A tail bound for the second class particle

In this section we prove that Assumption 3.2.2 holds for the TAEBLP model.

The difficulty comes from the fact that jump rates of the second class parti-cle, being the increments of the growth rates (3.7.2), are unbounded. First recall the coupling measure µ^λ,% of (3.2.23) and notice that it gives proba-bility one on pairs of the form (y, y) if λ=%. Define alsoµ^shock% by

µ^shock%(y, z) =

(µ^%(y), ifz=y+ 1, 0, otherwise.

With these marginals we define the shock product distribution µ^shock%: =O

i<0

µ^%+1,%+1·O

i=0

µ^shock%·O

i>0

µ^%,%, (3.7.24) a measure on a pair of coupled processes with a single second class particle at the origin.

Lemma 3.7.7. The first marginal of µ^shock% is the product distribution O

i<0

µ^%+1·O

i≥0

µ^%,

while the second marginal is O

i≤0

µ^%+1·O

i<0

µ^%. (3.7.25)

Proof. The first part of the statement and the second part, apart fromi= 0, follow from the definitions. The nontrivial part is

µ^%+1(z) =µ^%(z−1), z∈Z,

valid for the second marginal at i = 0. This is specific to the definition (3.2.13) ofµ^%, and of the exponential rates (3.7.2), and to prove it we write, withθ=θ(%),

µ^%(z−1) = f(z) e^θ · e^θz

f(z)!· 1

Z(θ) = e^(θ+β)z

f(z)! · 1 e^θ+β/2Z(θ).

Summing this up for allz ∈Z gives one on the left hand-side, hence leads to

Z(θ+β) = X∞ z=−∞

e^(θ+β)z

f(z)! = e^θ+β/2Z(θ), which also implies

%(θ+β) = d

dθlog(Z(θ+β)) =%(θ) + 1.

We conclude that

µ^%(z−1) = e^(θ+β)z

f(z)! · 1

Z(θ+β) =µ^%(θ+β)(z) =µ^%+1(z), which finishes the proof of the lemma.

The translation ofµ^shock% is denoted by τ_kµ^shock%: =O

i<k

µ^%+1,%+1·O

i=k

µ^shock%·O

i>k

µ^%,%.

The main tool we use is Theorem 1 from [108], which we reformulate here.

µS(t) will just denote the time evolution of a measure µunder the process dynamics:

Theorem 3.7.8. In the sense of bounded test functions on Ω×Ω, d

dt(τ_kµ^shock%)S(t) = e^θ(%+1)−e^θ(%)

·(τ_k+1µ^shock%−µ^shock%) + e^−θ(%)−e^−θ(%+1)

·(τ_k−1µ^shock%−µ^shock%).

(3.7.26) First some remarks. The exact role of the special exponential form of the rate function f can be seen here: Theorem 3.7.8 from [108] is only valid if the rates are of these form, and this theorem is crucial for us to show that the required tail bound for Q(t) holds. Without this and in the case of f with unbounded increments, we do not even have a good linear bound on the second class particle.

The first interesting consequence of this theorem is that the measure µ^shock% on a coupled pair evolves into a linear combination of its shifted versions. Second, notice that (3.7.26) is the Kolmogorov equation for an asymmetric simple random walk. Indeed, this theorem implies the following Corollary 3.7.9. Let the pair(ξ⁻(0), ξ(0))have initial distribution µ^shock%

defined by (3.7.24). Then its later distribution evolves into a linear combi-nation of translated versions ofµ^shock%: at time t the pair (ξ⁻(t), ξ(0)) has distribution

µ^shock%S(t) = X∞ k=−∞

P_k(t)·τ_kµ^shock%,

where P_k(t) is the transition probability at time t from the origin to k of a continuous time asymmetric simple random walk with jump rates

e^θ(%+1)−e^θ(%) to the right and e^−θ(%)−e^−θ(%+1) to the left.

In particular, Q^ξ(·), started from an environment µ^shock%, is a continuous time asymmetric simple random walk with these rates.

Although the corollary is quite natural, let us give a formal proof here.

First some notation. (ξ⁻(·), ξ(·)) will denote a pair of processes evolving under the basic coupling,gwill be a bounded function on the path space of such a pair, and for shortness we introduce Θ_t for the whole random path, shifted to time t: Θ_t = (ξ⁻(t+·), ξ(t+·)). Expectation of the process,

started fromτ_kµ^shock%, will be denoted by E^(k). Notice that under E^(k) we a.s. have a single positionQ^ξ(t) where the coupled pair differ by one, this is the position of the single conserved second class particle. With some abuse of notation we also use E^{(ξ−, ξ)} for the evolution of the pair (ξ⁻(·), ξ(·)), started from the specific initial state (ξ⁻, ξ).

We aim for proving the semigroup property ofS(·). The first step is Lemma 3.7.10. Given times 0< s < tand k∈Z,

E⁽⁰⁾

g(Θ_t)|Q^ξ(s) =k] =E^(k)[g(Θ_t−s)].

Proof. The left hand-side is E⁽⁰⁾

g(Θt) ; Q^ξ(s) =k]

P⁽⁰⁾{Q^ξ(s) =k}

= E⁽⁰⁾

E^{(ξ−(s), ξ(s))}g(Θ_t−s) ;Q^ξ(s) =k]

P⁽⁰⁾{Q^ξ(s) =k}

= P

j∈Z

P⁽⁰⁾{Q^ξ(s) =j}E^(j)

E^{(ξ−(0), ξ(0))}g(Θ_t−s) ;Q^ξ(0) =k]

P⁽⁰⁾{Q^ξ(s) =k}

= P⁽⁰⁾{Q^ξ(s) =k}E^(k)

E^{(ξ−(0), ξ(0))}g(Θt−s) ;Q^ξ(0) =k]

P⁽⁰⁾{Q^ξ(s) =k}

=E^(k)[g(Θ_t−s)],

where in the second equality we used that the distribution at time s is a linear combination of shifted versions ofµ^shock%.

Next we prove the Markov property forQ^ξ(·).

Lemma 3.7.11. Letn >0be an integer,ϕ_i,i= 0, . . . , nbounded functions onZ, and 0 =t₀< t₁ <· · ·< t_n. Then

E⁽⁰⁾ Yn i=1

ϕ_i Q^ξ(t_i)−Q^ξ(t_i−1)

= Yn i=1

E⁽⁰⁾ϕ_i Q^ξ(t_i−t_i−1) .

Proof. The statement is trivially true forn= 1. We proceed by induction,

and assume the statement is true forn−1. Then E⁽⁰⁾

Yn i=1

ϕ_i Q^ξ(t_i)−Q^ξ(t_i−1)

=X

j∈Z

P⁽⁰⁾{Q^ξ(t₁) =j}ϕ₁(j)·E⁽⁰⁾hYⁿ

i=2

ϕ_i Q^ξ(t_i)−Q^ξ(t_i−1)

|Q^ξ(t₁) =ji

=X

j∈Z

P⁽⁰⁾{Q^ξ(t₁) =j}ϕ₁(j)·E^(j) Yn i=2

ϕ_i Q^ξ(t_i−t₁)−Q^ξ(t_i−1−t₁)

=X

j∈Z

P⁽⁰⁾{Q^ξ(t₁) =j}ϕ₁(j)·E⁽⁰⁾ Yn i=2

ϕ_i Q^ξ(t_i−t₁)−Q^ξ(t_i−1−t₁)

=X

j∈Z

P⁽⁰⁾{Q^ξ(t₁) =j}ϕ₁(j)· Yn i=2

E⁽⁰⁾ϕ_i Q^ξ(t_i−t_i−1)

= Yn i=1

E⁽⁰⁾ϕ_i Q^ξ(t_i−t_i−1) .

The second equality uses Lemma 3.7.10, the third one uses the fact thatφ’s only depend onQ^ξ-differences, and the fourth one follows from the induction hypothesis.

Proof of Corollary 3.7.9. We know that at any fixed timet >0 the distribu-tion of (ξ⁻(t), ξ(t)) is a linear combination of shifted versions ofµ^shock%. The shift is traced by the second class particle Q^ξ(t), therefore the differential equation

d

dtP⁽⁰⁾{Q^ξ(t) =k}

= e^θ(%+1)−e^θ(%)

· P⁽⁰⁾{Q^ξ(t) =k+ 1} −P⁽⁰⁾{Q^ξ(t) =k} + e^−θ(%)−e^−θ(%+1)

·

P⁽⁰⁾{Q^ξ(t) =k−1} −P⁽⁰⁾{Q^ξ(t) =k} (3.7.27) follows from (3.7.26). In the above lemmas, we also proved that Q^ξ(t) is Markovian (annealed w.r.t. the initial distribution of (ξ⁻, ξ)). As there exists only one Markovian process with Kolmogorov equation (3.7.27) of the simple asymmetric random walk, we conclude that the process Q^ξ(·) with initial environmentµ^shock%is an asymmetric simple random walk with rates as stated in the Corollary.

Lemma 3.7.12. Let(ω⁻, ω)be a pair of processes in basic coupling, started from distribution (3.2.18), with second class particle Q(t). Then there exist constants 0< α0, C <∞ such that

P{|Q(t)|> K} ≤e^−CK

whenever K > α₀t and t is large enough.

Notice that this implies that Assumption 3.2.2 holds for the TAEBLP.

Proof. The proof uses auxiliary processes to connect the above arguments to the setting of Assumption 3.2.2. Define the pair

(λ_i, %_i) : =

((%, %+ 1), fori≤0, (%, %), fori >0.

Draw the pair (ζ(0), ξ(0)) from the product distribution of coupling

mea-sures (3.2.23) O

i∈Z

µ^λi,%i. Thenξ(0) has distribution

O

i≤0

µ^%+1·O

i<0

µ^%,

in agreement with (3.7.25).

Let now the pair (ζ(·), ξ(·)) evolve in the basic coupling, and let them play the role of (η(·), ω(·)) of Section 3.7.1. This results in the pair (ζ(·), ζ⁺(·)) with a second class particle Q^ζ(·) and the pair (ξ⁻(·), ξ(·)) with a second class particleQ^ξ(·) such thatQ^ζ(t)≤Q^ξ(t), see Lemma 3.7.3. Therefore the random walk result in Corollary 3.7.9 on Q^ξ(·) yields the desired estimate forQ^ζ(t). Finally, notice that the distribution ofω⁻(0) in Assumption 3.2.2 and ofζ(0) above only differ byω₀⁻(0)∼µb^%, while ζ0(0)∼µ^%. Therefore

P{Q(t)> K}= X∞ z=−∞

P{Q(t)> K|ω₀⁻(0) =z} ·µ^%(z)¹² bµ^%(z)² µ^%(z)

¹₂

= X∞ z=−∞

P{Q^ζ(t)> K|ζ₀(0) =z} ·µ^%(z)¹² bµ^%(z)² µ^%(z)

¹₂

≤h X^∞

z=−∞

P{Q^ζ(t)> K|ζ₀(0) =z} ·µ^%(z)i¹₂

·h X^∞

y=−∞

b µ^%(y)²

µ^%(y) i¹₂

=P{Q^ζ(t)> K}¹² ·h X^∞

y=−∞

b µ^%(y)²

µ^%(y) i¹₂

.

We are done as soon as we show that µb^%(y)/µ^%(y) is uniformly bounded in y. With the exponential rates (3.7.2) one obtains from (3.2.17)

b µ^%(y) µ^%(y) =C

X∞ z=y+1

(z−%)e^{−β2(z−βθ)2+β2(y−βθ)2} =C X∞ k=1

(k+y−%)e^{−β2k2−βky+θk}.

This is uniformly bounded for large y’s since then ye^−βy < 1. For large negativey’s one uses the equivalent form

b

µ^%(y) : = 1 Var^%(ω₀)

Xy z=−∞

(%−z)µ^%(z) of (3.2.17) and writes

b µ^%(y) µ^%(y) =C

Xy z=−∞

(%−z)e^{−β2(z−βθ)2+β2(y−βθ)2} =C X∞ k=1

(k−y+%)e^{−β2k2+βky−θk} which is again uniformly bounded for large negativey values.

To show a lower bound onQ(t), start with

(λi, %i) : =







(%, %), fori <0, (%−1, %), fori= 0, (%, %−1), fori >0, and the coupled pair (ζ(0), ξ(0)) in distribution

O

i∈Z

µ^λi,%i.

Now the roles of the pair (ζ(·), ζ⁺(·)) with a second class particleQ^ζ(·) and the pair (ξ⁻(·), ξ(·)) with a second class particleQ^ξ(·) are interchanged and we have Q^ζ(t) ≥ Q^ξ(t). The random walk estimate on Q^ξ and a Radon-Nikodym estimate similar to the one above completes the proof of the lower bound.

3.A Monotonicity of measures

In this first section of the Appendix we show that the measuresµ^% and bµ^% defined in (3.2.13) and (3.2.17), respectively, are stochastically monotone as functions of%. We start with a simple

Lemma 3.A.1. Fix a function ϕ(ω) onZ, bounded by a polynomial. Then E^θ(ϕ(ω)) is differentiable in θ on (θ,θ), and¯

d

dθE^θ(ϕ(ω)) =Cov^θ(ϕ(ω), ω).

Proof. Convergence of the series involved inE^θ(ϕ(ω)) can be verified via the ratio test, even after differentiating the terms. Sinceµ^θ is the exponentially

weighted version ofµ^θ0 for someθ₀, we have d

dθE^θϕ(ω)

= d dθ

E^θ0(ϕ(ω)·e^(θ−θ0)ω) E^θ0e^(θ−θ0)ω

= E^θ0(ϕ(ω)·ω·e^(θ−θ0)ω)

E^θ0e^(θ−θ0)ω −E^θ0(ϕ(ω)·e^(θ−θ0)ω)·E^θ0(ω·e^(θ−θ0)ω) [E^θ0e^(θ−θ0)ω]²

=Cov^θ(ϕ(ω), ω).

Corollary 3.A.2. For any θ < θ <θ, the state sum¯ (3.2.12) satisfies d

dθlogZ(θ) = 1 Z(θ)

ωX^max

z=ω^min

z e^θz

f(z)! =E^θ(ω) = :%(θ), (3.A.1) d²

dθ²logZ(θ) = d

dθ%(θ) =Var^θ(ω). (3.A.2)

The function %(θ) is strictly increasing and maps (θ,θ)¯ onto (ω^min, ω^max).

Proof. Everything is already covered except the last surjectivity statement.

Due to the monotonicity and continuity one only needs to show convergence at the boundaries θ, ¯θ to ω^min, ω^max. First let us consider the case when θ <¯ ∞. Then ω^max=∞and Fatou’s lemma implies

lim inf

θ%θ¯ Z(θ) = lim inf

θ%θ¯

X

z∈I

e^θz

f(z)! ≥X

z∈I

lim inf

θ%θ¯

e^θz

f(z)! =X

z∈I

e^θz¯ f(z)! =∞ since forz >0

e^θz¯ f(z)! =

Yz y=1

e^θ¯ f(y) ≥1

by definition of ¯θ and f being nondecreasing. This shows that logZ(θ) takes on arbitrarily large values asθ%θ. We also know that it is a smooth¯ and convex function on (θ,θ) (see (3.A.2)). This implies that its derivative¯ (3.A.1) is not bounded from above i.e., arbitrarily large % values can be achieved. The same reasoning works in case θ > −∞ for arbitrarily large negative% values.

When ¯θ =∞ then, regardless whether ω^max is finite or infinite, fix any 0≤y < ω^max and write

%(θ) =E^θ(ω·1{ω > y}) +E^θ([ω]⁺·1{ω≤y})−E^θ([ω]⁻·1{ω≤y})

≥(y+ 1)·P^θ(ω > y)−E^θ([ω]⁻·1{ω ≤y})

≥(y+ 1)−(y+ 1)·P^θ(ω≤y)− q

E^θ(([ω]⁻)²)· q

P^θ(ω ≤y)

≥(y+ 1)−(y+ 1)·P^θ(ω≤y)− q

E^θ0(([ω]⁻)²)· q

P^θ(ω ≤y) (3.A.3)

for a fixed θ < θ₀ < θ. Here [.]⁺ and [.]⁻ denotes the positive and the negative part of the expression in the brackets. The last inequality follows by monotonicity ofµ^θ inθ and ([ω]⁻)² being a nonincreasing function ofω.

For anyω^min−1< z ≤y and θ > θ, µ^θ(z)

µ^θ(y+ 1) = Yy x=z

µ^θ(x) µ^θ(x+ 1) =

Yy x=z

f(x+ 1)

e^θ ≤f(y+ 1) e^θ

y−z+1

.

Given 0≤y < ω^max and 1> ε >0, there is a large enough θ which makes the last fraction smaller thanε. With such a choice we have

P^θ{ω≤y}= Xy z=ω^min

µ^θ(z)≤µ^θ(y+ 1) Xy z=ω^min

ε^y−z+1≤ε·1−ε^y−ωmin+1 1−ε . Therefore, for the case of a finite ω^max, choosing y = ω^max−1 and large θ makes (3.A.3) arbitrarily close to ω^max. When ω^max=∞, the argument shows that%(θ)≥y+1 can be achieved for anyy≥0. A similar computation demonstrates that any density towardsω^min can be reached whenθ=−∞. Corollary 3.A.3. The measuresµ^% are stochastically nondecreasing in %.

Proof. Since % and θ are strictly increasing functions of each other, it is equivalent to show monotonicity of µ^θ. This follows if we can show 0 ≤

d

dθE^θ(ϕ(ω)) for an arbitrary bounded nondecreasing function ϕ. Lemma 3.A.1 transforms this derivative into the covariance ofϕ(ω) and ω, which is non-negative due toϕbeing nondecreasing.

Monotonicity ofµb^% requires somewhat more of a convexity argument.

Proposition 3.A.4. The family of measures µb^%, defined in (3.2.17), is stochastically nondecreasing in %.

Proof. Start by rewriting the definition:

b

µ^%(y) = E^% [ω−%]·1{ω > y}

Var^%(ω) = Cov^%(ω, 1{ω > y}) Cov^%(ω, ω)

=

d

dθP^θ{ω > y}

d dθ%(θ)

θ=θ(%)

= d

d%P^%{ω > y}.

Let us denote the bµ^%-expectation by Eb^%. Fix a bounded nondecreasing functionϕ. We need to show

0≤ d

d%Eb^%ϕ(ω).

We compute a different expression for this derivative. Passing the deriva-tive through the sum in the third equality below is justified because the series involved are dominated by certain geometric series, uniformly overθin small open neighborhoods. This follows from the definitions of θ and ¯θ and the assumptionθ < θ(%)<θ.¯

Eb^%ϕ(ω) =

ωX^max

y=ω^min

ϕ(y)· d

d%P^%{ω > y}

=

ωX^max

y=ω^min

ϕ(y)· d

d%[P^%{ω > y} −1{0≥y}]

= d d%

ωX^max

y=ω^min

ϕ(y)·[P^%{ω > y} −1{0≥y}]

= d d%E^%

ωX^max

y=ω^min

ϕ(y)·[1{ω > y} −1{0≥y}]

= d d%E^%

ωX^max

y=ω^min

ϕ(y)·[1{ω > y >0} −1{0≥y≥ω}]

= d

d%E^%h^ωX⁻¹

y=1

ϕ(y)− X0 y=ω

ϕ(y)i

= d

d%E^%Φ(ω).

Above we introduced the function Φ(x) =

x−1X

y=1

ϕ(y)− X0 y=x

ϕ(y),

with the convention that empty sums are zero. To conclude the proof, notice that Φ(x+ 1)−Φ(x) =ϕ(x). Thus a nondecreasing function ϕdetermines a (non-strictly) convex function Φ with Φ(1) = 0, and vice-versa. Hence the convexity theorem [103, Theorem 2.1 ] establishes that

d

d%Eb^%ϕ(ω) = d²

d%²E^%Φ(ω)≥0.

3.B Regularity properties of the hydrodynamic flux function

For the zero range process defined among the examples in Section 3.2.2, the hydrodynamic (macroscopic) flux function H : R≥0 → R≥0 of (3.2.14) is given by

H(%) =E^%f(ω).

The results of [103] forf now read as follows:

Proposition 3.B.1. If the jump rate f of the zero range process is con-vex (or concave), then the flux H is also convex (or concave, respectively).

Moreover, in this case H⁰⁰(%)>0 (or H⁰⁰(%)<0, respectively) for all % >0 if and only if f is not a linear function.

Parts of this proposition were proved with coupling methods in [95].

Next we show in the general case (i.e. not only for zero range processes) that H(%) is well defined, and is infinitely differentiable. (We use third derivatives in the proof of Theorem 3.2.3.) The functionH(%) is, in general, the expected net growth rate w.r.t.µ^% as defined in (3.2.14). We show that the series making up this expectation is finite, even after differentiating its terms. This will then lead to smoothness of H(%).

Lemma 3.B.2. Let g(y, z) ≥ 0 be any function on Z×Z, bounded by a polynomial in|y| and |z|. Then for any θ < θ <θ,¯

E^θ

(p(ω₀, ω₁) +q(ω₀, ω₁))g(ω₀, ω₁)

<∞.

Proof. We deal with the first part that contains p, the one with q can be treated analogously. The sum we are looking at is

ωX^max

y=ω^min+1 ω^maxX−1 z=ω^min

p(y, z)·g(y, z)· e^θ(y+z)

f(y)!·f(z)! · 1 Z(θ)².

These sums are certainly convergent ifω^minandω^maxare both finite. When this is not the case we split both summations at zero, and convergence is established on the four quadrants of the plane. We use (3.2.7) and the corollary

p(y, z) =p(z+ 1, y−1)· f(y)

f(z+ 1) forω^min < y≤ω^max, ω^min≤z < ω^max of (3.2.9), and we consider empty sums to be zero.

• y >0, z >0: In this case

p(y, z)≤p(y,0) =p(1, y−1)·f(y)

f(1) ≤p(1,0)·f(y) f(1), and the corresponding part of the summation is bounded by

p(1,0) f(1) ·

ωX^max

y=1

ω^maxX−1 z=1

g(y, z)· e^θ(y+z)

f(y−1)!·f(z)!· 1 Z(θ)².

• y≤0, z >0: In this case

p(y, z)≤p(1,0),

and the corresponding part of the summation is bounded by p(1,0)·

X0 y=ω^min+1

ω^maxX−1 z=1

g(y, z)· e^θ(y+z)

f(y)!·f(z)!· 1 Z(θ)².

• y≤0, z≤0: In this case

p(y, z)≤p(1, z) =p(z+ 1,0)· f(1)

f(z+ 1) ≤p(1,0)· f(1) f(z+ 1), and the corresponding part of the summation is bounded by

p(1,0)f(1)· X0 y=ω^min+1

X0 z=ω^min

g(y, z)· e^θ(y+z)

f(y)!·f(z+ 1)!· 1 Z(θ)².

• y >0, z≤0: In this case

p(y, z) =p(z+ 1, y−1)· f(y)

f(z+ 1) ≤p(1,0)· f(y) f(z+ 1), and the corresponding part of the summation is bounded by

p(1,0)·

ωX^max

y=1

X0 z=ω^min

g(y, z)· e^θ(y+z)

f(y−1)!·f(z+ 1)! · 1 Z(θ)².

Convergence of each of these bounds forθ < θ <θ¯is established e.g. by the ratio test.

Notice that a similar argument gives finite higher moments of the rates when log(f) is at most linear in both directions onZ.

Corollary 3.B.3. H(%) is infinitely differentiable at all % ∈ (ω^min, ω^max).

Proof. By the previous lemma the series F(θ) : =H(%(θ)) = 1

Z(θ)² ·

ωX^max

y, z=ω^min

(p(y, z)−q(y, z)) e^θ(y+z) f(y)!·f(z)!, is convergent and infinitely differentiable. Since H(%) = F(θ(%)) and % 7→

θ(%) is infinitely differentiable as well, the claim follows.

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In document J´ulia Komj´athy (Pldal 165-179)