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doi:10.7151/dmdico.1158

A VERSION OF NON-HAMILTONIAN LIOUVILLE EQUATION

Celina Rom

Department of Mathematics and Computer Science University of Bielsko-Bia la

Willowa 2, 43–309 Bielsko-Bia la, Poland e-mail: crom@ath.bielsko.pl

Abstract

In this paper we give a version of the theorem on local integral invariants of systems of ordinary differential equations. We give, as an immediate conclusion of this theorem, a condition which guarantees existence of an invariant measure of local dynamical systems. Results of this type lead to the Liouville equation and have been frequently proved under various assumptions. Our method of the proof is simpler and more direct.

Keywords: Liouville equation, invariant measure.

2010 Mathematics Subject Classification:34A99.

1. Introduction

The first results containing the Hamiltonian version of the Liouville equation have been derived from the early twentieth century. The Liouville equation de- scribes changes in time of the probability density function of the particle in phase space and it became an essential tool of classical and statistical mechanics. Non- Hamiltonian but the classical version of the Liouville equation appeared in later papers. It was also shown that the function which is the integral invariant of a dynamical system must satisfy this equation. These results play a fundamental role in the theory of certain stochastic differential equations. Various types of problems associated with the Liouville equation are still open issues and still new results concerning this subject appear [1, 2, 3, 4, 6]. Various stochastic versions of the Liouville equation are given in many of the present papers. These results are related to physical systems in which there is a white noise effect. In this paper a generalization of the classical result of the integral invariant of a dynamical

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system is given. We consider here systems of differential equations for which the Lipschitz condition is fulfilled only locally. We describe the way we understand the concept of the integral invariant in such case and then we show the equivalent form of the Liouville equation. As an immediate application of our result we ob- tain a condition under which invariant measures of local dynamical systems exist.

Our result can be useful to solve certain type of stochastic differential equations.

2. Preliminaries

We begin with the following notation and definitions. Let R denote the set of real numbers and let N denote the set of positive integer numbers. Given a, b ∈ R, a < b, the closed interval in R with ends a and b will be denoted by [a, b]. Let A× B be the cartesian product of sets A ⊂ Rm and B ⊂ Rn, n, m ∈ N. We will denote by Ln, n ∈ N, the family of Lebesgue measurable sets in Rn and by µ the Lebesgue measure defined on Ln. Moreover let δik = 1 for i = k and δik = 0 for i6= k, i, k ∈ N.

Let us fix a positive integer n, real numbers a, b such that a < b and an open set G ⊂ Rn. Let D = [a, b] × G. We will consider the system of ordinary differential equations of the form

y1 = F1(t, y1, . . . , yn) , . . . , yn= Fn(t, y1, . . . , yn) , (1)

with initial conditions

y1(t0) = y01, . . . , yn(t0) = y0n, (2)

where functions F1, . . . , Fn are defined on the set D, (y01, . . . , y0n) ∈ G and t0 ∈ [a, b].

We will assume that functions F1, . . . , Fn are continuous on D and have continuous partial derivatives ∂F∂yi

k on D for i, k = 1, . . . , n.

Let y = (y1, . . . , yn) ∈ G and y0 = (y01, . . . , y0n) ∈ G. Let us fix t0 ∈ [a, b]

and y0 ∈ G. We will denote by y (t, y0) = (y1(t, y0) , . . . , yn(t, y0)), t ∈ Jt0,y0, the saturated solution of system (1) with the initial condition (2), where Jt0,y0

denotes a maximal interval on which it is defined.

We are going to specify the way we understand the concept of the local integral invariant of system (1).

Definition. Let A ∈ Ln, A ⊂ G and t0 ∈ [a, b]. Let It0,A = T

y0∈AJt0,y0. Let φt0,t(y0) = y (t, y0), for t ∈ It0,A and y0∈ A. Let us assume that f : D → R is a function with nonnegative values, with nonpositive values or Lebesgue integrable with respect to the variable y ∈ G. We will call such a function a local integral

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invariants of system (1), if the following condition is satisfied Z

A

f(t0, y) dµ = Z

φt0,t(A)

f(t, y) dµ (3)

for all t0 ∈ [a, b], A ∈ Ln, A ⊂ G and for all t ∈ It0,A, where R

Af(t0, y) dµ and R

φt0,t(A)f(t, y) dµ are integrals with respect to the variable y ∈ G with respect to the Lebesgue measure in Rnon sets A and φt0,t(A), respectively.

Remark 1. Let t0 ∈ [a, b], A ∈ Ln and A ⊂ G are such that It0,A 6= {t0}. It follows from theorems on dependence of solution y (t, y0) to the system (1) on the initial condition y0 ∈ G, that if our assumptions for the system are satisfied then for t ∈ It0,A there exists an open set U ⊂ G such that

(a) A ⊂ U ,

(b) [t0, t] ⊂ It0,U for t > t0, (c) [t, t0] ⊂ It0,U for t < t0,

and φt0,t : U → φt0,t(U ) is a homeomorphism, φt0,t(A) ∈ Ln for every t ∈ It0,a

and our definition is correctly specified.

Moreover, if all saturated solutions of system (1) are defined on whole interval [a, b], then our definition of the local integral invariant of system (1) coincides with the classical definition of the local integral invariant of the dynamical system generated by system (1).

3. Our results

Under the notation of the previous section, we will prove the following theorem.

Theorem 2. Let an n be a fixed positive integer number. Let f : D → R be a function with nonnegative values, with nonpositive values or Lebesgue integrable with respect to a variabley∈ G for each fixed t ∈ [a, b]. Moreover let the function f and all its partial derivatives be continuous in D. Then the function f is the local integral invariant of system (1) if and only if the equation

∂f

∂t +

n

X

i=1

∂(f Fi)

∂yi

= 0 (4)

is satisfied for all(t, y) ∈ D.

Equation (4) is known as the Liouville equation. If this equation is satisfied then the function f is the integral invariant of system (1). This fact has been frequently proved under various assumptions. We will show that our version of

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this theorem is also valid. Our method of the proof is simpler and more direct then those given in earlier papers concerning this subject.

We will precede the proof of Theorem 2 by the following lemma.

Lemma 3. Let K ⊂ G be a bounded closed set, t0 ∈ [a, b] and It0,K 6= {t0}. Let t= t0+ ∆t ∈ It0,K.

Let L⊂ R be a closed interval such that 0 ∈ L and 0 is not an endpoint of L if t0 6= a, b. Moreover, t ∈ It0,K and t∈ It0,U for ∆t ∈ L, where U ⊂ G is some open set satisfying conditions (a), (b), (c) of Remark 1.

Then the following conditions hold

(1) yi(t, y0) = y0i + F (t0, y0) ∆t + ri(∆t, y0) ∆t, for y0 ∈ K, i = 1, . . . , n, where values ri(∆t, y0) are uniformly bounded for (∆t, y0) ∈ L × K and ri(∆t, y0) →∆t→00 for all y0 ∈ K, i = 1, . . . , n.

(2) The partial derivatives ∂φt0,t∂y(y0)

0i , i= 1, . . . , n, exist and the Jacobian of the functiony0→ φt0,t(y0), y0 ∈ U , is given by the formula

J acφt0,t(y0) = 1 +

n

X

i=1

∂Fi

∂yi (t0, y0) ∆t + γ (∆t, y0) ∆t,

where values γ(∆t, y0) are uniformly bounded for (∆t, y0) ∈ L × K and γ(∆t, y0) →∆t→00 for all y0 ∈ U .

Proof. Let y0 ∈ G, t0 ∈ [a, b], and

yi(t, y0) = Fi(t, y (t, y0)) , (5)

for i = 1, . . . , n, t ∈ Jt0,y0.

From the Lagrange theorem cf. [5], it follows that

yi(t, y0) = yi(t0, y0) + yi(t0, y0) ∆t + yi(θ, y0) − yi(t0, y0) ∆t (6)

with θ between t0 and t, t ∈ Jt0,y0, i = 1, . . . , n. Hence we get yi(t, y0) = y0i+ Fi(t0, y0) ∆t + ri(∆t, y0) ∆t, (7)

for y0 ∈ K, t0 ∈ It0,U and t = t0+∆t ∈ It0,U, where ri(∆t, y0) = Fi(θ, y (θ, y0))−

Fi(t0, y0) for some θ between t0 and t.

Note that y (t, y0) is continuous on It0,U × U . This fact is a consequence of theorems on dependence of solution of system (1) on the initial condition.

Moreover we can assume that (θ, y0) belongs to some compact set included in It0,K × K for (∆t, y0) ∈ L × K. Taking into account the continuity of functions Fi on D, i = 1, . . . , n, and formulas of ri, i = 1, . . . , n, we know that the values

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ri(∆t, y0), i = 1, . . . , n, are bounded as values of a continuous function on a compact set and that ri(∆t, y0) →∆t→0 0 for all y0 ∈ K, i = 1, . . . , n.

We are going to prove the property (2) now.

By hypotheses the partial derivatives ∂y∂yi

0k (t, y0) exist for i, k = 1, . . . , n, (t, y0) ∈ It0,U × U and they are continuous on It0,U × U . Moreover, we have

yi(t, y0) = y0i+ Z t

t0

Fi(s, y (s, y0)) ds for i = 1, . . . , n.

Computing a partial derivative with respect to y0k of both sides of the above equality we obtain

∂yi

∂y0k (t, y0) = δik+ ∂

∂y0k Z t

t0

Fi(s, y (s, y0)) ds

= δik+ Z t

t0

∂Fi

∂y0k (s, y (s, y0)) ds

= δik+ Z t

t0

n

X

j=1

∂Fi

∂yj (s, y (s, y0)) ∂yj

∂y0k(s, y0) ds

= δik+

n

X

j=1

∂Fi

∂yj (θ, y (θ, y0)) ∂yj

∂y0k (θ, y0) ∆t

= δik+∂Fi

∂yk(t0, y0) ∆t + hik(∆t, y0) ∆t (8)

where θ is between t0 and t = t0+ ∆t, and

hik(∆t, y0) =

n

X

j=1

∂Fi

∂yj (θ, y (θ, y0)) ∂yj

∂y0k (θ, y0) −∂Fi

∂yk(t0, y0) . (9)

Taking into account the continuity of functions y (t, y0) and ∂y∂yi

0k, i, k = 1, . . . , n, on It0,U × U , continuity of partial derivatives ∂F∂yi

k on D and the formula for hik(∆t, y0), i, k = 1, . . . , n, we obtain that values hi,k(∆t, y0), i, k = 1, . . . , n, as values of a continuous function on a compact set, are uniformly bounded for (∆t, y0) ∈ L×K. We assume that (θ, y0) belongs to some compact set. Moreover, we have ∂y∂yj

0k (θ, y0) →∆t→0 δjk, so hi,k(∆t, y0) →∆t→0 0 for all y0 ∈ K, i, k = 1, . . . , n.

The property (2) of Lemma 3 is now an immediate consequence of (8).

Now we are in a position to prove Theorem 2.

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Proof. We will first prove the equivalence of the condition (3) defining the local integral invariant and the condition (4) from Theorem 1 for a closed and bounded set K ⊂ G.

Let t0 ∈ [a, b], It0,K 6= {t0} and t ∈ It0,K. Let gK(t) =R

φt0,t(K)f(t, y) dµ.

Observe that the function gK is constant on It0,K if and only if gK (t) = 0 for all t∈ It0,K. Remark that if K ⊂ G is a closed and bounded set, then φt0,t(K) ⊂ G is also a closed and bounded set, for t ∈ It0,K. It is a consequence of the fact that functions φt0,t, t ∈ It0,K, are homeomorphisms of certain sets, cf. Remark 1.

From the above observation, the definition of function φt0,t for t ∈ It0,K and the definition of the derivative of the function gK, it follows that the following conditions are equivalent:

(a) gK(t0) = 0 for an arbitrary fixed t0 ∈ [a, b] and an arbitrary fixed closed bounded set K ⊂ G;

(b) gK(t) = 0 for t ∈ It0,K, t0 ∈ [a, b], for an arbitrary fixed closed bounded set K ⊂ G.

So let us fix a closed and bounded set K ⊂ G, t0 ∈ [a, b] such that It0,K 6= {t0} and let t = t0+ ∆t ∈ It0,K. Then, we have

gK (t0) = lim

∆t−→0

R

φt0,t(K)f(t, y) dµ −R

Kf(t0, y) dµ

∆t .

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Let us make a change of variables y → φt0,t(y) in the integral R

φt0,t(K)f(t, y) dµ where t ∈ It0,U, y ∈ U and U satisfy conditions (a), (b), (c) in Remark 1. Because of Remark 1 and property (2) of Lemma 3 we obtain for sufficiently small ∆t the following equality

Z

φt0,t(K)

f(t, y) dµ = Z

K

f(t, φt0,t(y)) Jacφt0,t(y) dµ

= Z

K

f(t, φt0,t(y)) · 1 +

n

X

i=1

∂Fi(t0, y)

∂yi

!

∆t + γ (∆t, y) ∆t

! dµ, (11)

where γ (∆t, y) is the function such as in the property (2) of Lemma 3.

From from the property (1) of Lemma 3, we infer

φt0,t(y) = (y1+ F1(t0, y) ∆t + r1(∆t, y) ∆t, . . . , yn+ Fn(t0, y) ∆t + rn(∆t, y) ∆t) for y = (y1, . . . , yn) ∈ K, where ri(∆t, y), i = 1, . . . , n are such as in the property (1) in Lemma 3. Taking into accout the Taylor formula, we have

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f(t, φt0,t(y)) = f (t0, y) +∂f

∂t (t0, y) ∆t +

n

X

i=1

 ∂f

∂yi(t0, y) · (Fi(t0, y) + ri(∆t, y)) ∆t + o (∆t, y) ∆t

 . (12)

By a reasoning as in the proof of property (1) of Lemma 3 we deduce that o(∆t, y) →∆t→0 0 for y ∈ K and that values of o (∆t, y) are uniformly bounded for sufficiently small ∆t and y ∈ K.

From equalities (11) and (12) we have Z

φt0,t(K)

f(t, y) dµ = (13)

Z

K

f(t0, y) +∂f

∂t (t0, y) ∆t +

n

X

i=1

∂(f Fi)

∂yi (t0, y) ∆t + R (∆t, y) ∆t

! dµ.

Taking into account properties of γ and ri(∆t, y) for i = 1, . . . , n and properties of o (∆t, y), we obtain that values R (∆t, y) are uniformly bounded for y ∈ K for sufficiently small ∆t and that R (∆t, y) →∆t→0 0 for y ∈ K.

From (10) and (13) it follows that

gK(t0) = lim

∆t−→0

Z

K

∂f

∂t (t0, y) +

n

X

i=1

∂(f Fi)

∂yi

(t0, y) + R (∆t, y)

! dµ.

Because of the continuity of partial derivatives of functions f and Fi, i = 1, . . . , n and properties of R (∆t, y), we can use the Lebegue dominated convergence the- orem to obtain

gK(t0) = Z

K

∂f

∂t (t0, y) +

n

X

i=1

∂(f Fi)

∂yi (t0, y)

! dµ.

Hence we deduce that the condition (3) defining local integral invariants for the closed and bounded set K ⊂ G is equivalent to the condition (4) from Theorem 2.

Now we are going to verify if the condition (4) in Theorem 2 implies the condition (3) for every bounded set A ∈ Ln, A ⊂ G.

Let t0 ∈ [a, b] and t ∈ It0,A. Let us represent the set A as a sum A = A1+∪ A2+∪ A1−∪ A2−, where

A1+ = {y ∈ A : f (t0, y) > 0, f (t, φt0,t(y)) > 0}, A2+ = {y ∈ A : f (t0, y) > 0, f (t, φt0,t(y)) < 0},

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A1− = {y ∈ A : f (t0, y) < 0, f (t, φt0,t(y)) > 0}, A2− = {y ∈ A : f (t0, y) < 0, f (t, φt0,t(y)) < 0}.

Sets A1+, A2+, A1− and A2− are pairwise disjoint and belong to Ln, because the function f is continuous and φt0,t is a homeomorphism. Moreover, these sets are bounded. If we prove that each of the sets A1+, A2+, A1− and A2− satisfy con- dition (3) then the set A also satisfies condition (3) due to the injectivity of φt0,t and the additivity of the Lebesque integral with respect to a set of integration.

First we will prove that the set A1+ satisfies the condition (3). Since A1+∈ Ln and A1+ is bounded we obtain that for all ǫ > 0 there exists a closed and bounded set K ⊂ A1+ such that µ (A1+\ K) < ǫ. Hence and from the injectivity of φt0,t , we get

Z

φt0,t(A1+)

f(t, y) dµ = Z

φt0,t(K)

f(t, y) dµ + Z

φt0,t(A1+\K)

f(t, y) dµ

= Z

K

f(t0, y) dµ + Z

φt0,t(A1+\K)

f(t, y) dµ

= Z

A1+

f(t0, y) dµ − Z

A1+\K

f(t0, y) dµ + Z

φt0,t(A1+\K)

f(t, y) dµ

= Z

A1+

f(t0, y) dµ − η + Z

φt0,t(A1+\K)

f(t, y) dµ,

where η is a nonnegative real number which can be sufficiently small while the set K is suitable chosen.

Hence and from the fact that R

φt0,t(A1+\K)f(t, y) dµ > 0 we have R

A1+f(t0, y) dµ 6 R

φt0,t(A1+)f(t, y) dµ. Because φt,t0t0,t(A1+)) = A1+ simi- larly we obtainR

A1+f(t0, y) dµ >R

φt0,t(A1+)f(t, y) dµ, so we haveR

A1+f(t0, y) dµ

=R

φt0,t(A1+)f(t, y) dµ for the set A1+.

Now we are going to show that the set A2+ satisfies condition (3). For the closed and bounded set K ⊂ φt0,t(A2+), we get

Z

φt0,t(A2+)

f(t, y) dµ = Z

K

f(t, y) dµ + Z

φt0,t(A2+)\K

f(t, y) dµ

= Z

K

f(t, y) dµ − η = Z

φt,t0(K)

f(t0, y) dµ − η,

where η is a nonnegative number which can be sufficiently small while the set K is suitable chosen. As before fromR

φt0,t(A2+)f(t, y) dµ 6 0 andR

φt,t0(K)f(t0, y) dµ >

0, we obtainR

φt0,t(A2+)f(t, y) dµ = 0.

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For the closed and bounded set K ∈ A2+, we also get Z

A2+

f(t0, y) dµ = Z

K

f(t0, y) dµ + Z

A2+\K

f(t0, y) dµ

= Z

φt0,t(K)

f(t, y) dµ + η,

where η is a nonnegative number which can be sufficiently small while the set K is suitable chosen, R

A2+f(t0, y) dµ > 0 and R

φt0,t(K)f(t, y) dµ 6 0. Hence R

A2+f(t0, y) dµ = 0 and consequently also the set A2+ satisfies the condition (3).

Arguing as for the sets A1+ and A2+ we deduce that the sets A1− and A2−

satisfy condition (3), too. Hence the condition (3) is valid for every bounded set A∈ Ln, A ⊂ G.

If the condition (4) of Theorem 1 holds and the set A ∈ Ln, A ⊂ G is unbounded then the fact that A satisfies the condition (3) is a conclusion drawn from the following facts

(a) The space Rncan be represented as a countable sum of pairwise disjoint sets from Ln.

(b) Every bounded set A ⊂ G, A ∈ Ln, A ⊂ G satisfies the condition (3).

(c) The Lebesque integral is additive with respect to a set of integration.

This complets the proof of Theorem 2.

Let µ1 be an absolutely continuous (with respect to the Lebesque measure) mea- sure given by the formula

µ1(A) = Z

A

g(y) dµ f or A ∈ Ln, (14)

where g : Rn → R is a continuous function with nonnegative values and having continuous partial derivatives on Rn.

The following corollary is an immediate consequence of the definition of the local integral invariant of system (1), Theorem 2 and the condition ∂g∂t(y) = 0 for each y ∈ Rn and t ∈ [a, b].

Corollary 4. The measure defined by the formula (14) is an invariant measure of the local dynamical system generated by (1) if and only if Pn

i=1

∂(gFi)

∂yi = 0.

References

[1] F. Calgero and A. Degasperis, Spectral Transform and Solitons: Tools to Slove and Investigate Nonlinear Evotion Equations (New York, North-Holland, 1982).

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[2] D.G. Crowdy, General solutions to the 2D Liouville equation, Int. J. Engng Sci. 35 (1997) 141–149. doi:10.1016/S0020-7225(96)00080-8.

[3] R. Kubo, Stochastic Liouville equations, J. Math. Phys. 4 (1963) 174–183.

doi:10.1063/1.1703941.

[4] Y. Matsumo, Exact solution for the nonlinear Klein-Gordon and Liouville equations in four – dimensional Euklidean space, J. Math. Phys. 28 (1987) 2317–2322.

doi:10.1063/1.527764.

[5] P.K. Sahoo and T. Riedel, Mean Value Theorems and Functional Equations (World Scientific Publishing, Singapore, 1998).

[6] V.E. Tarasov, Stationary solutions of Liouville equations for non-Hamiltonian sys- tems, Ann. Phys. 316 (2005) 393–413. doi:10.1016/j.aop.2004.11.001.

Received 8 May 2013

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