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Inhomogeneous Continuity Equation with Application to Hamiltonian ODE

(joint work with L. Chayes & W. Gangbo) Helen K. Lei

California Institute of Technology

October 12, 2011

Mathematical Background

✆ Evolution of Measure ✆ Continuity Equation

“Physical” Motivations

✆ Hamiltonian ODE with Interaction ✆ Mass Reaching Infinity in Finite Time ✆ Regularization: Fade With Arc Length Inhomogeneous Continuity Equation ✆ Inhomogeneous Continuity Equation ✆ Deficient Hamiltonian ODE Limiting Equation and Dynamical Considerations ✆ Dynamical Hypothesis ✆ Closeness of Trajectories & Representation Formula ✆ Validity of Regularization: Convergence of Mass

Evolution of Measure

vt µ0 µ1

Xt

Given vt, have flow equation: ✩ ✫ ✪ ✾ Xt ✏ vt♣Xtq X0 ✏ id Eulerian:

Continuity Equation I

❇ρ ❇t ∇ ☎ ♣ρvq ✏ 0

V

△ in mass = flux in/out of infinitesimal volume:

dMV dt

✏ ➩

V ❇ρ ❇t dx ✏ ✁

➩

V ∇ ☎ ♣ρvq dx ✏ ✁

➩

❇V ρ v ☎ ˆ

n dS

ρ = (probability) density v = velocity field

Integrated version for macroscopic volume:

Continuity Equation II

Mass of particle constant along trajectories (incompressible): d dt rρ♣Xt, tqs ✏ ❇ρ ❇t ∇ρ ☎ v ✏ 0. Therefore, ∇ρ ☎ v ✏ ∇ ☎ ♣ρvq ù ñ ∇ ☎ v ✏ 0 and have weak formulation for measures : ❇tµt ∇ ☎ ♣µtvtq ✏ 0 means ➺ T ➺ ❇tϕ ①vt, ∇ϕ② dµt dt ✏ 0 ❅ϕ P C ✽

c ♣Rd ✂ ♣0, Tqq

Weak Formulation

Define µt ✏ Xt#µ0

(Here T#µ ✏ ν if for any measurable A ν♣Aq ✏ µ♣T ✁1♣Aqq

• r for any test function ϕ P L1♣dνq

➩ ϕ♣yq dν♣yq ✏ ➩ ϕ♣T♣xqq dµ♣xq )

Then (formally) ❇tµt ∇ ☎ ♣vtµtq ✏ 0: ϕ P C ✽

c ♣Rd ✂ ♣0, Tqq;

Ψ♣x, tq ✏ ϕ♣Xt♣xq, tq ➺ T ➺

Rd ❇tϕ♣xq ①vt♣xq, ∇ϕ♣xq② dµt♣xq dt

✏ ➺ T ➺

Rd ❇tϕ♣Xt♣xq, tq ①vt♣Xt♣xq, ∇ϕ♣Xt♣xqq② dµ0♣xq dt

✏ ➺ T ➺

Rd

dΨ dt ♣x, tq dµ0♣xq dt ✏ ➺

Rd ϕ♣XT♣xq, Tq ✁ ϕ♣x, 0q dµ0♣xq

✏ 0

Hamiltonian Dynamics I

Let R2d ◗ x ✏ ♣p, qq ✏ ♣momentum, positionq H♣p, qq ✏ 1 2⑤p⑤2 Ψ♣qq ✏ kinetic potential Then ✾ x ✏ ✄ ✾ p ✾ q ☛ ✏ ✄ ✁Id Id ☛ ✄ Hp Hq ☛ ✏ J∇H Start with measure, infinite dimensional Hamiltonian system?

H ♣µq ✏ 1 2 ➺ ⑤p⑤2 dµ ➺ Φ♣qq dµ 1 2 ➺ ♣W ✝ µq♣qq dµ ✾ Xt ✏ Jr∇H ♣µqs♣p, qq ✏ ♣✁∇♣W ✝ µ Φq♣qq, pq ✍ interaction means velocity field has non–trivial dependence on µt ✍

Finite Range Interactions

Hamiltonian Dynamics II

✆ Infinitesimal conservation of mass certainly holds ✆ ∇H ❑ J∇H ù ñ ∇ ☎ ♣J∇H q ✏ 0 Should describe by continuity equation: ❇tµt ∇ ☎ ♣J∇H ♣µtqµtq ✏ 0. ✆ Energy not pointwise conserved: dH ♣µtq dt ♣p, qq ✏ ✒ ①∇H , J∇H ② ❇H ❇t ✚ ♣p, qq ✏ 1 2❇t♣W ✝µtq.

✍ Formally, using continuity equation and supposing ⑤∇W ⑤ ↕ B ⑤❇t♣W ✝ µtq⑤ ✏ ⑤ d dt ➺ W ♣x ✁ yq dµt♣yq⑤ ↕ B ➺ ⑤J∇H ♣µtq⑤ dµt is locally bounded ✍

Total energy (integrated over µt) should still be conserved.

Hamiltonian ODE on Wasserstein Space

• L. Ambrosio and W. Gangbo.

Hamiltonian ODE’s in the Wasserstein Space of Probability

• Measures. Comm. in Pure and Applied Math.,

61, 18–53 (2007).

• W. Gangbo, H. K. Kim, and T. Pacini. Differ-

ential forms on Wasserstein space and infinite dimensional Hamiltonian systems. To appear in Memoirs of AMS.

Definition (Hamiltonian ODE). H : P2♣R2dq Ñ ♣✁✽, ✽s (proper, lowersemicontinuous). A.C. curve tµt✉r0,Ts is a Hamiltonian ODE w.r.t. H if ❉vt P L2♣dµtq, ⑥vt⑥L2♣dµtq P L1♣0, Tq such that ✩ ✫ ✪ ❇tµt ∇ ☎ ♣Jvtµtq ✏ 0, t P ♣0, Tq vt P Tµt P2♣R2dq ❳ ❇H ♣µtq for a.e., t

• Theorem. (Ambrosio, Gangbo) Suppose H : P2♣R2dq Ñ R satisfies

♣⑤∇H ♣xq⑤ ↕ C♣1 ⑤x⑤q ✆ If µn ✏ ρnL 2d, µ ✏ ρL 2d and µn á µ then ∇H ♣µnk qµnk á ∇H ♣µqµ Then given µ0 ✏ ρ0L 2d: ✆The Hamiltonian ODE admits a solution for t P r0, Ts ✆ t ÞÑ µt is L♣T, µ0q–Lipschitz (with respect to the Wasserstein distance) ✆ If H is λ–convex, then H ♣µtq ✏ H ♣µq.

Wasserstein Distance

Φ#ρ0 ✏ ρ Ð Φ : Π

ρ0 ρ Φ

M (density) M✝ (flow map) s ✏ ✁∇ ☎ ♣ρ∇pq; gρ♣s1, s2q ✏ ➩ ρ∇ρ1 ☎ ρ2 (non-flat) g✝

Φ ♣v1, v2q ✏

➩ ♣v1 ☎ v2qρ0 (flat)

(Induced distance: d♣x0, x1q2 ✏ inft ➩1

0 gx♣tq♣ dx dt , dx dt q dt : t ÞÑ x♣tq P M✶, x♣0q ✏ x0, x♣1q ✏ x1✉

) x0 x1

Upshot: d♣ρ0, ρq2 ✏ infΦ:ρ✏Φ#ρ0 ➩ ρ0⑤id ✁ Φ⑤2

• F. Otto.

The geometry of dissipative evolution eqns: the porous medium equation.

• Comm. PDE, 26 (2001), 101-174.

A.C. Curves and the Continuity Equation

• Definition. Let

P2♣Rd, W2q

denote the space of probability measures with bounded second moment equipped with the Wasserstein distance W 2

2 ♣µ, νq ✏ min

★➺

Rd ✂Rd ⑤x ✁ y⑤2 dγ♣x, yq : γ P Γ♣µ, νq

✰ and Γ♣µ, νq ✏ tγ : γ♣A ✂ Rd q ✏ µ♣Aq and γ♣Rd ✂ Bq ✏ ν♣Bq, for all measurable A and B✉

• Theorem. There is a correspondence:

tA.C. curves in P2♣Rd, W2q✉ ð ñ tvelocity fields vt P L2♣dµtq✉ via ❇tµt ∇ ☎ ♣vtµtq ✏ 0 and lim

hÑ0

1 ⑤h⑤W2♣µth, µtq♣↕q ✏ ⑥vt⑥L2♣µtq Thus W 2

2 ♣µ0, µ1q ✏ min

✧➺ 1 ⑥vt⑥2

L2♣dµtq : ❇tµt ∇ ☎ ♣vtµtq ✏ 0

✯ and TµP2♣Rd, W2q ✏ t∇ϕ : ϕ P C ✽

c ♣Rdq✉ L2♣dµq

Mass Reaching Infinity in Finite Time

Condition (♣). We are solving ❇tµt ∇ ☎ ♣J∇H µtq ✏ 0; vt :✏ J∇H ♣µtq Recall characteristics ✾ Xt ✏ vt♣Xtq; X0 ✏ id ⑤vt♣xq⑤ ↕ C♣1 ⑤x⑤q ù ñ ⑤Xt⑤ ➚ eCt♣1 ⑤X0⑤q: preserves compact support, second moment... What about other Hamiltonians? E.g.,

q Φ♣qq

Explicit Computation. ⑤vt♣Xtq⑤ ✏ C♣1 ⑤Xt⑤qR, R → 1

✄ ⑤Xt⑤ ⑤X0⑤ ☛R✁1 ✏ 1 1 ✁ t♣R ✁ 1q⑤X0⑤R✁1

x ✽ at time τ♣xq ✏ 1 ♣R ✁ 1q⑤x⑤R✁1 ➔ ✽

Continuity Equation in “Finite Volume”

Particles that have ever been in finite region during r0, ts: blue = good pink = negligible red = bad yellow = gone.

• Expect. Under reasonable dynamical conditions, still have

❇tµt ∇ ☎ ♣J∇H ♣µtqµtq ✏ 0 distributionally.

Example: Quadratic Velocity in 1D

Consider the velocity field and associated trajectories vt♣xq ✏ x2, xt ✏ x0 1 ✁ tx0 and densities ρ0 ✏ 1r0,1s, ρt ✏ xt#ρ0. By change of variables, have ρt♣yq ✏ ρ0♣x✁1

t

♣yqq♣x✁1

t

q✶♣yq ✏ 1 ♣1 ytq2 . We have then ❇tρt ✏ ✁2y ♣1 ytq3 and ♣ρtvtq✶ ✏ 2y ♣1 ytq3 and so ❇tρt ♣ρtvtq✶ ✏ 0.

Regularization: Fade With Arc Length

✾ Xt ✏ vt♣Xtq Mt ✏ M0e✁

➩t

0 Cs♣Xsq⑤vs♣Xsq⑤ ds

For simplicity, Cs ✑ ε; later, send ε Ñ 0.

Inhomogeneous Continuity Equation

♣♠q ❇tµε

t ∇ ☎ ♣vtµε t q ✏ ✁ε⑤vt⑤µε t

Given µ0, vt, define ♣µε

t q✝ ✏ X ε t #µ0

Rε

t ♣X ε t q ✏ exp♣✁ε

➺ t ⑤vt♣X ε

s q⑤ dsq

then µε

t ✏ Rε t ♣µε t q✝

satisfies ♣♠q.

µ0 µ✝

t

µt

• Proposition. (♠) preserves α–exponential moments for α ↕ ε, since

distance traveled ↕ arclength

☞

✍ directly gives global (in space) regularization ✍

Existence of ε–Dynamics

• Lemma. Let µ0 P M✽,ε. Suppose we have prescribed (time–dependent) velocity

fields vε

t satisfying

⑤vε

t ♣xq⑤ ↕ C♣1 ⑤x⑤qR

for some constants C, R → 0. Then for 0 ➔ T ➔ ✽ ✆ ❉ distributional solution ♣µε

t qtPr0,Ts to

❇tµε

t ∇ ☎ ♣vε t µε t q ✏ ✁ε⑤vε t ⑤µε t

❅ϕ P C ✽

c ♣R2d ✂ r0, Tsqq,

➩T ➩

R2d ♣❇tϕ ①vt, ∇xϕ②q dµt dt ✏ ✁ε

➩T ➩

R2d ⑤vε t ⑤ϕ dµt dt

realized as a linear functional such that ➺

R2d ϕ dµε t ✏

➺

Sε

t

♣Rε

t ϕq ✆ X ε t dµ0,

❅ϕ P Cc♣R2dq. ✆ ♣µε

t qtPr0,Ts is narrowly continuous.

✆ Preservation of moments. ☞

Topologies of Convergence

♣C ✽

c ⑨qCc

cpctly supported Ó distributional ❸ C0 vanishing at ✽ Ó weak✝ ❸ Cb bounded Ó narrow

✆ finite measures ù ñ Banach–Alaoglu gives some limit point in weak✝ topology ✆ distributional convergence + moment control ù ñ narrow convergence We have Radon measures so if µn á µ and A is a Borel set µ♣A✵q ↕ lim inf

n

µn♣Aq ↕ lim sup

n

µn♣Aq ↕ µ♣Aq

Technical Remarks

• Continuity. Let ϕ P C ✽

c ♣R2dq and suppose t Ñ t✍.

Then, with Yτ ✏ Xt✍ ✆ X ✁1

τ

, ⑤ ➺ ϕ dµε

t ✁

➺ ϕ dµε

t✍⑤ ✏ ⑤

➺ ✓ ϕ ✁ ϕ♣Yt✍q exp♣✁ε ➺ t✍

t

⑤ ✾ Yτ⑤ dτq ✛ dµε

t ⑤

➚ϕ,vε

t ⑤t ✁ t✍⑤ ε

Limiting Measures. Suppose ❇tµε

t ∇ ☎ ♣v ε t µε t q ✏ ✁ε⑤v ε t ⑤µε t for

t P r0, Ts and v ε

t uniformly locally bounded on r0, Ts.

For tk P Q ❳ r0, Ts, have by Banach–Alaoglu µε

tk á µtk

Continuity gives extension to all t. Limiting dynamics later...

Deficient Hamiltonian ODE I

• Theorem. Let µ0 P M✽,ε and 0 ➔ T ➔ ✽. Let

H ♣µq ✏ 1 2 ➺ ⑤p⑤2 dµ ➺ Φ♣qq dµ 1 2 ➺ ♣W ✝ µq♣qq dµ such that ⑤Φ♣qq⑤ ➚ ⑤q⑤R, some R → 0. Then there exists a narrowly continuous path t ÞÑ µε

t P M✽,ε such that

❇tµε

t ∇ ☎ ♣J∇H ♣µε tqµε tq ✏ ✁ε⑤J∇H ♣µε tq⑤µε t.

“Proof”. Time discretization: h ✏ 1④n, vk ✏ J∇H ♣µtkqµtk µε,n vε,n µε,n

1

vε,n

1

...

Deficient Hamiltonian ODE II

Get

❇tµε,n

t

∇ ☎ ♣J∇H ♣µε,n

tn qµε,n t ⑤q ✏ ✁ε⑤J∇H ♣µε,n tn q⑤µε,n t .

Want to take all n Ñ ✽:

✆ Limiting measure for each t by Banach–Alaoglu ✆ Only dependence of velocity field on measure is the term ∇W ✝ µ

q p

✌ Have tightness by Markov’s inequality: ➺

Bc

r ✂Rd ⑤∇W ♣¯

q ✁ qq⑤ dµε,n

t

♣p, qq ➚W e✁εrMε♣µ0q ù ñ ∇W ✝ µε,n

t

Ñ ∇W ✝ µε

t

• unif. on cpct sets

✌ By deficient continuity equation ⑤∇W ✝ µε,n

tn

✁ ∇W ✝ µε,n

t

⑤ ➚ h

☞

Uniform in ε Control on Velocity Field

To take ε Ñ 0 the previous logic can be applied if we can control the velocity field. Idea: Use the potential Φ to rid us of red particles. Enforce that there exists rings of no return tending to infinity...

Example: Spherically Symmetric Potential

Consider H♣p, qq ✏ 1 2⑤p⑤2 Υ♣⑤q⑤q. Define ✍–ring by Υ♣qq ➔ Υ♣L✍q, for all ⑤q⑤ → ⑤L✍⑤.

✍–rings are rings of no return, since ˜ H ✏ 1 2 ✞ ✞ ✞ ✞ d⑤q⑤ dt ✞ ✞ ✞ ✞

2

Υ♣qq is increasing for t ➙ t✍: d ˜ H dt ✏ d⑤q⑤ dt ✂d2⑤q⑤ dt2 ✁ d2q dt2 ☎ ˆ q ✡ ➙ 0 and at t✍ radial velocity is positive.

More General Potentials

If H♣p, qq ✏ 1 2⑤p⑤2 Φ♣qq Ψ♣t, qq, with ⑤∇Ψ♣t, qq⑤ ↕ B, for all t, q, consider bounding potential u♣rq, such that u✶♣rq ➙ B max

⑤q⑤✏r①∇Φ, ˆ

q②. Then ✍–rings of u are rings of no return for original dynamics. Postulate that u has infinitely many ✍–rings of no return:

Rings of No Return

✍ c.f., renewal points for random walks... ✍

Estimates on Velocity Field I

S ✏ blue, G ✏ yellow, O ✏ pink Choose Q a ➔ L✍♣✦ rq, then G does not contribute to ♣∇W ✝ µε,n

t

q♣qq for q P BQ:

❇BL✍ ❇BQ

z ❇Ba♣zq ✌ Tightness: ➺

Bc

r ✂Rd ⑤∇W ♣p, q ✁ qq⑤ dµε,n

t

➚W µ0♣♣S ❨ Oq ❳ tpt ➙ r✉q ↕ µ0♣Rd ✂ Bc

L✍q µ0♣S ❳ tpt ➙ r✉q.

Estimates on Velocity Field II

On S, ⑤qs⑤ ↕ L, for all 0 ↕ s ↕ t so r ↕ ⑤pt⑤ ↕ ➺ ✞ ✞ ✞ ✞ dps ds ✞ ✞ ✞ ✞ ds ⑤p0⑤ ↕ ➺ ⑤♣∇Φ ∇W ✝ µε,n

t

q♣qsq⑤ ds ↕ ML✍t ⑤p0⑤, so S ❳ tpt ➙ r✉ ⑨ Bc

r✁ML✍ t ✂ BL✍.

✁Q Q L✍ ✁♣r ✁ ML✍tq r r ✁ ML✍t ✁r ✁L✍

✌ Time evolution: Formally, ❇tF ε,n

t

♣p, qq ✏ ➺

R2d ♣p ☎ ∇2W ♣q ✁ qq ✁ ε⑤vε,n t

♣p, qq⑤∇W ♣q ✁ qqq dµε,n

t

♣p, qq. This can be estimated a similar way, now invoking moment bound on µ0...

Hamiltonian ODE I

• Theorem. Let µ0 P M✽,α, some α → 0 and 0 ➔ T ➔ ✽. Let

H ♣µq ✏ 1 2 ➺ ⑤p⑤2 dµ ➺ Φ♣qq dµ 1 2 ➺ ♣W ✝ µq♣qq dµ such that ⑤Φ♣qq⑤ ➚ ⑤q⑤R, some R → 0. Then there exists distributional limit ♣µtqtPr0,Ts of ♣µε,n

t qtPr0,Ts along some

subsequence ♣εk, nkq such that ✆ t ÞÑ µt P M is distributionally continuous and ✆ ♣µtqtPr0,Ts satisfies the continuity equation: ❇tµt ∇ ☎ ♣J∇H ♣µtqµtq ✏ 0. ☞

Representation Formula

✆ µε,n

t

defined by pushforward: µε,n

t

✏ X ε,n

t

#µ0, so have representation formula: ➺

R2d ϕ♣yq dµε,n t

♣yq ✏ ➺

Sε,n

t

♣ϕ ☎ Rε,n

t

q ✆ X ε,n

t

dµ0♣xq, ❅ϕ P C ✽

c ♣RDq,

where Sε

t ✏ tx P RD : ❉! solution to ✾

X ε

s ✏ vε s ♣X ε s q, X ε 0 ✏ x, ❅s P r0, ts✉.

✆ µε

t , µt obtained abstractly, so need to retrieve representation formula...

Need to show ➺

Sε,n

t

♣ϕ ☎ Rε,n

t

q ✆ X ε,n

t

dµ0 Ñ ➺

Sε

t

♣ϕ ☎ Rε

t q ✆ X ε t dµ0 ✌ If x P Sε

t , then x P Bε t ♣Lq for L sufficiently large and show pointwise convergence.

✌ If x ❘ Sε

t , argue that ♣Rε,n t

✆ X ε,n

t

q♣xq Ñ 0 as n Ñ ✽.

Both cases follow from finite volume convergence of trajectories:

Finite Volume Closeness of Trajectories I

• Lemma. Let T → 0. Suppose

vn Ñ v uniformly on K ✂ r0, Ts for any compact K ⑨ RD and sup

n

✓ sup

tP♣0,Tq

sup

xPK

⑤vn

t ♣xq⑤ Lip♣vn t , Kq

✛ :✏ fK ➔ ✽. Then given any δ → 0 sup

xPBt♣Lq

sup

sPr0,ts

⑤X n

s ✁ Xs♣xq⑤ ➔ δ

for n sufficiently large, where BL♣tq :✏ tx : Xs♣xq P BL, ❅s P r0, ts✉ ♣⑨ supp♣µ0qq.

Finite Volume Closeness of Trajectories II

✌ For n sufficiently large so that ⑤vn ✁ v⑤ ➔ σ and Xs P BL, X n

s P BLδ,

d ds ⑤X n

s ✁ Xs⑤ ↕ ⑤vn s ♣X n s q ✁ vs♣Xsq⑤

↕ ⑤vn

s ♣X n s q ✁ vn s ♣Xsq⑤ ⑤vn s ♣Xsq ✁ vs♣Xsq⑤

↕ ⑥vn

s ⑥Lip ☎ ⑤X n s ✁ Xs⑤ σ

↕ fBLδ⑤X n

s ✁ Xs⑤ σ.

✌ By Gronwall and choosing σ sufficiently small (n sufficiently large) ⑤X n

T ✁ XT ⑤ ↕

σ fBLδ ☎ e

fBLδ T ➔ δ.

✌ Result follows by a bootstrapping argument.

☞ Representation formula holds for µε

t and can directly take ε Ñ 0:

Hamiltonian ODE II

• Theorem. Let µ0 P M✽,α, some α → 0 and 0 ➔ T ➔ ✽. Let

H ♣µq ✏ 1 2 ➺ ⑤p⑤2 dµ ➺ Φ♣qq dµ 1 2 ➺ ♣W ✝ µq♣qq dµ such that ⑤Φ♣qq⑤ ➚ ⑤q⑤R, some R → 0. Then there exists distributional limit ♣µtqtPr0,Ts of ♣µε

tqtPr0,Ts along some

subsequence ♣εkq such that ✆ t ÞÑ µt P M is distributionally continuous and ✆ ♣µtqtPr0,Ts satisfies the continuity equation: ❇tµt ∇ ☎ ♣J∇H ♣µtqµtq ✏ 0. ☞

Phase Space Regions of No Return

Let L✍ correspond to ✍–ring and define ¯ ΩL✍♣tq ✏ BL✍♣a✍ηqt ✂ BL✍,

time

where η → 0, a✍ ✏ supqPBL✍ ⑤∇Φ⑤ ⑤∇W ⑤, so that

d ds ⑤ps⑤ ↕ a✍, ❅s P r0, ts.

Then:

Monotonicity of Mass

Let 0 ↕ t1 ➔ t2. Given any δ → 0, let L✍ be such that µ0♣BL✍q ➔ δ. Then can show for all ε ➙ 0, µε

t1♣¯

ΩL✍♣t1qq ➙ µε

t2♣¯

ΩL✍♣t2qq ✁ δ. ✍ Could also directly obtain representation formula for µt by invoking no return property... ✍

Mass Convergence?

“mass difference = mass “burned” at ✽ by ε regularization” M0 ✁ Mt ✏ M0 ✁ lim

εÑ0 Mε t

✆ Since the function 1 ✑ f ❘ Cc, mass convergence not immediate. ✌ Without interaction W , trajectories same for all ε ù ñ mass convergence: Have Mε

t Õ M✝ t

is well defined. Let δ → 0. (i) Choose L such that µt♣Bc

Lq ➔ δ. Then

Mt ↕ µt♣B✵

Lq ↕ lim inf µε t ♣BLq δ ↕ M✝ t δ.

(ii) For any ε → 0, choose Lε such that µε

t ♣BLεq ➔ δ. Then

Mt ➙ µt♣BLεq ➙ µε

t ♣BLεq ➙ Mε t ✁ δ.

Presence of W ù ñ non–trivial dependence of trajectories on measure so a priori:

“Counterexample” to Mass Convergence I

Varying ε: distance = λ ε✁1 mass = 1 at time 0 mass = e✁λ at time 1

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Mt: Mε

t : ✍ Mass does not converge at point of discontinuity... ✍

“Counterexample” to Mass Convergence II

Varying ε: mass = 1 at time 0 mass = e✁λ at time t ✁ τ mass → 0 at time t

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Mt: Mε

t : Mass not tending to ✽ fast enough:

Stronger Dynamical Condition

✌ ℓ♣Lq ➔ L ring of no return, ℓ, L Ñ ✽; ✆ EL♣tq ✏ tq0 : qt P ❇BL, qt✶ P Bℓ♣Lq for some t✶ ➔ t✉; ✆ θL♣tq ✏ suptτ : q0 P EL♣tq, ⑤qtτ⑤ ➔ ✽✉; ✆ τL ✏ sup

t

θL♣tq; ✌ Require: lim

LÑ✽ τL ✏ 0.

Example: Super–quadratic Potential

Consider Υ♣qq ✒ ✁⑤q⑤1R, R → 1. Recall ˜ H ✏ 1 2 ✞ ✞ ✞ ✞ d⑤q⑤ dt ✞ ✞ ✞ ✞

2

Υ♣qq is increasing provided d⑤q⑤

dt

→ 0. Therefore (for ⑤q⑤ ✧ 1) d⑤q⑤ dt → ❜ 2♣˜ H0 ✁ Υ♣qqq ✒ ⑤q⑤

1R 2

:✏ C♣1 ⑤q⑤qs, s → 1. Suppose at time t, ⑤qt⑤ ✏ L✍, d⑤qt⑤

dt

→ 0. Direct integration of differential inequality: ♣1 ⑤qtτ⑤qs✁1 ➙ ♣1 ⑤qt⑤qs✁1 1 ✁ Cτ♣s ✁ 1q♣1 ⑤qt⑤qs✁1 . We conclude the particle reaches infinity by time t τL✍, where τL✍ ✒ 1 Ls✁1

✍

Ñ 0 as L✍ Ñ ✽

Mass Convergence Almost Everywhere

• Theorem. Suppose the stronger dynamical condition holds and suppose µε

t á µt. Let

M✁

t

✏ lim

t✶Õt Mt✶,

M

t

✏ lim

t✶×t Mt✶

M✌

t ✏ lim Mε t ,

M✵

t ✏ lim Mε t .

Then M

t

↕ M✵

t ↕ M✌ t ↕ M✁ t .

In particular, the mass converges at all points of continuity of Mt. “Proof.” Already have M

t

↕ M✵

t . To show M✌ t ↕ M✁ t :

✆ Let δ → 0 and ℓ ✏ ℓ♣Lq be such that µ0♣¯ Ωℓ♣0qcq ➔ δ. ✆ For any ε → 0 let 0 ➔ Lε ➔ ✽ be such that µε

t ♣¯

ΩLε♣tqcq ➔ δ. Mε

t ↕ µε t ♣¯

ΩLε♣tqq δ ↕ µε

t✁τL♣¯

ΩL♣t ✁ τLqq 2δ:

“One Ring to Rule Them All”

ε Ñ 0 L Ñ ✽ ☞

Questions and Extensions.

✆ Meaningful physical systems of relevance? ✆ Different inhomogeneous equation? ✆ Uniqueness of limiting measures? (under investigation) ✆ Stronger topology?

Thank you