Gomes D.A., Valdinoci E. - Entropy penalization methods for Hamilton-Jacobi equations(54).pdf

ENTROPY PENALIZATION METHODS FOR HAMILTON-JACOBI

EQUATIONS

DIOGO A. GOMES, ENRICO VALDINOCI

Abstrat. In this paper we present a new entropy penalization problem and we

disuss its relations with approximate solutions of Hamilton-Jaobi equations, the

onvergene of assoiated disrete shemes, as well as several appliations, suh as: a

generalization of the Hopf-Cole transformation whih onverts non-linear Hamilton-

Jaobi equations into linear evolution equations, the study of xed point problems,

approximation of ertain linear evolution equations, and the onstrution of entropy

penalized Mather measures.

Contents

1. Introdution 2

2. Motivation for the sheme 5

3. Existene of a minimizing measure and equivaleny between (4) and (7) 6

4. Generalized Hopf-Cole transformation 7

5. Some properties of G 9

6. Some properties of L and

^

L 16

7. Formal Asymptotis 18

8. Fixed point problems 25

9. Convergene issues 26

10. Entropy penalized Mather measures 32

11. Convergene to Mather measures 43

Appendix A. A Banah-Ca

ioppoli-type Theorem 49

Appendix B. Some elementary properties of the semionave funtions 51

Referenes 54

Supported in part by FCT, POCTI/FEDER, POCI/FEDER/MAT/55745/2004 and

MIUR Variational Methods and Nonlinear Dierential Equations

Subjet Classiation: 70H20, 37M25, 37J50

1

2 DIOGO A. GOMES, ENRICO VALDINOCI

``A quel tempo, di numeri e n'erano soltanto due: il numero e e il numero pi greo. Il Dea-

no fa un alolo a o

hio e roe, e risponde: - Crese di e elevato a ti. Bravo furbo! Fin

l i arrivano tutti.''

(Italo Calvino, Le Cosmiomihe).

1. Introdution

The objetive of this paper is to study a new approximation proedure for visosity

solutions of the Hamilton-Jaobi equations arising in optimal ontrol. This proedure

onsists in the disretization of the ontrol problem and in the addition of a suitable

entropy funtional. This funtional regularizes the evolution of the disrete value fun-

tion by ating as a visosity term. The entropy penalization method is motivated by

onsiderations in statistial mehanis, as well as by the works of [JKO98, [FS86,

[Ana04, and [Eva04.

1

In this paper, we introdue an entropy penalized sheme related to Lagrangian and

Hamiltonian dynamis and Hamilton-Jaobi-type equations, we prove its onvergene,

we introdue a related generalized Hopf-Cole transformation, and we investigate the

onnetions between xed points of the sheme, the vanishing visosity method, and

the theory of Mather measures.

In further detail, the organization of this paper and its main results are the following.

In x 2, we introdue and motivate the entropy sheme

7! G[ = ln

Z

R

N

e

hL(x;v)+(x+hv)

dv

= inf

Z

R

N

[hL(x; v) +

n

(x + hv) + ln (v) (v) dv ;

where the inmum above is taken over all the probability densities on R

N

. Here, h

plays the ro^le of a time step disretization and is the entropy penalization parameter.

We notie from the above formulas that there are two equivalent ways of dening suh

a sheme (see formulas (4) and (7) here below). The equivalene between these two

denitions is shown in x 3.

Then, a linear sheme

7! L[ =

Z

R

N

e

hL(x;v)

(x + hv) dv

1

It may be useful to give a short sketh about the motivations of entropy methods in mathematial

physis. Rougly speaking, in genetral, an entropy penalized metod onsists in perturbing a given

problem by adding an \entropy term" of the type S(f) =

R

f log f . In statistial mehanis, suh

entropy S \ounts", in a logaritmi sale, the number of mirosopi ongurations of a physial

system whih are ompatible with a given marosopi behavior, while f has the meaning of the

frequeny of any state. We refer to [Vil03 for a more detailed statistial motivation of the entropy

funtional.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI 3

is introdued in x 4, whih turns out to be equivalent to the (non-linear) entropy

penalized method, via a generalized Hopf-Cole transformation of the type

7! e

=

:

An analogous linear sheme has been introdued in [Ana04 for the study of Mather

measures on path spaes, and, althought not dened expliitly there, the sheme G

also arises in that paper. In fat, although our starting point is dierent, there are

some deep onnetions between this work and the ones by Nalini Anantharaman; in

this introdution we will try to spell out some similarites and dierenes between some

of our results and the ones in [Ana04.

We devote x 5 to the study of some properties of the non-linear sheme G. In partiular,

the sheme is proved to enjoy a ontration property in the spae of funtions \where

onstants are quotiented out" (see Theorem 9 for details). Though this feature is

weaker than the standard strit ontration property, it will be still suient to solve

the related xed point problem by iteration.

We also study the semionavity properties of the sheme. In partiular, a uniform

semionavity estimate is obtained in Theorem 13, from whih a uniform Lipshitz

bound follows. More preisely, we show that the Lipshitz and semionavity moduli

of G[ are bounded by the maximum between the semionavity modulus of and a

universal onstant.

Some elementary properties of linear sheme L are outlined in x 6.

We arry out some asymptotis in x 7, a

ording to dierent hoies of the parameters

involved. Namely, the linear sheme is related to a paraboli equation (see Propo-

sition 23), while the non-linear sheme asymptoti is related to the Hamilton-Jaobi

equation, as shown in Proposition 24.

In x 8, the nth iteration of the non-linear sheme is proved to onverge uniformly. This

result is obtained by a xed point proedure whih makes use of the non-standard

ontration property of the sheme. We show that G

n

[ onverges, as n ! +1, to

a xed point of G \when onstants are quotiented out" (we refer to Theorem 26 for

further details). In fat, in this, our results improve the ones in [Ana04, as we an

avoid Shauder's theorem for the onstrution of a xed point, and, in partiular, we

obtain a very simple proof for the uniqueness of xed points.

The onvergene of the sheme for small time steps is disussed in x 9, by follow-

ing [Sou85 and [BS91. We prove that, when the time step h goes to zero, the solutions

of the suitably saled linear and non-linear shemes onverge uniformly to solutions of

the heat equation and of the visous Hamilton-Jaobi equation, respetively. The el-

lipti term in these equations is provided by the entropy (see Theorems 28, 29 and 30).

In x 10, we investigate the onnetion between the xed points of our sheme and

suitably penalized Mather measures. In partiular, the generalized Hopf-Cole transfor-

mation provides a orrespondene between xed points of the sheme and minimizing

measures for an entropy penalized ation (see Theorem 32), whih, in turn, onverge to

the usual Mather measures when the penalization vanishes (see Theorem 40). We will

also prove uniqueness of the entropy penalized Mather measures (see Theorem 36). We

also remark that an \expliit" representation of entropy penalized Mather measures


in terms of xed points of the non-linear sheme holds (namely, formula (108)). In

partiular, we show that, if is the unique (up to onstants) solution of

G[ = 2 R

and is the unique probability density solving

Z

R

N

(x hv)e

hL(xhv;v)+(x)(xhv)

dv = (x) ;

then

(x; v) = (x)e

hL(x;v)+(x+hv)(x)

is the unique minimizer for the entropy penalized ation. Our results in this setion,

although similar to the ones in [Ana04, avoid ompletely the use of path spaes, and

only use elementary tehniques. In the paper [Ana04 the analog to entropy penalized

Mather measures, Gibbs measures, are measures on the spae of all paths on (T

n

)

Z

,

our approah is simpler as we only need to work with the stationary version of these

measures whih are supported on T

n

R

n

, whih simplies onsiderably the problem.

In partiular, we present a self-ontained proof of the minimization property, as well

as uniqueness of entropy penalized Mather measures.

We devote x 11 to the analysis of the onvergene of the penalized problem to the lass

ial problem of Mather. In partiular, both the penalized measure and the penalized

\eetive Hamiltonian" onverge to the ones of Mather's problem (see Theorems 38

and 40 for preise statements). In partiular, both the minimizing measures and the

minimal values of the entropy penalized ation onverge to the analogous objets in

the non-penalized setting. More preisely, if L is the Lagrangian, the entropy penalized

ation of a \holonomi" (or \ow invariant", see (91)) measure is given by

Z

T

N

R

N

L(x; v)d(x; v) +

h

S[; :

Then, we onstrut the densities whih minimize suh penalized funtional and we prove

their onvergene to Mather measures for small entropy. Again, a similar result was

proved in [Ana04 for path spae measures, but our approah is onsiderably simpler

and self-ontained.

The paper ends with two appendies. The rst one is devoted to the \abstrat" xed

point argument whih is used in the proof of Theorem 26. The seond appendix ollets

some elementary results on semionave funtions.

The paper is, essentially, self-ontained, exept for some basi fats about Mather

theory, for whih we refer to [Mat91, [MF94 and [Gom05, and for some general fat

about visosity solutions, whih may be found, for instane, in [CIL92. In general,

we devoted some eort to using elementary arguments in the proofs, rather than other

ones involving ner tehnologies.


2. Motivation for the sheme

Let H(p; x) be a Hamiltonian, whih has L(x; v) as onjugated Lagrangian (the preise

assumptions on whih will be listed in x 3):

(1) L(x; v) = sup

p

p v H(p; x):

Let (x; t) be a visosity solution of the Hamilton-Jaobi equation

t

+H(r; x) = 0;

with presribed terminal ondition (x; T ). The solution admits the following repre-

sentation formula:

(2) (x; t) = inf

Z

T

t

L(q(); _q()) d + (q(T ); T ) ;

for any T > t, where the above inmum is taken over all Lipshitz urves q : [t; T ! R

N

satisfying q(t) = x (see, for instane,

2

Theorem 6.4.5 in [CS04).

A disrete sheme with time step h > 0 to approximate onsists in the iteration

n+1

(x) := inf

v2R

N

[hL(x; v) +

n

(x+ hv) :

By elementary onsiderations, the previous formula an be written as

(3)

n+1

(x) = inf

2P

Z

R

N

[hL(x; v) +

n

(x+ hv) d(v) ;

where P denotes the spae of the probability measures on R

N

. To make the sheme

smoother, we replae (3) by the \entropy penalized" sheme

(4)

n+1

(x) := inf

2D

Z

R

N

[hL(x; v) +

n

(x + hv) + ln (v) (v) dv;

where > 0 (here and in the rest of the paper) is a small parameter and we denoted

by D the set of probability densities on R

N

, i.e.

(5) D :=

2 L

1

(R

N

) j (v) 0 a:e: ;

Z

R

N

(v) dv = 1

:

The idea of penalizing a linear optimization problem with a non-linear term an be

traed bak to [JKO98, where some nie appliations to Foker-Plank equations are

presented.

For a given (measurable) funtion : R

N

! R, we dene G[ as

(6) G[(x) := ln

Z

R

N

e

hL(x;v)+(x+hv)

dv

:

Note that G is well dened, for instane, if is bounded and G[ is also bounded.

In Theorem 2 below, we show that (4) is equivalent to the expliit iteration sheme

(7)

n+1

:= G[

n

:

2

Beware of a sign hange both in the time diretion and in (1) between our notation and the one

in [CS04.


Then, in Theorem 3, we will establish the equivalene with a linear sheme, whih

generalizes the Hopf-Cole transformation. In relation with that, we reall that if u is

a solution to the Hamilton-Jaobi equation

u

t

+

jDuj

2

2

= u;

then the Hopf-Cole transform v = e

u

is a solution to the heat equation

v

t

= v ;

thus it is oneivable that similar exponential transformations happen to be useful in

our framework. We now provide a formal setting of the above disussions.

3. Existene of a minimizing measure and equivaleny between (4)

and (7)

From now on, we assume that the Lagrangian is a suitably smooth (e.g., Lipshitz)

funtion that has the form

L(x; v) = K(v) U(x) ; for v 2 R

N

; x 2 R

N

;

in whih K, the \kineti enery", is stritly onvex in v and superlinear at innity, and

U is the \potential energy" whih is bounded, Z

N

-periodi and semionvex, that is,

there exists C

U

> 0 suh that

(8) sup

x;y2R

N

y 6=0

U(x + y) + U(x y) 2U(x)

jyj

2

C

U

:

We suppose further that K is semionave, i.e., that there exists C

K

suh that

(9) sup

v;w2R

N

w 6=0

K(v + w) +K(v w) 2K(v)

jwj

2

C

K

:

Proposition 1. Let D as in (5). Fix x 2 R

N

. Then, there exist

?

2 D realizing

(10) inf

2D

Z

hL(x; v) + (x+ hv) + ln (v)

(v) dv :

More expliitly,

?

is given by

?

:= e

^

L(x;v)+

;

with

^

L(x; v) := hL(x; v) + (x + hv);

and

:= ln

Z

e

^

L(x;v)

dv :

Also, the quantity in (10) is equal to .


Proof. By denition,

?

is a probability density satisfying

(11) hL(x; v) + (x+ hv) + ln

?

(v) = :

We laim that

?

is optimal. Indeed, by the onvexity of the funtion t 7! t ln t, we

have, for any other probability density , that

Z

R

N

(hL(x; v) + (x+ hv) + ln ) (v)dv

Z

R

N

(hL(x; v) + (x + hv) + ln

?

)

?

(v)dv+

Z

R

N

(hL(x; v) + (x + hv) + ln

?

+ ) ((v)

?

(v))dv:

From (11) and the fat that both (v) and

?

are probability densities, we onlude

that the last integral vanishes, and so

?

is optimal.

The last laim in Proposition 1 follows from (11) and the fat that

?

is a probability

density.

From Proposition 1, we easily onlude that:

Theorem 2. We have that

inf

2D

Z

[hL(x; v) + (x+ hv) + ln (v) (v) dv

= ln

Z

R

N

e

hL(x;v)+(x+hv)

dv

:

In partiular, the shemes in (4) and (7) are equivalent.

4. Generalized Hopf-Cole transformation

In this setion, we disuss a hange of variables whih transforms the non-linear evo-

lution operator G into a linear evolution operator L, this operator has in fat been

introdued in [Ana04, without an expliit mention to G, although this one has in fat

been used there too. This proedure is related with the lassial Hopf-Cole transfor-

mation (see, for instane, x4.4.1 of [Eva98).

For this, we dene the linear operator

(12) L[ (x) =

Z

R

N

e

hL(x;v)

(x + hv) dv :

We will relate L and G through the exponential transformation:

(13) E [(x) := e

(x)

:

Then, it is easy to see that:

Theorem 3. We have

(14) L E = E G :

In partiular, if

n

(x) := e

n

(x)

;


the shemes in (4) and (7) are equivalent to the sheme

n+1

= L[

n

:

We omit the details of the elementary alulation needed to prove (14) and hene

Theorem 3.

It is also onvenient to introdue the resaled linear operator

^

L[ (x) :=

L[ (x)

R

R

N

e

hL(x;v)

dv

=

R

R

N

e

hK(v)

(x+ hv) dv

R

R

N

e

hK(v)

dv

:

A

ordingly, we may onsider the iteration

(15)

^

n+1

=

^

L[

^

n

:

Notie that

^

L is independent on the potential U .

As an appliation of the generalized Hopf-Cole transformation we now show the fol-

lowing property, whih is somehow related with the fat that the minimum of visosity

solutions is a visosity solution.

Proposition 4. Assume that

n

and '

n

are solutions to the iterative sheme

n+1

= G[

n

'

n+1

= G['

n

:

Then so is

n

= ln

e

n

+ e

'

n

:

Proof. This follows from (14) and the linearity of the sheme L.

We observe that

ln

e

n

+ e

'

n

! minf

n

; '

n

g;

as ! 0

+

, whih is onsistent with the fat that the minimum of visosity solutions of

the terminal value problem is a visosity

3

solution.

Note that the fat that L[ = L[ reets into G[ + = G[ + , that is G

ommutes with addition of onstants. This elementary fat will be further stressed

and used in the sequel.

3

Indeed, if

1

and

2

are solutions of (2), so is := minf

1

;

2

g. To onrm this, note that

Z

T

t

L(q(); _q()) d +

i

(q(T ); T )

Z

T

t

L(q(); _q()) d + (q(T ); T ) ;

for i = 1; 2 and thus, by taking the inmum as in (2),

i

(x; t) inf

Z

T

t

L(q(); _q()) d + (q(T ); T ) ;


5. Some properties of G

We now point out some features of G, suh as algebrai properties, smoothness, ana-

lyti bounds and semionavity estimates. In partiular, a ne ontration property is

established in Theorem 9 and a uniform bound on the semionavity modulus is given in

Theorem 13. The bounds for the semionavity modulus are essential to prove results

suh as existene of xed points and onvergene of the sheme, among others.

The rst three properties follow diretly from the denition and we thus omit the proof:

Proposition 5. If

0

is Z

N

-periodi, then so is

n

:= G

n

[

0

for any n. If U and

0

belong to C

1

(R

N

), then so does

n

for any n.

Proposition 6 (Monotoniity). If

, then G[ G[

.

Proposition 7. G[+ r = G[ + r, for any r 2 R.

We now point out that G is a (non-strit) ontration on L

1

(R

N

):

Proposition 8. For any ;

2 L

1

(R

N

),

kG[ G[

k

L

1

(R

N

)

k

k

L

1

(R

N

)

:

Proof. Fix x 2 R

N

. Without loss of generality we may assume that

G[(x) G[

(x) ;

thus

jG[(x) G[

(x)j = ln

R

R

N

e

L(x;v)+

(x+hv)

dv

R

R

N

e

L(x;v)+(x+hv)

dv

ln

R

R

N

e

L(x;v)+(x+hv)

e

k

k

L

1

(R

N

)

dv

R

R

N

e

L(x;v)+(x+hv)

dv

= k

k

L

1

(R

N

)

:

The ontration property an be improved in the following way:

for i = 1; 2. On the other hand, xed any a > 0, there exists a suitable q

a

so that, possibly intehanging

the indexes of

i

,

a+ inf

Z

T

t

L(q(); _q()) d + (q(T ); T )

Z

T

t

L(q

a

(); _q

a

()) d + (q

a

(T ); T )

=

Z

T

t

L(q

a

(); _q

a

()) d +

1

(q

a

(T ); T )

1

(x; t) (x; t) :

These omputations show that is a solution of (2).


Theorem 9. Suppose that both and

are Z

N

-periodi and Lipshitz funtions,

with Lipshitz onstant bounded by . Then, there exists a onstant C > 0, possibly

depending on h, , and other universal quantities, suh that

(16) kG[ G[

k

(1 C k

k

N

) k

k

;

in whih we used the norm

(17) k k

:= inf

2R

k + k

L

1

(R

N

)

:

Proof. Let us observe that

(18) k k

k k

L

1

(R

N

)

and

(19) k k

= k + rk

;

for any funtion and any r 2 R. Also, there exists a onstant 2 R suh that

k+

k

L

1

(R

N

)

= k

k

:

By (19) and the fat that G ommutes with onstants (reall Propopsition 7), possibly

replaing with + inf

and

with

inf

, we may assume, without loss

of generality, that

(20) k

k

L

1

(R

N

)

= k

k

:

and that

(21) inf

= 0 :

Notie that, by (21) and the periodiity assumption,

k

k

k (0)

k

L

1

k (0)k

L

1

+ k

k

L

1

2

p

N :

(22)

Let x

be a point maximizing

(23) jG[() G[

()j ;

and let

the optimal probability measure on R

N

that yields:

(24) G[

(x

) =

Z

R

N

hL(x

; v) +

(x

+ hv) + ln

dv :

Reall indeed that, by Proposition 1,

= e

hL(x

;v)+

(x

+hv)+

;

with

(25)

= ln

Z

R

N

e

hL(x

;v)+

(x

+hv)

dv :

Also, due to (4),

(26) G[(x

)

Z

R

N

hL(x

; v) + (x

+ hv) + ln

dv :


Dene, for v 2

1

h

T

N

,

(27)

(x; v) :=

X

k2Z

N

e

hL(x;v+k=h)+

(x+h(v+k=h))+

:

Note that, sine

is bounded and L grows superlinearly in v by assumption, the series

above onverges. Furthermore,

(28)

Z

1

h

T

N

(x; v) dv = 1 ;

sine

is a probability measure on R

N

.

We laim that there exists a positive onstant suh that

(29)

(x; v)

for any x 2 R

N

and v 2

1

h

T

N

. Indeed, rst note that, by the Lipshitz properties of

and (21), we have that

(30) 0

C

1

and therefore, by (25),

(31) e

=

C

2

;

where C

i

> 0 denote here suitable quantities possibly depending on N , , h and .

Also, if v 2

1

h

T

N

(and thus jvj

p

N=h), using again (30), we see that

(32) e

hL(x;v)+

(x+hv)

C

3

:

Then, from (27), (31) and (32), we have that

(x; v) e

hL(x;v)+

(x+hv)+

C

4

;

for any (x; v) 2 R

N

1

h

T

N

. This proves (29).

Now, we use the Z

N

-periodiity of and

, (24), (26) and (27) to dedue that

G[(x

) G[

(x

)

Z

R

N

[(x

+ hv)

(x

+ hv)

(v) dv

=

Z

1

h

T

N

[(x

+ hv)

(x

+ hv)

(x

; v) dv(33)

=

1

h

N

Z

x

+T

N

[(w)

(w)

x

;

w x

h

dw :

Note also that

(34) inf

= sup

= k

k

;

thanks to (20).

We laim that there exists a set N x

+ T

N

of measure larger than

^

Ck

k

N

,

for a suitable

^

C > 0, in whih

k

k

=8. For proving this, note that

if =

the laim is obvious; otherwise, by (34), there is a point x 2 x

+ T

N

so


that (

)(x) < 0 and k

k

2j(

)(x)j. By the Lipshitz property of

the funtions involved, it follows that

k

k

=8 in the ball of radius

k

k

=(4) entered at x. Thus, if we take

N := B

k

k

=(4)

(x) \ (x

+ T

N

) ;

the above laim follows from (22).

Then, by suh a laim and (29), we dedue that

Z

N

x

;

w x

h

dw C k

k

N

;

for a suitable C > 0. Combining this with (33) and (28), we onlude that

G[(x

)G[

(x

)

1

h

N

Z

(x

+T

N

)nN

(w)

(w)

x

;

w x

h

dw

k

k

h

N

Z

(x

+T

N

)nN

x

;

w x

h

dw

k

k

h

N

"

h

N

Z

1

h

T

N

(x

; v) dv C k

k

N

#

= (1 Ck

k

N

) k

k

;

for a suitable C > 0. Then, by the hoie of x

performed in (23), possibly interhanging

the ro^les of and

, we gather that

kG[ G[

k

L

1

(R

N

)

(1 Ck

k

N

) k

k

:

This and (18) imply (16).

We now dene

;h

:= sup

x2R

N

ln

Z

R

N

e

hL(x;v)

dv

:

Note that, in many ases,

;h

may be estimated expliitly; e.g., if L(x; v) = K(v) =

jvj

2

, then

;h

= (N=2)j ln(=h)j.

By taking

= 0 in Proposition 8, we gather the following

Proposition 10 (L

1

-bound).

kG[k

L

1

(R

N

)

kk

L

1

(R

N

)

+

;h

:

In partiular,

k

n

k

L

1

(R

N

)

k

0

k

L

1

(R

N

)

+ n

;h

:

We point out that, in general, it is not possible to obtain a bound of k

n

k

L

1

whih

is uniform on n: even in the ase of L = jvj

2

and

0

= 0, whih an be worked out

expliitly, one obtains

(35) k

n

k

L

1

=

Nn

2

j ln

h

j :


We now study the semionavity of

n

. The semionavity modulus of a funtion is

(36)

:= sup

x;y2R

N

y 6=0

(x+ y) + (x y) 2(x)

jyj

2

:

We say that a ontinuous funtion is semionave if

< +1.

We will prove that

n

is uniformly bounded (see Theorem 13). Sine

n

is ontinuous

if so is

0

, this yields that

n

is uniformly semionave.

The estimate on

n

will be obtained in three steps: we rst get a rst rough estimate

on

n

; we then prove a semionavity \improvement" estimate (that is, knowing a

bound on

n

, we show that atually a slightly better bound is possible); nally, by

iterating the proedure, we will be able to obtain a uniform bound on

n

. Let us now

x a semionave funtion

0

and work out the details:

Lemma 11. Let C

U

and C

K

be as in (8) and (9). Then,

n

C

K

2h

+

hC

U

2

;

for any n 1.

Proof. Fix w 2 R

N

. By using (9) and the Cauhy-Shwarz Inequality, we get that

n+1

(x) hU(x)

ln

Z

R

N

e

h

2

(K(v+w)+K(vw)C

K

jwj

2

)

e

n

(x+hv)

dv

= hU(x)

hC

K

2

jwj

2

ln

Z

R

N

e

hK(v+w)+

n

(x+hv)

2

e

hK(vw)+

n

(x+hv)

2

dv

hU(x)

hC

K

2

jwj

2

2

ln

Z

R

N

e

hK(v+w)+

n

(x+hv)

dv

Z

R

N

e

hK(vw)+

n

(x+hv)

dv

= hU(x)

hC

K

2

jwj

2

2

ln

Z

R

N

e

hK(v+w)+

n

(x+hv)

dv

2

ln

Z

R

N

e

hK(vw)+

n

(x+hv)

)

dv

:


Thus, hanging variables of integration,

n+1

(x) hU(x)

hC

K

2

jwj

2

2

ln

Z

R

N

e

hK(~v)+

n

(xhw+h~v)

d~v

2

ln

Z

R

N

e

hK(~v)+

n

(x+hw+h~v)

d~v

=

hC

K

2

jwj

2

hU(x) +

h

2

U(x + hw) + U(x hw)

+

1

2

n+1

(x + hw) +

n+1

(x hw)

;

thus, from (8),

n+1

(x+ hw) +

n+1

(x hw) 2

n+1

(x)

C

K

2h

+

hC

U

2

(hjwj)

2

;

from whih the desired laim follows.

Notie that the bound proven in Lemma 11 does not really look satisfatory for small

h. We now show how to improve it.

Lemma 12 (Semionavity improvement). Fix n 2 N. Assume that

(37)

n

:

Then,

4

(38)

n+1

hC

U

+

C

K

C

K

+ h

:

Proof. Fix y 2 R

N

and 2 (0; 1) and set t := 1 . Then, hanging variable in the

integration, we gather that

n+1

(x hy) = hU(x hy)

ln

Z

R

N

e

h

K(wy)

e

1

n

(x+hwthy)

dw

;

4

Note that, if

hC

U

+

q

h

2

C

2

U

+ 4C

U

C

K

=2 ;

then the right hand side of (38) is less than , thus we improved the semionavity bound from the

nth step to the (n+ 1)th step. If, on the other hand,

hC

U

+

q

h

2

C

2

U

+ 4C

U

C

K

=2 ;

then the right hand side of (38) provides a uniform bound for

n+1

. Observations of this type will

play a ro^le in the proof of Theorem 13 here below.


so that, by means of (8) and using the Cauhy-Shwarz Inequality, we infer that

n+1

(x+ hy) +

n+1

(x hy) 2

n+1

(x)

hC

U

(hjyj)

2

+ 2 ln

Z

R

N

e

h

K(w)

e

1

n

(x+hw)

dw

ln

Z

R

N

e

h

K(wy)

e

1

n

(x+hw+thy)

dw

Z

R

N

e

h

K(w+y)

e

1

n

(x+hwthy)

dw

hC

U

(hjyj)

2

+ 2 ln

Z

R

N

e

h

K(w)

e

1

n

(x+hw)

dw

2 ln

Z

R

N

e

h

2

(K(wy)+K(w+y))

e

1

2

(

n

(x+hw+thy)+

n

(x+hwhty))

dw

:

Therefore, by (9) and (37),

n+1

(x + hy) +

n+1

(x hy) 2

n+1

(x)

hC

U

(hjyj)

2

+ hC

K

(jyj)

2

+ (htjyj)

2

;

hene, sine t = 1 ,

n+1

hC

U

+

C

K

h

2

+ (1 )

2

:

The desired result follows now by hoosing := h=(C

K

+ h).

We now get the uniform ontrol on

n

by a suitable iteration of Lemma 12 (thus

improving Lemma 11 in the ase of small h):

Theorem 13 (Uniform semionavity). Let

?

:=

hC

U

+

p

h

2

C

2

U

+ 4C

U

C

K

2

:

Then, for any n 2 N,

n+1

max

n

; h C

U

+

?

:

In partiular,

n

max

0

; h C

U

+

?

:

Proof. The seond laim follows from the rst one, by a simple iteration. To prove the

rst laim, we distinguish two ases: either

n

?

or

n

?

. Let us deal rst

with the ase

n

?

. Note that, if

?

, then

hC

U

+

C

K

C

K

+ h

:

This observation and Lemma 12 imply that

n+1

hC

U

+

C

K

n

C

K

+ h

n

n

;


proving the desired result when

n

?

. If, on the other hand,

n

?

, Lemma 12

implies that

n+1

hC

U

+

C

K

n

C

K

+ h

n

hC

U

+

n

hC

U

+

?

;

whih proves the desired result.

We are now in position to dedue a uniform Lipshitz bound on

n

.

Theorem 14 (Uniform Lipshitz bound). Let h 2 (0; 1. Assume that

0

is Z

N

-

periodi and semionave. Then, there exists C, depending only on

0

, N , C

U

and

C

K

so that

j

n

(x)

n

(y)j C jx yj ; 8x; y 2 R

N

; 8n 1:

Proof. By Proposition 5 and Theorem 13,

n

is Z

N

-periodi and uniformly semionave.

Then, the result follows from Theorem 44.

6. Some properties of L and

^

L

In analogy with the results in x 5, we now point out some properties of the linear

operators L and

^

L. The following ones are quite easy to hek:

Proposition 15. If

0

is Z

N


n

:= L

n

[

0

for any n. If

^

0

is

Z

N


^

n

:=

^

L[

^

0

for any n. If U and

0

belongs to C

1

(R

N

), then

so does

n

for any n. If

^

0

belongs to C

1

(R

N

), then so does

^

n

for any n.

Proposition 16 (Monotoniity). If

, then L[ L[

. If

^

^

, then

^

L[

^

^

L[

^

.

Proposition 17. L[ + r = L[ + r

R

R

N

e

hL(x;v)

dv, and

^

L[

^

+ r =

^

L[

^

+ r, for

any r 2 R.

Also, it is easy to see that

^

L is a (non-strit) ontration on L

1

(R

N

):

Proposition 18. For any

^

;

^

2 L

1

(R

N

),

k

^

L[

^

^

L[

^

k

L

1

(R

N

)

k

^

^

k

L

1

(R

N

)

:

In partiular:

Proposition 19 (Uniform L

1

-bound).

k

^

L[

^

k

L

1

(R

N

)

k

^

k

L

1

(R

N

)

:

Analogously,

Proposition 20 (Uniform C

k

andW

k;1

-bound). If

^

2 C

k

(R

N

), then

^

L[

^

2 C

k

(R

N

)

and

k

^

L[

^

k

C

k

(R

N

)

k

^

k

C

k

(R

N

)

:

An analogous statement holds by replaing C

k

with W

k;1

.


We now obtain a (non-uniform in h and ) C

1

-bound of L[ for the dependene of the

L

1

-norm of :

Proposition 21 (Improvement of regularity). Assume that

(39)

Z

R

N

e

hK(v)

jDK(v)j dv < +1 :

Then,

D(

^

L[ )

L

1

(R

N

)

k k

L

1

(R

N

)

R

R

N

e

hK(v)

jDK(v)j dv

R

R

N

e

hK(v)

dv

;

for any 2 L

1

(R

N

).

Remark. For K(v) = jvj

2

, the result in Proposition 21 yields that

D(

^

L[ )

L

1

(R

N

)

onst k k

L

1

(R

N

)

p

h

:

Proof. By hanging variable, we have that

^

L[ (x) =

R

R

N

e

h

K((wx)=h)

(w) dw

h

N

R

R

N

e

hK(v)

dv

:

Therefore,

D

x

(

^

L[ )

R

R

N

e

h

K((wx)=h)

jDK((w x)=h)j j (w)j dw

h

N

R

R

N

e

hK(v)

dv

=

R

R

N

e

h

K(v)

jDK(v)j j (x+ hv)j dv

R

R

N

e

hK(v)

dv

;

whih implies the desired result.

Further regularity results an be also dedued in an analogous way from Proposition 21.

Also, it is straightforward to see that that

^

L preserves the average over T

N

of Z

N

-

periodi funtion, namely:

Proposition 22. If 2 L

1

(T

N

), then

Z

T

N

^

L[ =

Z

T

N

:

We remark that the above bounds and the Theorem of Asoli imply the uniform onver-

gene, up to a subsequene, of the iteration

^

L

n

[

0

. In partiular, if

0

is Z

N

-periodi,

Propositions 20 (or 21) and 22 imply that

^

L

n

[

0

onverges, when n ! +1, up to a

subsequene, to a xed point of

^

L with the same T

N

-average of

0

.


7. Formal Asymptotis

In this setion, we disuss the formal asymptoti behavior of the shemes orresponding

to

^

L, L and G. These omputations play a ruial ro^le in the rigorous onvergene

results given in x 9.

We deal with dierent hoies of the parameters and h. Ellipti, paraboli and

Hamilton-Jaobi-type equations will show up in these asymptotis.

Proposition 23 (Asymptotis for

^

L). Let

b

i

:=

R

R

N

e

hK(v)

v

i

dv

R

R

N

e

hK(v)

dv

; a

ij

:=

h

R

R

N

e

hK(v)

v

i

v

j

dv

2

R

R

N

e

hK(v)

dv

:

Suppose that jb

i

j+ ja

ij

j < +1. Let

^

n+1

=

^

L[

^

n

:

Then,

(40)

^

n+1

^

n

h

= b

i

D

x

i

^

n

+ a

ij

D

2

x

i

x

j

^

n

+ error terms :

The error terms an be estimated by:

onst h

2

kD

3

^

n

k

L

1

(R

N

)

R

jvj

3

e

hK(v)

dv

R

e

hK(v)

dv

:

Remark. The left hand side of (40) may be thought as a disrete approximation

of

t

^

as h ! 0. Though the oeients in (40) depend on the spei form of the

kineti energy and on the relative sale of and h, it may be useful to work out the ase

K(v) = jvj

2

somehow expliitly. In this ase, the expressions in Proposition 23 beome

b

i

= 0, a

ij

= onst

ij

and the error term is bounded by onst h

1=2

3=2

kD

3

n

k

L

1

.

Proof. We have

^

n+1

^

n

h

=

R

e

hK(v)

h

^

n

(x+ hv)

^

n

(x)

i

dv

h

R

e

hK(v)

dv

=

;h

Z

e

hK(v)

v

i

D

x

i

^

n

(x)dv +

h

2

Z

e

hK(v)

v

i

v

j

D

2

x

i

x

j

^

n

(x)dv

+O

h

2

;h

kD

3

n

k

1

Z

jvj

3

e

hK(v)

dv

;

with

;h

=

Z

e

hK(v)

dv

1

:


As an appliation of this theorem, we study the xed points

5

of the sheme (15).

Assume, for the moment, that h = . Observe that if we dene

a

ij

=

1

2

Z

e

K(v)

v

i

v

j

;

the operator

a

ij

i

j

is (possibly degenerate) ellipti, as

a

ij

i

j

=

1

2

Z

e

K(v)

v

i

i

v

j

j

=

1

2

Z

e

K(v)

jv j

2

0:

Furthermore, ifK is even, then b

i

= 0. In this ase, the xed points of the sheme (15)

satisfy

a

ij

i

j

= O(hkD

3

k

2

);

as it follows from the estimate of Proposition 23.

Proposition 24 (Asymptotis for G). Let 2 C

3

(R

N

). Assume that K(v) = jvj

2

and

that

(41) h kD

2

k

L

1

(R

N

)

is smaller than a suitable onstant.

Let

(42) a :=

R

R

N

e

jwj

2

jwj

2

dw

2

R

R

N

e

jwj

2

dw

:

Then,

(43)

G[

h

= H(D(x); x) + a

N

2h

ln

h

+ error terms :

The error terms an be estimated by

C

?

(h+

3=2

h

1=2

) ;

where C

?

is a suitable positive quantity depending only on kD

j

k

L

1

for 1 j 3.

5

Note that any onstant is a xed point for

^

L. Moreover, if K is even in any of eah variables,

given a quadrati polynomial

p(x) = b+ ` x+Mx x ;

with b 2 R, ` 2 R

N

and M 2 Mat(N N), one easily sees that p is a xed point for

^

L if and only if

a

j

2

j

p = 0 ;

where

a

j

:=

R

R

N

e

hK(v)

v

2

j

dv

R

R

N

e

hK(v)

dv

:

In partiular, for K(v) = jvj

2

, p is a xed point of

^

L if and only if it is harmoni.


Remark. There are several heuristi ways of interpreting the result in Proposition 24

a

ording to the sheme

n+1

:= G[

n

. The rst onsists in taking h! 0 and =h! 0,

so that, formally, (43) goes to the Hamilton-Jaobi equation

t

= H(D

x

(x); x) :

Another possibility is taking > 0 to be xed and sending h! 0. Then, if we dene

(44)

n

:=

n

+

Nn

2

ln

h

;

it follows that (43) goes formally to

t

= H(D

x

(x); x) + a

;

that is, a Hamilton-Jaobi equation of \vanishing visosity" type. The reader may

ompare this ase with the example disussed in (35). This fat will be made rigorous

in Theorem 30 below and it may also be rephrased as follows. A diret omputation

shows that

G[0(x) = hU(x)

N

2

ln

h

:

Therefore, if we dene

(45)

^

G[ := G[ G[0 ;

then Proposition 24 gives that

(46)

^

G[

h

= K(D(x)) + a + error terms :

The latter interpretation is related with Proposition 8, whih gives that

(47) k

^

G[k

L

1

(R

N

)

kk

L

1

(R

N

)

;

that is, a uniform bound on the

^

G-iterations. We will use these observations in Theo-

rem 29 below, where a formal proof of the onvergene of the

^

G-sheme for h! 0 will

be provided.

Proof. We have

G[ = ln

Z

e

hL(x;v)+(x+hv)

dv(48)

= hU(x) ln

Z

e

hjvj

2

+(x+hv)

dv:

Dene v

= v

(x; h) to be a minimizer of the funtion

h jvj

2

+ (x+ hv) :

Note that the minimizing property of v

yields that

h jv

j

2

+ (x+ hv

) (x)

and so

h jv

j

2

h kDk

L

1

jv

j ;


that is

(49) jv

j kDk

L

1

:

By the minimizing property of v

, it also follows that

(50) 2v

+D

x

(x+ hv

) = 0

and thus

hjvj

2

+ (x+ hv)

(51)

=

[hjv

j

2

+ (x+ hv

) + h jv v

j

2

+

1

(x; v) ;

with

(52)

1

(x; v) :=

(x + hv) (x+ hv

) hD

x

(x+ hv

) (v v

)

:

Note that, by onstrution,

(53) j

1

j C

h

2

kD

2

k

L

1

jv v

j

2

;

where C, here and in the remainder of the proof, we will denote an appropriate universal

onstant (possibly taking a dierent value at dierent steps of the omputation). We

will also use the following short hand notation:

Z

f(v) dv :=

R

R

N

e

hjvv

j

2

f(v) dv

R

R

N

e

hjvj

2

dv

:

Let also

(54)

2

(x) :=

Z

(e

1

1 +

1

) :

Then, by (53) and (41), we get that

j

2

j

Z

1

X

k=2

j

1

j

k

k!

Z

j

1

j

2

1

X

k=0

j

1

j

k

k!

=

Z

j

1

j

2

e

1

(55)

C h

4

kD

2

k

2

L

1

2

Z

jv v

j

4

e

C h

2

kD

2

k

L

1

jvv

j

2

C h

2

kD

2

k

2

L

1

R

R

N

e

jwj

2

8

jwj

4

dw

R

R

N

e

jwj

2

dw

C h

2

kD

2

k

2

L

1

:


We also dene

(56)

1

(x; v) := hU(x) + hjv

j

2

+ (x+ hv

) :

Using this denition, (48), (51) and (54), we obtain that

G[ =

1

(x; v) ln

Z

R

N

e

h jvv

j

2

e

1

(x;v)

dv

=

1

ln

Z

e

1

(x;v)

dv

ln

Z

R

N

e

hjvj

2

dv

(57)

=

1

ln

Z

(1

1

) dv +

2

(x)

N

2

ln

h

=

1

ln

1

Z

1

dv +

2

(x)

N

2

ln

h

:

We now dene

(58)

3

(x) :=

2

(x)

Z

1

(x; v) dv :

Note that

(59) j

3

j C h kD

2

k

L

1

;

thanks to (53), (55) and (41). Let also

(60)

4

:= ln(1 +

3

)

3

:

Then,

(61) j

4

j C h

2

kD

2

k

2

L

1

;

due to (59). Also, using (57), (58) and (60), we onlude that

G[ =

1

N

2

ln

h

3

4

(62)

=

1

N

2

ln

h

+

Z

1

(

2

+

4

):

Sine

2

and

4

have the same order of magnitude, it is onvenient to dene

5

:=

2

+

4

;

so that (55) and (61) give

(63) j

5

j C h

2

kD

2

k

2

L

1

;

and we obtain from (62) that

(64) G[ =

1

N

2

ln

h

+

Z

1

dv

5

:


We now dene

6

(x; v) :=

1

(x; v)

h

2

2

ij

(x) (v

i

v

i

)(v

j

v

j

) and

7

(x) :=

Z

6

(x; v) dv :(65)

Note that, from (52) and (49),

j

6

j

1

h

2

2

ij

(x+ hv

) (v

i

v

i

)(v

j

v

j

)

+

Ch

3

jv

j kD

3

k

L

1

jv v

j

2

Ch

3

kD

3

k

L

1

(jv v

j+ kDk

L

1

) jv v

j

2

and so

(66) j

7

j C h

2

kD

3

k

L

1

(

p

=h + kDk

L

1

) :

Moreover,

Z

1

dv = ah +

7

;

due to (65), (42) and a parity argument. Therefore, we dedue from (64) that

G[ =

1

N

2

ln

h

+ ah +

7

5

:

A

ordingly, if

8

:=

7

5

, we have that

(67) G[ =

1

N

2

ln

h

+ ah +

8

:

Also, by (63) and (66),

(68) j

8

j Ch

2

h

kD

2

k

L

1

+ kD

3

k

L

1

(

p

=h+ kDk

L

1

)

i

:

We now dene

2

(x; v

) =

h

4

jD(x)j

2

jD(x+ hv

)j

2

3

(x; v

) = (x)

h

4

jD(x+ hv

)j

2

hjv

j

2

(x + hv

) ;

so that

6

, by (56),

(69)

1

= hH(D(x); x) + (x) +

2

3

:

Note that

j

2

j 2h kDk

L

1

D(x+ hv

)D(x)

C h

2

kDk

L

1

kD

2

k

L

1

jv

j

C h

2

kDk

2

L

1

kD

2

k

L

1

;

6

Of ourse, if K(v) = jvj

2

, one has that H(p; x) = jpj

2

=4 + U(x).


thanks to (49). Furthermore, by using (49) and (50), one has that

j

3

j j(x) (x+ hv

) + hD(x+ hv

) v

j

+h

1

4

jD(x+ hv

)j

2

+ jv

j

2

+D(x+ hv

) v

C h

2

kD

2

k

L

1

jv

j

2

+ 0

C h

2

kDk

2

L

1

kD

2

k

L

1

:

Therefore, setting

(70)

:=

2

3

+

8

;

the estimates above and (68) yield that

j

j C h

2

n

kDk

2

L

1

kD

2

k

L

1

(71)

+

h

kD

2

k

L

1

+ kD

3

k

L

1

(

p

=h + kDk

L

1

)

io

:

Also, by olleting the identities in (67), (69) and (70), we have that

G[ = hH(D(x); x) + (x)

N

2

ln

h

+ ah +

:

This and (71) yield the desired result.

The previous result immediately applies to a xed point

7

problem:

Corollary 25. Suppose that

(72) G[ = +

?

:

Then, we have

aH(D

x

; x) =

?

h

+

N

2h

ln

N

h

+ error terms:

The oeient a and the error terms have the same form as in Proposition 24.

Remark. Heuristially, as h! 0 and =h! 0, the result in Corollary 25 means that

?

=h plays the ro^le of the \eetive Hamiltonian" H in the equation

H(D

x

; x) +H = 0:

A formal justiation for this will be given in Theorem 38.

7

For the existene of funtions satisfying (72), see Theorem 26. Note also that, if satises (72),

so does G

n

[, for any n 2 N, due to Proposition 7.


8. Fixed point problems

We now derive from the above estimates a xed point result on our iteration sheme

G. Namely, any iteration on the sheme G onverges, in the k k

dened in (17), to a

funtion whih solves a xed point type problem, up to additive onstants.

Theorem 26. There exists

?

2 R

N

and a Z

N

-periodi Lipshitz semionave funtion

?

so that

(73) G[

?

=

?

+

?

:

Furthermore, given any Z

N

-periodi semionave funtion

0

, we have that

(74) lim

n!+1

kG

n

[

0

?

k

= 0 :

The Lipshitz onstant and the semionavity modulus of

?

are bounded by a onstant

depending only on N , C

U

and C

K

.

Also,

?

and

?

enjoy the following uniqueness properties: if there exist 2 R

N

and a

Lipshitz Z

N

-periodi funtion so that G[ = + , then =

?

+ for some 2 R

and =

?

.

Finally, if

(75)

Z

R

N

e

hK(v)

jDK(v)j dv < +1 ;

then the above uniqueness property holds even if we replae the assumption that is

Lipshitz with the one that it is bounded.

Proof. Let us onsider the quotient spae Y := C

0

(R

N

)=R, that is, let us identify

ontinuous funtions if and only if they dier by a real number. Then, Y is a Banah

spae, with norm k k

as dened in (17): see, e.g., Theorem 3.14-A on page 105

of [Tay58 for a proof of this fat. Note that the semionavity denition in (36) passes

to the quotient in Y : thus we may take S to be the (lass of the) funtions in Y with

semionavity modulus bounded by an appropriately large . It is easily seen that S is

losed in Y . The results in Propositions 5 and 7 also show that G is well dened on Y

and, by Theorem 13, G sends S into itself. Funtions in S are also uniformly Lipshitz

(see Theorem 44), due to their semionavity properties.

Therefore, by Theorem 9 and Theorem 41, we dedue that G has a unique xed point

in S Y and that (74) holds. Unfolding bak the quotient spae Y to the over spae

X, we get (73).

Let us now deal with the uniqueness property. Observe that, by hanging variable of

integration in (6), we have that

G[ = ln

1

h

N

Z

R

N

e

hL(x;(wx)=h)+(w)

dw


and therefore, if G[ = + , it follows that is Lipshitz, sine so is K (reall

also (75)). Therefore, by Theorem 9,

k

?

k

= kG[ G[

?

+

?

k

= kG[ G[

?

k

1 C k

?

k

N

k

?

k

;

showing that k

?

k

= 0 and thus =

?

+ .

But then, by Proposition 7,

?

= G[ G[

?

+

?

= 0 :

We remark that a xed point result may also be proved applying the Shauder Fixed

Point Theorem (see, e.g., x 9.2.2 in [Eva98) to

G[(x) G[(0) ;

seen as ating on the spae of funtions with suitably bounded L

1

and Lipshitz norm

(and suh spae is onvex and ompat by the Theorem of Asoli). The estimates

needed for making suh an argument work are given by Proposition 5 and Theorem 14.

This approah, however, only gives the xed point property (73) but it does not give

information either on the onvergene of the iterations of G or on the uniqueness of the

xed points, while Theorems 26 and 41 do.

Corollary 27. Fix x 2 R

N

. Under the assumptions and the notation of Theorem 26,

let the measure

?

be dened by

(76) d

?

:= e

hL(x;v)+

?

(x+hv)

?

(x)

?

dv :

Then,

?

is a probability measure on R

N

. Furthermore, if

?

:= E [

?

, then

?

(x) = e

?

L[

?

:

Proof. Straightforward from (73).

Other features of the xed points for G will be disussed in x 10, where we will relate

the term

?

in (73) and the measure

?

in (76) with an entropy penalized Mather

measure.

9. Convergene issues

In this setion, we use a variation of the results by [Sou85 and [BS91 to prove the

onvergene of the entropy penalized sheme for small time step h. We will prove a

rigorous relation between our linear (resp., non-linear) sheme and the heat equation

(resp., the Hamilton-Jaobi equation).


We rst deal with the onvergene of the

^

L-sheme as h ! 0, for a xed

8

> 0. For

this, we stress that

^

L depends on h by writing

^

L

h

(the dependene on is not expliitly

written). Consider u 2 W

1;1

(T

N

) and assume, for simpliity,

9

that 1=h 2 N . We dene

u

h

: T

N

[0; 1 ! R in the following way. Let

u

h

(x; 0) := u(x)

and then, iteratively,

u

h

(x; t) :=

^

L

tih

[u

h

(; ih)(x)

if t 2 (ih; (i + 1)h, for i = 0; : : : ; (1=h) 1. Then, we have the following onvergene

result:

Theorem 28. Let K(v) = jvj

2

and

(77) a :=

R

R

N

e

jvj

2

jvj

2

dv

2

R

R

N

e

jvj

2

dv

:

Let u = u(x; t) be the unique (visosity) solution of

(78)

t

u(x; t) = a

x

u(x; t)

in T

N

(0; 1, with u(x; 0) = u(x). Then, u

h

onverges uniformly to u.

Proof. The proof is a minor variation of the one given in Theorem 2.1 of [BS91 (see

also [Sou85). Sine, in our ase, the arguments involved are elementary, we provide

full details, for the reader's onveniene.

We note that, if t 2 (ih; (i+ 1)h and x 2 T

N

ju

h

(x; t)j sup

x2T

N

ju

h

(x; ih)j;

and so

sup

x2T

N

; t2(ih;(i+1)h

ju

h

(x; t)j sup

x2T

N

; t2((i1)h;ih

ju

h

(x; t)j kuk

L

1

(T

N

)

;

that is

ku

h

k

L

1

(T

N

[0;1)

kuk

L

1

(T

N

)

:

Therefore, we may dene

(79) u

(x; t) := lim inf

(y;s)!(x;t); h!0

u

h

(y; s) :

We show that u

is a visosity supersolution of (78). For this, x (x

0

; t

0

) 2 T

N

(0; 1)

and let be a smooth funtion, so that u

has a strit minimum at (x

0

; t

0

). Let

(x

h

; t

h

) 2 T

N

[0; 1 be minimizers for the funtion u

h

. We show that we may

hoose a sequene h

k

! 0 suh that

(80) lim

k!+1

(x

h

k

; t

h

k

) = (x

0

; t

0

) :

8

The ase when ! 0 too is atually easier and an be dealt with by a modiation of the proof of

Theorem 28. In this irumstane, Theorem 28 holds with = 0 in (78), namely u(x; t) = u(x). We

omit the details of the proof, sine it losely follows the one of Theorem 28.

9

The reader may onvine herself that, with minor modiations, it would be possible to onsider

more general partitions of an interval, instead of the uniform h-mesh that, for simpliity, we deal with.


For this, let (y

k

; s

k

)! (x

0

; t

0

) and h

k

! 0 be a sequene so that

u

(x

0

; t

0

) = lim

k!1

u

h

k

(y

k

; s

k

) ;

a

ording to (79). We may also assume, up to subsequenes, that (x

h

k

; t

h

k

)! (x;

t) 2

T

N

[0; 1 as k ! +1. Then, (80) is proved if we show that

(81) (x;

t) = (x

0

; t

0

) :

For this, let us observe that, by onstrution,

(u

)(x

0

; t

0

) = lim

k!1

(u

h

k

)(y

k

; s

k

)

lim

k!1

(u

h

k

)(x

h

k

; t

h

k

)

(u

)(x;

t) :

Therefore, sine (x

0

; t

0

) was assumed to be a strit minimum for u

, the above

estimate proves (81) and, thene, (80).

We now onsider the sequene h

k

in (80), and we denote h := h

k

! 0 for short. The

fat that

u

h

u

h

(x

h

; t

h

) (x

h

; t

h

) ;

together with Propositions 16 and 17, implies that

^

L

[u

h

(; t)(x) u

h

(x

h

; t

h

)

^

L

[(; t)(x) (x

h

; t

h

)

for any x 2 T

N

, any t 2 [0; 1 and any > 0. Thene, if i

h

2 f0; : : : ; (1=h) 1g is so

that t

h

2 (i

h

h; (i

h

+ 1)h and we set

h

:= t

h

i

h

h ;

we have that

lim

h!0

h

= 0

and

0 =

^

L

h

[u

h

(; t

h

h

)(x

h

) u

h

(x

h

; t

h

)

^

L

h

[(; t

h

h

)(x

h

) (x

h

; t

h

) :(82)

Also, sine is smooth, we infer from Proposition 23 that

(83) lim

h!0

^

L

h

[(; t

h

h

)(x

h

) (x

h

; t

h

)

h

= a

x

(x

0

; t

0

)

t

(x

0

; t

0

) :

By olleting (82) and (83), we get that

a

x

(x

0

; t

0

)

t

(x

0

; t

0

) 0

if u

has a minimum at (x

0

; t

0

). That is, u

is a visosity supersolution of (78).

Analogously, if we dene

(84) u

+

(x; t) := lim sup

(y;s)!(x;t); h!0

u

h

(y; s) ;

we have that u

+

is a visosity subsolution of (78).


Also, if s 2 (0; h),

ju

h

(y; s) u(x)j = j

^

L

s

[u(y) u(x)j

R

R

N

e

s jvj

2

ju(y + sv) u(y)j dv

R

R

N

e

s jvj

2

dv

+ ju(y) u(x)j

kDuk

L

1

p

s + jx yj

;

and so

u

(x; 0) = u

+

(x; 0) = u(x) :

We laim that

u

(resp., u

+

) is(85)

lower semiontinuous (resp., upper semiontinuous).

For proving this, x z := (x; y) 2 T

N

[0; 1 and onsider a sequene z

j

2 T

N

[0; 1,

so that z

j

! z as j ! +1. For eah j 2 N , let also h

j;k

> 0 and

j;k

2 T

N

[0; 1 be

so that

lim

k!+1

h

j;k

= 0 ; lim

k!+1

j;k

= z

j

and

u

(z

j

) = lim

k!+1

u

h

j;k

(

j;k

) ;

for any xed j 2 N . Fix now > 0. Let k

0

(; j) be so that

ju

h

j;k

(

j;k

) u

(z

j

)j

for any k k

0

(; j). Let also k

1

(j) suh that

jh

j;k

j+ j

j;k

z

j

j

1

j

;

for any k k

1

(j). We dene

k

?

(; j) := max

n

k

0

(; j) ; k

1

(j)

o

;

h

j

:= h

j;k

?

(;j)

and

j

:=

j;k

?

(;j)

:

Then, by onstrution, h

j

! 0 and

j

! z as j ! +1. Therefore,

u

(z) lim

j!+1

u

h

j

(

j

) lim

j!+1

u

(z

j

) + :

Sine is arbitrary, we onlude that

u

(z) lim

j!+1

u

(z

j

) ;

for any sequene z

j

! z, and thus u

is lower semiontinuous. Analogously, one sees

that u

+

is upper semiontinuous, thus onrming (85).


Thanks to (85), the Comparison Priniple (see, e.g., Theorem 8.2 in [CIL92) yields

that u

+

u

. Sine, by (79) and (84), the opposite inequality also holds, we obtain

that

u

= u

+

=: u

and thus u is the unique

10

(ontinuous visosity) solution of (78).

To omplete the proof of the desired result, we show the uniform onvergene of u

h

to u, by arguing as follows. If, by ontradition, u

h

did not onverge uniformly to u,

there would exist > 0, and z

h

2 T

N

[0; 1 so that

ju

h

(z

h

) u(z

h

)j

for innitely many h ! 0. Then, either u

h

(z

h

) u(z

h

) or u(z

h

) u

h

(z

h

) for

innitely many h! 0. Let us assume the latter (the other ase being analogous) and

assume also that z

h

! z 2 T

N

[0; 1 for this set of h's. Then,

u(z) = u

(z)

lim

h!0

u

h

(z

h

)

u(z) ;

whih is a ontradition.

We now deal with the sheme G as h! 0. We dene

^

G as in (45).

We will x > 0 and take h! 0. We will expliilty write

^

G

h

to stress the dependene

on h in

^

G.

We x w 2 C

3

(T

N

), we assume

11

that 1=h 2 N and we dene w

h

: T

N

[0; 1 ! R as

follows: rst, we set

w

h

(x; 0) := w(x)

and then, reursively,

w

h

(x; t) :=

^

G

tih

[w

h

(; ih)(x)

if t 2 (ih; (i + 1)h, for i = 0; : : : ; (1=h) 1. Then, the following onvergene result

holds:

Theorem 29. Let a be as in (77). Then, as h ! 0, w

h

onverges uniformly to w,

where w satises

(86)

t

w(x; t) +K(D

x

w(x; t)) = a

x

w(x; t)

in the visosity sense, with w(x; 0) = w(x).

Proof. As in Theorem 28, the proof is a variation of the arguments in [Sou85 and [BS91.

First, we observe that

(87) kw

h

k

L

1

(T

N

[0;1)

k wk

L

1

(T

N

)

;

10

The results we use about visosity sub/super/solutions may be found, for instane, in [CIL92.

11

For the sake of simpliity, we assumed w to be smooth enough, in order to use (46) in estimate (88)

here below. We remark that, by applying the lower order arguments in Proposition 24 diretly to w,

less smoothness may be required.


due to (47). Thus, we may dene

w

(x; t) := lim inf

(y;s)!(x;t); h!0

w

h

(y; s) :

We show that w

is a visosity supersolution of (86). For this, x (x

0

; t

0

) 2 T

N

(0; 1)

and let be a smooth funtion, so that w

has a strit minimum at (x

0

; t

0

). Let

(x

h

; t

h

) 2 T

N

[0; 1 be minimizers for the funtion w

h

. As shown in (80), we may

and do assume, up to a subsequene, that (x

h

; t

h

)! (x

0

; t

0

).

Sine w

h

w

h

(x

h

; t

h

) (x

h

; t

h

), by exploiting Proposition 6, we gather that

^

G

[w

h

(; t)(x) w

h

(x

h

; t

h

)

^

G

[(; t)(x) (x

h

; t

h

)

for any x 2 T

N

, any t 2 [0; 1 and any > 0. Thene, if i

h

2 f0; : : : ; (1=h) 1g is so

that t

h

2 (i

h

h; (i

h

+ 1)h and we set

h

:= t

h

i

h

h,

0 =

^

G

h

[w

h

(; t

h

h

)(x

h

) w

h

(x

h

; t

h

)

^

G

h

[(; t

h

h

)(x

h

) (x

h

; t

h

) :

Then, by means of (46), we have that

lim

h!0

^

G

h

[(; t

h

h

)(x

h

) (x

h

; t

h

)

h

= K(D

x

(x

0

; t

0

)) + a

x

(x

0

; t

0

)

t

(x

0

; t

0

) :

Therefore,

a

x

(x

0

; t

0

)

t

(x

0

; t

0

) K(D

x

(x

0

; t

0

))

if w

has a minimum at (x

0

; t

0

). That is, w

is a visosity supersolution of (86).

Analogously, if we dene

w

+

(x; t) := lim sup

(y;s)!(x;t); h!0

w

h

(y; s) ;

we have that w

+

is a visosity subsolution of (86). Also, if s 2 (0; h),

(88) j

^

G

s

[ w wj C h ;

thanks to (46), where C depends only on k wk

C

3

(T

N

)

, and so

jw

h

(y; s) w(x)j = j

^

G

s

[ w(y) w(x)j

j

^

G

s

[ w(y) w(y)j+ j w(y) w(x)j(89)

C

h + jx yj

:

This implies that w

(x; 0) = w

+

(x; 0) = w(x). Thus, by using the Comparison Prini-

ple and arguing as in the proof of Theorem 28, we obtain that w

h

onverges uniformly

to w

= w

+

.


Theorem 29 may also be adapted to obtain a Hamilton-Jaobi equation with a potential

term, a

ording to the following sheme, related to (44). We dene

e

G[ := G[ +

N

2

ln

h

=

^

G[ hU(x) :

We then dene w

h

(x; 0) := w(x) and then, reursively,

w

h

(x; t) :=

e

G

tih

[w

h

(; ih)(x)

if t 2 (ih; (i + 1)h, for i = 0; : : : ; (1=h) 1. Then, the following onvergene result

holds:

Theorem 30. Let a be as in (77). Then, as h ! 0, w

h

onverges uniformly to w,

where w satises

t

w(x; t) +H(D

x

w(x; t); x) = a

x

w(x; t)

in the visosity sense, with w(x; 0) = w(x).

Proof. One sees by indution that

sup

x2T

N

t2[0;jh

jw

h

(x; t)j k wk

L

1

(T

N

)

+ jh kUk

L

1

(T

N

)

;

for j = 0; : : : ; 1=h. From this a uniform estimate as in (87) follows.

The proof of Theorem 29 may then repeated verbatim, but substitutingK() withH(; x),

^

G with

e

G, and taking C in (88) and (89) to be also depending on kUk

L

1

(T

N

)

.

10. Entropy penalized Mather measures

In Mather's theory (see, e.g., [MF94 and referenes therein), one looks for probability

measures on T

N

R

N

that minimize the ation

(90)

Z

T

N

R

N

L(x; v)d(x; v)

and satisfy the \holonomy" (or \ow invariany") onstraint

(91)

Z

T

N

R

N

['(x + hv) '(x) d(x; v) = 0 ;

for all ' 2 C(R

N

). In this setion, we disuss an entropy penalized version of Mather's

problem, and we present a solution in terms of xed points of the operator G. The

entropy penalized Mather problem onsists in minimizing

(92)

Z

T

N

R

N

L(x; v)d(x; v) +

h

S[;

in whih

S[ =

Z

T

N

R

N

(x; v) ln

(x; v)

R

R

N

(x; w)dw

dxdv


is the \entropy term". The minimization in (92) is performed over the spae of prob-

ability densities on T

N

R

N

(93)

2 L

1

(T

N

R

N

) ; 0 a:e: ;

Z

T

N

R

N

d = 1

that satisfy

12

the onstraint (91). In this setting, we study the minimizers of the

funtional in (92) in the spae given by (93) under the onstraint in (91), that is the

\entropy penalized Mather problem". This problem is the stationary version of the

problem studied in [Ana04, whih avoids the use of measures on path spaes. We will

show that these penalized minimal measures always exist (see Theorem 32 here below)

and that they are unique (see Theorem 36). An expliit formula for this measure will

be provided by using the xed point struture of G (see formula (108)). In proving

these results, an important ro^le is played by the T

N

-projetion of penalized measures

and on their analyti bounds (see Propositions 34 and 35).

For this, we need the following preliminary result:

Lemma 31. Let 2 W

1;1

(T

N

) satisfy

(94) G[ = + :

Then, there exists 2 L

1

(T

N

) satisfying

(95) 0 ;

(96)

Z

R

N

(x hv)e

hL(xhv;v)+(x)(xhv)

dv = (x)

and

(97)

Z

T

N

(x) dx = 1 :

Proof. The following elementary observation will be used throughout this proof: if 2

L

1

(T

N

) and 2 R

N

, then

Z

T

N

=

Z

T

N

(x + ) dx :

Further, we note that

(98)

Z

R

N

e

hL(x;v)+(x+hv)(x)

dv = 1 ;

for any x 2 T

N

, due to (94). Now, dene for # 2 L

1

(T

N

),

(99) F [#(x) :=

Z

R

N

#(x hv)e

hL(xhv;v)+(x)(xhv)

dv :

Let us observe that (98) and (99) imply that

(100)

Z

T

N

F [# =

Z

T

N

# :

12

As ustomary, with a slight abuse of notation, given a funtion 2 L

1

(T R), we denote

d(x; v) = (x; v) dx dv.


Let now

0

:= 1 and, reursively,

(101)

n+1

:= F [

n

;

for any n 2 N . Note that

n

is Z

N

-periodi for any n 2 N , beause so are

0

, U and .

Analogously,

n

0, beause so is

0

. Thene, by (100), we have that d

n

(x) =

n

(x) dx

is a sequene of probability mesures on T

N

, that is

(102)

Z

T

N

d

n

= 1 ;

for any n 2 N . We may thus suppose that, up to subsequene, d

n

weakly onverges

to a Radon measure d on T

N

(see, e.g., page 55 of [EG92). Also, d is a probability

measure, sine T

N

is ompat (see, e.g., Theorem 1-(ii) on page 54 of [EG92). Our

objetive now is to show that d is absolutely ontinuous with respet to the Lebesgue

measure. For this, let 2 C(T

N

) be so that 0 1. Then, exploiting (101), we get

that

Z

T

N

(x) d

n+1

(x) dx

=

Z

T

N

(x)

n+1

(x) dx

=

Z

T

N

(x)F [

n

(x) dx(103)

=

Z

T

N

Z

R

N

(x)

n

(x hv) e

hL(xhv;v)+(x)(xhv)

dv dx :

We now denote by

i

some appropriate positive quantities, whih depend only on N ,

, L, , and h (but not on ). Then, by the fat that K is superlinear and (102), we

dedue that

h

N

Z

R

N

n

(x hv) e

hK(v)

dv

=

Z

R

N

n

(w) e

hK((xw)=h)

dw

=

X

k2Z

N

Z

T

N

n

(w) e

hK((xwk)=h)

dw

1

X

k2Z

N

Z

T

N

n

(w) e

2

jkj

dw

3

Z

T

N

n

(w) dw

=

3

:

This estimate, together with (103), gives that

Z

T

N

(x) d

n+1

(x)

4

Z

T

N

(x) dx :


Taking now the limit as n! +1, using the weak onvergene of

n

, we gather that

(104)

Z

T

N

(x) d(x)

4

Z

T

N

(x) dx ;

for any 2 C(T

N

).

Take now any measurable set A T

N

, with small Lebesgue measure, say jAj , for

> 0. We also denote the -measure of A by (A). By standard results on the Radon

measure approximation with open and ompat sets (see, e.g., Theorem 4 on page 8

of [EG92), we have that there exist a ompat set K and an open set U so that

K A U ;

(A) (K) + and jU j jAj+ 2 :

Exploiting the lassial Urysohn's Lemma (see, e.g., Theorem 2 on page 15 of [DS58),

we see that there exists a funtion 2 C(T

N

) so that 0 1, (x) = 0 for any

x 2 T

N

n U and (x) = 1 for any x 2 K.

A

ordingly,

(A) (K) +

Z

T

N

(x) d(x) +

4

Z

T

N

(x) dx +

4

jU j +

(2

4

+ 1) ;

by means of (104).

Thene, d is absolutely ontinuous with respet to the Lebesgue measure.

Thus, we write

d(x) = (x) dx ;

with (x) 2 L

1

(T

N

). Sine d is, by onstrution, a probability measure on T

N

, we

have that satises (95) and (97).


Moreover, given any 2 C(T

N

), we dedue from (103) and the weak onvergene of

d

n

that

Z

T

N

(x) (x) dx

=

Z

T

N

(x) d(x)

= lim

n!+1

Z

T

N

(x) d

n+1

(x)

= lim

n!+1

Z

T

N

Z

R

N

(x)

n

(x hv) e

hL(xhv;v)+(x)(xhv)

dv dx

= lim

n!+1

Z

R

N

Z

T

N

(y + hv)

n

(y) e

hL(y;v)+(y+hv)(y)

dy dv

= lim

n!+1

Z

R

N

Z

T

N

(y + hv) e

hL(y;v)+(y+hv)(y)

d

n

(y) dv

=

Z

R

N

Z

T

N

(y + hv) e

hL(y;v)+(y+hv)(y)

d(y) dv

=

Z

R

N

Z

T

N

(y + hv) (y) e

hL(y;v)+(y+hv)(y)

dy dv

=

Z

R

N

Z

T

N

(x) (x hv) e

hL(xhv;v)+(x)(xhv)

dx dv :

Sine here above is arbitrary, we get that satises (96), as desired.

Suh plays a deisive ro^le in the onstrution of penalized Mather measures, as we are

now going to show. We also remark that the regularity and uniqueness of the funtion

will be dealt with in Propositions 34 and 35 below.

Let us now deal with the entropy penalized Mather measures:

Theorem 32. Let 2 W

1;1

(T

N

) satisfy

(105) G[ = + :

Let 2 L

1

(T

N

) be so that

13

0,

(106)

Z

R

N

(x hv)e

hL(xhv;v)+(x)(xhv)

dv = (x)

and

(107)

Z

T

N

(x) dx = 1 :

Let

(108) (x; v) := (x)e

hL(x;v)+(x+hv)(x)

:

Then, minimizes the funtional (92) over the spae (93) under the onstraint (91).

13

The existene of a satisfying (106) and (107) is assured by Lemma 31 here above. The existene

(and essential uniqueness) of and satisfying (105) is assured by Theorem 26.


Remark. The result in Theorem 32 here above thus says that

d(x; v) = (x)e

hL(x;v)+(x+hv)(x)

dx dv

is an \entropy penalized Mather measure". The uniqueness of the measure will be

proved in Theorem 36.

Proof. Exploiting (106) and (98), we dedue that

Z

T

N

R

N

'(x + hv) d(x; v)

=

Z

T

N

R

N

'(x + hv) (x) e

hL(x;v)+(x+hv)(x)

dx dv

=

Z

T

N

R

N

'(x) (x hv) e

hL(xhv;v)+(x)(xhv)

dx dv

=

Z

T

N

'(x) (x) dx

=

Z

T

N

R

N

'(x) (x) e

hL(x;v)+(x+hv)(x)

dx dv

=

Z

T

N

R

N

'(x) d(x; v) ;

for any funtion ' 2 L

1

(T

N

), whih shows that the onstraint (91) is satised by .

In addition, from (108) and (98), we get that

(109)

Z

R

N

(x; v) dv = (x) ;

for any x 2 T

N

. Therefore, by (107),

Z

T

N

R

N

d(x; v) =

Z

T

N

(x) dx = 1 ;

and so belongs to the spae given by (93).

To prove the minimizing property of it sues to show that for any other density ~

in (93) that satises (91) and any 0 1, one has that the funtion

(110) I[ =

Z

T

N

R

N

Ld

+

h

S[

;

with

:= (1 )+ ~, is onvex and that I

0

[0 = 0. Denote by _

:= ~ Then,

I

0

[ =

Z

T

N

R

N

Ld _

+

h

ln

R

dw

_

h

R

dw

Z

_

dw

;(111)

where we have used the fat that

Z

T

N

R

N

_

t

=

Z

T

N

R

N

(~ ) = 1 1 = 0 :


Further, notie that, sine both and ~ are probability measures,

Z

T

N

R

N

R

R

N

dw

Z

R

N

(~ )dw

dx dv

=

Z

T

N

R

R

N

dv

R

R

N

dw

Z

R

N

(~ )dw

dx

=

Z

T

N

Z

R

N

(~ )dw

dx(112)

=

Z

T

N

R

N

(~ )

= 1 1 = 0 :

Moreover, making use of (109) and (108), we dedue that

Z

T

N

R

N

hL d(~ )

+

Z

T

N

R

N

ln

R

R

N

dw

(~ ) dx dv

=

Z

T

N

R

N

hL(x; v) + ln

(x; v)

R

R

N

(x; w) dw

(~(x; v) (x; v)) dx dv

=

Z

T

N

R

N

hL(x; v) + ln

(x; v)

(x)

(~(x; v) (x; v)) dx dv(113)

=

Z

T

N

R

N

(x+ hv) + (x) +

(~(x; v) (x; v)) dx dv

=

Z

(x) (x+ hv)

d~

+

Z

(x+ hv) (x)

d

+

Z

d~

Z

d

= 0 + 0 + (1 1) = 0 ;

where we have also used again that both and ~ are probability measures satisfy-

ing (91).

Colleting (111), (112) and (113), it follows that

I

0

[0 = 0:

Thus, to prove that is minimal, it is suient to show that

I

00

[ 0:


To hek this, we take a further derivative in (111), thus obtaining

h

I

00

[ =

Z

T

N

R

N

_

2

2

_

R

_

dw

R

dw

+

R

_

dw

2

R

dw

2

(114)

=

Z

T

N

R

N

_

R

_

dw

R

dw

2

0 :

As a onsequene of Theorem 32, we provide the following variational haraterization

for the xed point problem of Theorem 26.

Corollary 33. Let 2 W

1;1

(T

N

) and 2 R satisfy G[ = +. Then, the inmum

of

Z

T

N

R

N

hL(x; v) d(x; v) + S[

when is in the spae given by (93) and satises the onstraint (91), is equal to

(whih, in turn, is equal to G[ ).

Proof. Theorem 32 says that the above inmum is attained at the measure given by

(108), for whih, by a diret omp

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