 2, ... be a sequence of iid copies of ξ; by an elementary property of convolution, Sn = Pn
i=1ξi is a random vector with the density Qn∗ for each n. Then the renewal function R(x, t) as sums of probability densities defines the renewal measure
H(B) :=
∞
X
n=0
P (Sn∈ B) = Z
B
R(x, t)dxdt (1.10)
for B ⊂ R2 × R+, which is finite for every bounded measurable set B if (see [4, p.654]) either
1. The expectation Eξ 6= 0 exists.
2. Eξ = 0, the second moment matrix EξTξ exists and has rank not less than 3.
In our case,
Eξ = Z
(x, y, t)Q(x, y, t)dxdydt = (0, 0,
√ 2πθ
g ) 6= 0.
In fact, since Q(x, t) = 0 for t ≤ 0, the third component ξ(3) > 0 almost surely and hence Sn(3) > 0 almost surely for each n. This gives a further structure of the renewal measure H. To give this third component a special treatment, we name it τ and write ξ = (ζ, τ ).
As the random variable τ is positive, the finite sums of its iid copies form a
sequence with strong order, which means it can be treated as an index and is referred as the renewal times. Some stochastic processes index by time t are defined later, which has a structure concerning these renewal times.
1.2 Numerical Computation
For small time t, the flux ρ(x, t) can be achieved numerically by computing the renewal function R(x, t) =P∞
n=0Qn∗(x, t).5 This subsection aims for gaining some intuitions by visualizing our problem.
For demonstration, the dimension of x is reduced to one (it’s easier to plot the result) and coefficients are scaled and normalized. Let
Q(x, t) = Q(x, t) := 2
√πexp (−x2 t2 − t2), fτ(t) :=
Z ∞
−∞
Q(x, t)dx = 2t exp (−t2).
The function Q chosen here leads to the same fτ as the function Q defined by (1.6) up to a scaling.6 It’s an easy observation to see that
Z
R(x, t)dx = XZ
Qn∗(x, t)dx =X
fτn∗(t).
Under suitable initial data7, the initial flux f+(x, t) can be chosen to be the Dirac delta function, so that the flux ρ(x, t) = P∞
n=0Qn∗(x, t) and the flow rate on R is R∞
−∞ρ(x, t)dx = P∞
n=0fτn∗(t). The density ρ(x, t) and Qn∗(x, t) are shown by contours in Figure 1,2, while Figure 3 shows the flow rate R∞
−∞ρ(x, t)dx and fτn∗(t), n = 1, 2, 3, ..., for x = 0.
5”small” depends on one’s computation power and required accuracy.
6Note that fτ can be treated as a probability density, which defines a random variable τ con-
Figure 1: Contour of ρ(x, t) for 0 < t < 12 and −3 < x < 3. Started quite concentrated at the origin, it gradually spreads out as t grows.
Figure 2: Contour of Qn∗(x, t) for n = 1, 2, ..., 8, 0 < t < 20 and −5 < x < 5.
The center of Qn∗(x, t) shifts with a unit pace as n grows; meanwhile, their shapes gradually deform to a normal distribution.
From Figure3, one can see the flow rateR ρ(x, t)dx becomes almost a constant after oscillating a few periods (with the fixed period Eτ ). The convergence of the
Figure 3: The flow rate R∞
−∞ρ(x, t)dx and fτn∗(t), n = 1, 2, 3, ..., for t < 20. The flow rateR ρ(x, t)dx converges to a constant 1/Eτ = 2/√
π ≈ 1.128. The peak point of fτn∗(t) is reached when t = nEτ .
flow rate can be shown by an elementary renewal theory (see e.g. [6, Sec 2.2]) that
t→∞lim N (t)/t = 1/Eτ a.s. (1.11)
where N (t) and τ are defined in Definition 4.1.
For small time t, the flux ρ(x, t) is dominated by a finite number of Qn∗(x, t) for n near t/Eτ i.e. letting m = bt/Eτ c, ρ(x, t) is well approximated by Pm+N
n=0 Qn∗(x, t) for some N < ∞ (with the error term of O(exp(−N ))) estimated by exponential Chebyshev’s inequality). This is why the numerical approximation is feasible. How- ever, the required N (upon an accuracy) grows as t grows.8 This is when the limit theorems kicks in.
8As can be seen that Qn∗(x, t) spreads out as n grows so more effective overlaps of Qn∗(x, t) on the t-large side of Figure3.
2 Local Limit Theorem
This section aims for the asymptotics of the renewal measure H defined in (1.10).
If f+ is the Dirac delta function, then ρ(x, t) is the density of H and it’s regarded as the homogeneous case. For general f+, the inhomogeneous renewal measure H1 is defined as H besides that among the iid sequence of random variables ξi, ξ1 is no longer being identical. Recall that ||f+|| := R f+dxdt = ||f || and assign the probability density of ξ1 with f+/||f ||, then ρ(x, t)/||f || becomes the density of H1.9 Let us begin with a more general problem formulation. Set ξ be a nondegenerate random vector in Rd, with distribution F , recall that Sn = Pn
i=1ξi, where ξi, i = 1, 2, ... is a sequence of iid copies of ξ.
2.1 Local Limit Theorem for Sums of Independent Random Vectors
Before going to the asymptotics of H, we should first study the asymptotics of Sn. Let
ψ(λ) := Eeλ·ξ, λ ∈ Rd
be the Laplace transform of ξ and A := {ψ < ∞} be its effective domain. For λ ∈ A, let F denotes the distribution of ξ, its Cram´er transform is defined such that
Fλ(dy) = eλ·y
ψ(λ)F (dy). (2.1)
9The normalization of f+ can be omit i.e. ξ1 need not be a random variable, doing so is just for an easier interpretation.
Denoted by ξ(λ) is the random vector with this tilted, or re-weighted distribution and Sn(λ) is defined accordingly. An easy computation gives10
Eξ(λ) = (ln ψ(λ))0, V ξ(λ) = (ln ψ(λ))00.
The key point of the local limit theorem of Sn is choosing the right λ of the Cram´er transform such that the center of Sn(λ) shifts to the point of interest, so the distribution around this point can be approximated by the central limit theorem.
Given v ∈ Rn, by properties of the Laplace transform, we have
eλ·v
ψn(λ)P (Sn∈ dv) = P (Sn(λ) ∈ dv), P (Sn∈ dv) = e−λ·v+n ln ψ(λ)
P (Sn(λ) ∈ dv).
We can replace P (Sn(λ) ∈ dv) with (2πn)−d/2det(V ξ(λ))−1/2(1 + o(1)) by the central limit theorem if λ is chosen such that ESn(λ)= v. This is achieved when (ln ψ(λ))0 = Eξ(λ) = v/n. Let α = v/n and assume such λ can be achieved and determined according to this value, i.e. λ = λ(α), we rewrite the above equation as
P (Sn∈ dv) = e−n α·λ(α)−ln ψ(λ(α))
p(2πn)ddet((ln ψ(λ(α))00)(1 + o(1))dv (2.2)
as n → ∞.
The deviation function corresponding to ξ is defined as
Λ(α) := sup
λ
{λ · α − ln ψ(λ)}, (2.3)
which is the conjugate of the convex function ln ψ(λ). Let L denotes the domain of analyticity of Λ.
Theorem 2.1 (Local limit theorem for sums of independent random vectors [3]).
Given a compact set K ⊂ L, for α = v/n ∈ K, the equation (ln ψ(λ))0 = α can be solved uniquely so the function λ(α) described in (2.2) is well defined; furthermore, the function α · λ(α) − ln ψ(λ(α)) coincides with Λ(α). Thus, rewrite (2.2),
P (Sn∈ dv) = e−nΛ(α)
p(2πn)ddet((ln ψ(λ(α))00)(1 + o(1))dv (2.4)
as n → ∞, where the remainder o(1) is uniform in K.
2.2 Local Limit Theorem for the Renewal Measure
The local limit theorem of the renewal measure has a similar form to (2.4) but with a different rate function. The second rate function (see [4])
D(α) := inf
r>0rΛ(α/r),
in terms of convex analysis, is the positively homogeneous convex function generated by Λ. Let D denotes its domain of analyticity. Let ψ1 denotes the Laplace transform of ξ1, A1 := {ψ1 < ∞} be its effective domain and AK := {D0(α) | α ∈ K}.
Theorem 2.2 (Integro-local limit theorem for the renewal measure [2][4]). For x ∈ Rd and T ∈ R, let α := x/T . Suppose α ∈ K ⊂ D for a compact set K separated from the origin and AK ⊂ A1, then
H(dx) = dx
T(d−1)/2C(α)e−T D(α)(1 + o(1)), (2.5) H1(dx) =ψ1(D0(α))H(dx)(1 + o(1)) (2.6)
as T → ∞, where the remainder o(1) is uniform in K.
A brief outline of the proof is given here. For x ∈ Rd,
H(dx) :=
∞
X
n=1
P (Sn∈ dx).
Depending on the structure of the distribution of ξ, there’re several ways showing that this series is approximately the partial sums
X
N2
P (Sn∈ dx),
where N2 := {c1T ≤ n ≤ c2T } and c1, c2 are constants depending on α. Let r := n/T and L(r) := rΛ(α/r), by Theorem 2.1and the Laplace method,11
X
N2
P (Sn∈ dx) =X
N2
C1(x/n)
nd/2 e−nΛ(x/n)(1 + o(1))dx
= X
r∈[c1,c2]
C1(α/r)
(T r)d/2 e−T rΛ(α/r)(1 + o(1))dx
=Z c2
c1
T C1(α/r)
(T r)d/2 e−T L(r)dr
(1 + o(1))dx
=T C1(α/rα) (T rα)d/2
s 2π
T L00(rα)e−T D(α)(1 + o(1))dx.
For the inhomogeneous case (ξ1 6=dξ), we need the following lemma.12
Lemma 2.3. Under the assumptions in Theorem 2.2, there exists c > 0 and C0, T0 < ∞ such that
H1(dx) − E
H d(x − ξ1); |ξ1| ≤ ln2T
≤ C0e−T D(α)−c ln2T (2.7)
for T ≥ T0.
11For details, see [2, p. 580].
12In fact, H1(B) = E H(B − ξ1) so this lemma means if we put a restriction on ξ1, then there is a corresponding error.
For |ξ1| ≤ ln2T ,
−T D(α − ξ1/T ) = −T D(α) + D0(α) · ξ1+ O(ln4T /T ).
Thus,
E
H d(x − ξ1); |ξ1| ≤ ln2T
=E
dx
T(d−1)/2C(α − ξ1/T )e−T D(α−ξ1/T )(1 + o(1)); |ξ1| ≤ ln2T
=E
dx
T(d−1)/2C(α)e−T D(α)+D0(α)·ξ1
(1 + o(1))
=ψ1(D0(α))H(dx)(1 + o(1))
as T → ∞.
3 The Second Rate Function
The valid region of the asymptotic in Theorem 2.2 depends on the second rate function D and the distribution of the initial step ξ1. Recall that
Q(x, t) = Q(x, y, t) :=
g2
8πθ2texp (−x2θt2+y22 −g8θ2t2), t > 0
0, t ≤ 0
,
which is now assigned as the density function of ξ, determines D. On the other hand, f+/||f || is assigned to be the density function of ξ1.
For simplicity, we consider this nonparametrized version
Q(x, t) = Q(x, y, t) :=
C1t exp (−x2+yt2 2 − t2), t > 0
0, t ≤ 0
,
where C is the normalizing constant. First, the Laplace transform of ξ is computed.
ψ(s, u1, u2) = Z ∞
0
Z
R2
C1
t exp (−x2+ y2
t2 − t2+ st + u1x + u2y)dxdydt
=C Z ∞
0
πt exp (t2(u21 + u22)
4 − t2+ st)dt.
Note that ψ is analytic in its effective domain
A := {ψ < ∞} = {(s, u1, u2) ∈ R3 | u21+ u22 < 4}.
Write ψ(s, µ) = ψ(s, u1, u2) for µ =pu21+ u22. For µ ∈ (µ−, µ+) := (−2, 2), define
A(µ) := − sup{s : ψ(s, µ) ≤ 1},
A∞(µ) := − sup{s : ψ(s, µ) ≤ ∞} = −∞.
Fix µ ∈ (µ−, µ+), ψ(s, µ) strictly increases as s grows so λ = A(µ) is the unique solution for the equation ψ(−λ, µ) = 1. By implicit function theorem, A is analytic in the region where A(µ) > A∞(µ). This region happens to be (µ−, µ+). For α ∈ R, define
D1(α) := D(1, α).
Some convex analysis arguments show that
D1(α) = sup
µ
{µα − A(µ)},
which is analytic for α ∈ (A0(µ−+ 0), A0(µ+− 0)). Combining the facts
• The graph (A(µ), (µ)) is a contour of ψ.
• ψ(s, µ) is strictly increasing in s-component.
• The effective domain in µ-component of ψ is bounded.
We conclude that (A0(µ−+ 0), A0(µ+− 0)) = (−∞, ∞) in our problem. Thus, for the homogeneous case, the local limit theorem for the renewal measure can be applied without restriction.
For the inhomogeneous case, the constraint on ξ1 actually depends on the region of interest. Given α ∈ R+× R2, it’s sufficient that D0(α) ∈ (A1), which means, by selecting K a sufficiently small neighborhood of D0(α), let
λ+j := sup{D0(α)j, α ∈ K},
λ−j := inf{D0(α)j, α ∈ K}, j = 1, 2, 3,
then it’s sufficient for ξ1 to satisfy: there exists r, C > 0 such that
P ((ξ1)j ≥ β) ≤Ce−λ+jβ, P ((ξ1)j ≤ −β) ≤Ce−λ−jβ
for β ≥ r.
3.1 Normal and Moderate Large Deviations
When α = x/t → 0, the asymptotic is said to be in the moderate large deviation region. In this case, D1(α) → 0, D01(α) → 0 and D100(α) → (EζEτ2)−1 =: Σ−1; also, ψ1(D10(α)) → 1 and C(α) → (2πEτ |Σ|)−1/2; thus, we have the following asymptotic
ρ(x, t)
||f || → ((2π)2Eτ |Σ|)−1/2exp (−x2Eτ tTΣ−1x)
t ,
which is uniform in any compact set separated from the origin. Combining Theo- rem 4.4, we have
Theorem 3.1 (Local limit theorem of the boundary flux in its central part [1, p.401]).
ρ(x, t)
||f || → ((2π)2Eτ |Σ|)−1/2exp (−x2Eτ tTΣ−1x)
t (3.1)
as t → ∞. This asymptotic is uniform when x = o(t2/3).
Note that the initial data has no contribution to the asymptotic in this region.13
4 Stochastic Processes
Physical quantities satisfying conservation laws of the Boltzmann equation (1.1) or the free transport equation (1.3), which are of physical interest, can be treated as stochastic processes. What are conservation laws? For the transport equation, these physical quantities are required to be invariant along characteristic curves.
For example, f, (v1, v2)f and (|v|22 + gz)f give the distributions of the position, the horizontal momentum and the total energy of the gas in the phase-space respectively.
Let us denote these processes by Y0(t), Y1(t) and Y2(t). Given B ⊂ H+× R3,
P (Y0(t) ∈ B) = C0 Z
B
f (x, t, v)dxdv, P (Y1(t) ∈ B) = C1
Z
B
(v1, v2)f (x, t, v)dxdv, P (Y2(t) ∈ B) = C2
Z
B
(|v|2
2 + gz)f (x, t, v)dxdv,
where Ci are normalizing constants.
Similarly, R f dv, R (v1, v2)f dv and R (|v|22 + gz)f dv give the distributions of the position, the horizontal momentum and the total energy of the gas in the upper-
13Though the tails of the initial f+ are still required to be subject to exponential decay.
space H+ respectively. Denote these processes by X0, X1 and X2. For U ⊂ H+,
P (X0(t) ∈ B) = C0
Z
U
Z
R3
f (x, t, v)dvdx, P (X1(t) ∈ B) = C1
Z
U
Z
R3
(v1, v2)f (x, t, v)dvdx, P (X2(t) ∈ B) = C2
Z
U
Z
R3
(|v|2
2 + gz)f (x, t, v)dvdx,
The stochastic formulation of these quantities enable us to reach their limiting be- haviours.
Additionally, these processes can be mapped onto the plane {z = 0} and forms some two-dimensional processes. Among them, one of the simplest processes is the last hitting position Z(t) of a single gas particle, which is obtained by mapping Y0(t) along its characteristic curve to {z = 0}. This process admits the following definition, being the main object studied in [1].
Definition 4.1 (Compound renewal processes [1]). Consider a sequence of random vectors ξn := (τn, ζn), which are iid to ξ = (τ, ζ) for n = 2, 3, ... and independent of ξ1; τ > 0 and τ1 ≥ 0. For each n, ζn can depend on τn while being independent of τm for all m 6= n. Let Tn:=Pn
j=1τj, a renewal process is
N (t) :=
∞
X
n=0
1(Tn ≤ t) (4.1)
Let Z(0) = 0, then
Z(t) :=
N (t)
X
j=1
ζj (4.2)
is a compound renewal process.
The compound renewal process can be associated with the renewal measure H1 directly. Let us interpret Qn∗(x, t) as the probability density of a chicken walking n steps to move a displacement x while spending exactly t amounts of time (or
reaching (x, t) from the origin (0, 0)). Suppose further that with every step it takes, it lays an egg. If the chicken was at the origin at time t = 0, then R(x, t) becomes the increasing rate (with time) of the expected number of eggs per unit area at x and at time t. Writing probabilistically the chicken’s behaviour, we get exactly the process defined in Definition 4.1.
As moments of f or sums of mappings of f along the characteristic curve, these processes are all belong to a class of stochastic process.
Definition 4.2 (Processes with regenerative increments [6, p. 193]). Let τn, n = 2, 3, ... be a sequence of iid random variables with distributions identical to τ and τ1 be another random variable independent to random variables τn; τ > 0 and τ1 ≥ 0.
Tn:=Pn
j=1τj is called a renewal time. An increment of a process G(t) is given by
gn := τn, {G(t + Tn−1) − G(Tn−1) : 0 ≤ t < τn}.
A process G(t) is said to have regenerative increments if gn are iid for n = 2, 3, ...
and g1 is independent of increments gn (not necessarily identical).
These processes are all closely related to the renewal measure H1. Their distribu- tions can be written explicitly into the form as ”a convolution of H1 and a function F0 plus another function F1”. Indeed,
P (Z(t) ∈ U ) − 1{0∈U }P (T1 > t) = P[∞
n=1
{Zn∈ U, Tn≤ t, Tn+1> t}
=
∞
X
n=1
P (Zn ∈ U, Tn ≤ t) − P (Zn ∈ U, Tn+1≤ t)
= H1([0, t] × U ) −
∞
X
n=1
P (Zn∈ U, Tn+ τ ≤ t)
= H1([0, t] × U ) − H1([0, t] × U ) ∗ P (τ ≤ t)
= H1([0, t] × U ) ∗ P (τ > t),
characteristic method used in deriving (1.5), the distribution of Xi(t), Yi(t) can also be written into this form.
With this connection to the renewal measure, the limiting behaviour of these physical quantities are revealed. Like the inhomogeneous case to the homogeneous case, an extra term arises in front of the renewal measure, which is the Laplace transform of the function F0.
Theorem 4.3 (Integro-local limit theorems for the compound renewal process).
Under the condition of Theorem 2.2,
P (Z(t) ∈ dx) =
Z ∞ 0
eλ(α)yP (τ > y)dy
H1(d(t, x))(1 + o(1)) +1(x = 0)P (T1 > t)
for t → ∞.
There’s a version of central limit theorem for a process with regenerative incre- ment, which can be found in e.g. [6] (in this book, it’s proved by using a functional central limit theorem)
Theorem 4.4 (Regenerative CLT (one dimensional) [6, Sec 2.13]). In the context of Definition 4.2, suppose
µ := Eτ,
a := E[G(τ )]/µ, Mn:= sup
0≤t<τn
{G(t + Tn−1) − G(Tn−1)}, n ≥ 1, σ2 := V [G(τ ) − aτ ]
are finite. Then for the case G(0) = 0,
G(t) − at
√t
−d
→ N (0, σ2/µ), as t → ∞.
When modified to its multi-dimensional version, this theorem matches our esti- mate of the boundary flux in its central part (see Theorem 3.1).
References
[1] Borovkov, A. A., and Mogulskii, A. A. F. (2018). Integro-local limit theorems for compound renewal processes under Cramer’s condition. I. Siberian Mathematical Journal, 59(3), 383-402.
[2] Borovkov, A. A., and Mogulskii, A. A. F. (2018). Integro-local limit theorems for compound renewal processes under Cramer’s condition. II. Siberian Mathe- matical Journal, 59(4), 578-597.
[3] Borovkov, A. A. (2013). Large deviation probabilities for sums of independent random variables. In Probability Theory (pp. 239-276). Springer, London.
[4] Borovkov, A. A., and Mogulskii, A. A. F. (1996). The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimen- sional random walks. Siberian Mathematical Journal, 37(4), 647-682.
[5] Liu, T. P., and Yu, S. H. (2008). Diffusion under gravitational and boundary effects. Bulletin of Institute of Mathematics Academia Sinica, 3(1), 167-210.
[6] Serfozo, R. (2009). Basics of applied stochastic processes. Springer Science and Business Media.
