ON THE NOTION OF L

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OF A STOCHASTIC FLOW ON A MANIFOLD

YU. E. GLIKLIKH AND L. A. MOROZOVA Received 14 June 2002

We introduce the notion ofL¹-completeness for a stochastic flow on manifold and prove a necessary and suﬃcient condition for a flow to be L¹-complete.

L¹-completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort ofL¹-functional space, natural for manifolds where no Riemannian metric is specified.

1. Introduction

The main goal of the paper is to investigate the conditions under which solutions of stochastic diﬀerential equations are well posed up to infinity, that is, complete.

At the moment it is a well-known fact that partial differential equations of parabolic type are related to stochastic differential equations. Thus, the completeness results for stochastic differential equations are useful for parabolic equations.

Stochastic differential equations may be considered as a special generalization of ordinary differential equations, and for the latter a necessary and sufficient condition for completeness was found in [5] (see also [6]). Consider a smooth vector field (right-hand side of the equation) on a finite-dimensional manifold M. It was proved that all solutions are well posed fort∈(−∞,∞) if and only if on the manifoldR×Mthere exists a complete Riemannian metric such that the vector field (1,X) is uniformly bounded with respect to it. Recall that the char- acteristic feature of the complete Riemannian metric is that its distance function is proper (seeDefinition 2.3below).

For stochastic differential equations a certain role of the vector filedX, mentioned above, is played by the second order differential operator called the generator of equation. We cannot apply the technique of Riemannian metrics to second order operators but we try to replace it by Elworthy’s methods from [3], where a certain sufficient condition for completeness of a stochastic differential equation in terms of values of the generator on proper functions was found.

2000 Mathematics Subject Classification: 58J65, 58J35, 60H10 URL:http://dx.doi.org/10.1155/S1085337502206053

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628 L -completeness of a stochastic flow

We introduce the notion ofL¹-completeness for a stochastic flow on a manifold that is a certain modification of ordinary stochastic completeness providing the property that the flow lives in some sort ofL¹-functional space, natural for manifolds where no Riemannian metric is specified. We prove that for this sort of completeness a certain condition of Elworthy’s type onR+×Mbecomes necessary and sufficient. As well as for ordinary differential equation, the use of productR+×Mis essential. As a corollary we obtain a necessary and sufficient condition ofL¹-stochastic completeness for a Riemannian manifold in terms of L¹-completeness of Wiener process on the manifold.

InSection 2, we discuss our notion ofL¹-completeness and give general definitions.Section 3is devoted to the proof of the main results.

2. Discussion and general definitions

LetMbe a finite-dimensional manifold. Consider a stochastic dynamical system (SDS) onM (see [3]) with the generatorᏭ(t,x). In local coordinates it is described in terms of a stochastic differential equation with smooth coefficients in It ô or in Stratonovich form. Since the coefficients are smooth we can pass from Stratonovich to It ô equation and vice versa. Denote byξ(s) :M→M,s∈[0,T], the stochastic flow of the SDS. For any pointx∈M and timetthe trajectory ξ_t,x(s),s∈[t,T], of this flow is the unique solution of the above equation with initial conditionsξt,x(t)=x. As the coefficients of equation are smooth, this is a strong solution and a Markov diffusion process given on a certain random time interval. Below we denote the probability space, where the flow is defined by (Ω,Ᏺ,P) and suppose that it is complete. We also deal with separable realiza- tions of all processes.

Definition 2.1. The flowξ(s) is complete on [0,T] ifξt,x(s) exists for any couple (t,x) and for alls∈[t,T].

Definition 2.2. The flowξ(s) is complete if it is complete on any interval [0,T]⊂ R.

Let the flowξ(s) be given on the vector spaceRⁿ. In this case it is natural to say that the flowξ(s) isL¹-complete on [0,T] if the following assumptions hold:

(i)ξ(s) is complete on [0,T];

(ii)Eξt,x(s)<∞for allt,x, ands∈[t,T] (i.e.,ξt,x(s) belongs to the space L¹for anys);

(iii) property (ii) holds for anys∈[t,T] if it holds for at least one s, for example, for initial instanttor final instantT(i.e., ifEξt,x(t) = x<

∞orEξ_t,x(T)<∞);

(iv) for any sthe trajectoryξt,x(s) is smooth enough in mean (i.e., inL¹) with respect to initial valuest,x, that isEξt,x(s)is smooth enough in tand inx. Some conditions providing such a smoothness are described, for example, in [1,4].

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On an arbitrary finite-dimensional manifold M, in the definition of L¹- completeness the norm should be replaced by another object, satisfying the properties of norm inRⁿsuch as its properness and smoothness.

Definition 2.3. The function f :M→Ris called proper if the preimage f⁻¹(A) of any compact setA⊂Ris compact inM.

LetMbe a complete Riemannian manifold. Denote byρthe distance function generated by the Riemannian metric of manifold. Specify a certain pointx0∈ M. For an arbitrary pointx∈M, the distanceρ(x0,x) is a natural analogue of the norm in vector space. SinceMis complete, it follows from the Hopf-Rinow theorem (see, e.g., [2]) that the functionρ(x0,·) is proper. Then an analogue of, for example, (ii) takes the form

Eρx0,ξ_t,x(s)<∞. (2.1) Unlike the norm inRⁿ, the functionρmay not be smooth at the points of cut locus. Neverthelessρ can be approximated by a smooth function with similar properties as follows.

Takeε >0 small enough and denote byUx(ε) a neighborhood ofx∈Mthat is the intersection of the metric ball with radiusε(with respect to distanceρ), centered atx, and of some chart containingx. SinceM is paracompact, we can select a countable locally finite subcovering{Uxi(ε)}from the covering{Ux(ε)}. Let{ϕ_i}^∞_i₌1be partition of unity adapted to this subcovering. Introduce the func- tionρ_ε(x)=_∞

i=1ϕ_i(x)ρ(x0,x_i). By the construction, functionρ_ε(x) is smooth (unlike the distanceρ(x0,x)).

Show that

ρx0,x−ρε(x)< ε, (2.2)

wherexis an arbitrary point inM. Indeed, from the triangle inequality we get

|ρ(x0,x)−ρ(x0,x_i)|< ρ(x_i,x)< εfor anyiand from the fact that the covering is locally finite it follows that

ρ(x0,x)−ρε(x)=

ρ(x0,x)− m i=p

ϕi(x)ρ(x0,xi)

≤

ρ(x0,x)− max

p≤i≤mρ(x0,xi) m i=p

ϕi(x)

=

ρ(x0,x)− max

p≤i≤mρ(x0,x_i)< ε.

(2.3)

From (2.2) one can easily see that the functionρ_ε(x) is also proper. Moreover, from (2.2) it follows thatρ_ε(x)< ρ(x0,x) +ε, so Eρ_ε(ξ)< Eρ(x0,ξ) +εfor any

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random elementξsuch thatEρ(x0,ξ) exists. Thus, if (2.1) holds, Eρε

ξt,x(s)<∞. (2.4)

So, if a complete Riemannian metric is given on a manifoldM and inequality (2.1) holds for the distance, there exists a smooth proper function onM for which (2.4) holds. Thus, it is more natural (and more general since a metric may not be given on the manifold a priori) to introduce the notion ofL¹- completeness using some proper function onMwith values inRinstead of the distance so that the function has some additional properties leading to require- ments analogous to (i), (ii), (iii), and (iv).

Definition 2.4. A flow ξ(s) on a finite-dimensional manifold M is calledL¹- complete on [0,T] if the following conditions are fulfilled:

(i)ξ(s) is complete on [0,T];

(ii) there exists a smooth proper positive function v:M→R¹ such that Ev(ξt,x(T))<∞for allx∈M,t∈[0,T];

(iii) for anyK >0 there exists a compactCK,T⊂M, depending onK andT, such that the inequalityEv(ξ_t,x(T))< Kyieldsx∈C_K,T;

(iv) the function f(t,x)=Ev(ξt,x(T)) is smooth.

Notice that the conditions of Definition 2.4are weakened analogues of re- quirements (i), (ii), (iii), and (iv). Namely, (ii) and (iii) together mean a reduced form of assumption, analogous to (ii) and (iii) together, and smoothness, pos- tulated in (iv), is analogous to that of (iv).

Definition 2.5. The flowξ(s) is L¹-complete if it is L¹-complete on any finite interval [0,T]⊂R+=[0,∞).

3. Main statements

Consider the direct productM^T =[0,T]×M and denote byπ^T:M^T →M the natural projection:π^T(t,x)=x. On the manifoldM^T consider the flowη(s)= (s,ξ(s)). Obviously, for any point (t,x)∈M^T the trajectory ofη(t,x)(s) satisfies the relationπ^T(η(t,x)(s))=ξ_t,x(s). It is clear thatη(s) is the flow generated by SDS with generatorᏭ^T determined by the formula

Ꮽ^T(t,x)=Ꮽ(t,x) + ∂

∂t. (3.1)

Let u:M^T →Rbe a proper function. Consider the sequence of compacts Wn=u⁻¹([0,n]) inM^T. Specify a point (t,x)∈M^T and fornsuch that (t,x)∈ W_ndenote byτ_nthe first exit time ofη(t,x)(s) fromT_n.

Theorem3.1. The flowξ(s)onMisL¹-complete on[0,T],T >0if and only if there exists a smooth proper positive functionuonM^Tsuch that for all(t,x)∈M^T,

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the equalityᏭ^T_(t,x)u=Cholds whereCis a certain constant, and for all(t,x)∈M^T the random variablesu(η(t,x)(T∧τn))are uniformly integrable.

Proof

Suﬃciency. Without loss of generality, we can suppose thatC=0. Indeed, let C >0, then construct f(t,x) :M→R, by the formula f(t,x)=u(t,x)−Ct+CT.

Evidently f(t,x)>0 for all (t,x)∈M^T, f(t,x) is smooth, proper, andᏭ^T(t,x)f = 0. IfC <0 construct f(t,x)=u(t,x)−Ct.

So, let there exist a smooth proper positive functionu(t,x) onM^T such that Ꮽ^T(t,x)u=0 at all points ofM^T. Then from [3, Theorem IX. 6A] it follows that η(s) is complete, that is, the trajectoriesη(t,x)(s)=(s,ξ_t,x(s)) do exist for all points (t,x)∈M^T and for alls∈[t,T]. Obviously this means that the flowξ(s) is also complete.

Consider the sequence of compactsW_n=u⁻¹([0,n]) inM^T and the corresponding sequence of first exit timesτnintroduced above.

Taking into account It ˆo formula and the definition of generator and applying the machinery of [3] (see Corollary III.7C and the proof of Theorem IX. 6A there) one easily obtains that

Euη(t,x)

T∧τn

=u(t,x) +E1 2

_T_∧_τ_n

t Ꮽ^T_η_(t,x)(s)uds. (3.2)

SinceᏭ^T_η_(t,x)(s)u=0 by the hypothesis, we have Euη(t,x)

T∧τ_n=u(t,x). (3.3)

Immediately from the construction, it follows that the random variables u(η(t,x)(T∧τn)) a.s. converge tou(η(t,x)(T)). Since by the hypothesis the variablesu(η(t,x)(T∧τ_n)) are uniformly integrable, this means that

Euη(t,x)(T)=u(t,x). (3.4) Consider the function

v(x)=u(T,x). (3.5)

From (3.4) and (3.5) it follows that

Evξ_t,x(T)=EuT,ξ_t,x(T)=Euη(t,x)(T)=u(t,x). (3.6) Thus,Ev(ξ_t,x(T))<∞andEv(ξ_t,x(T)) is smooth as a function of variables (t,x).

Suppose thatEv(ξt,x(T))< K, thenEu(η(t,x)(T))=u(t,x)< K. Sinceuis positive, this means that the value u(t,x) belongs to the compact [0,K]⊂Rand sinceuis proper, the setu⁻¹([0,K])⊂M^T is compact. Thus ifEv(ξ_t,x(T))< K, the pointxbelongs to the compactπ^T(u⁻¹([0,K])).

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So, conditions (i), (ii), (iii), and (iv) ofDefinition 2.4are fulfilled and the flow ξ(s) isL¹-complete. This proves the suﬃciency.

Necessity. Let the flowξ(s) onMbeL¹-complete on [0,T]. Denote byv:M→R the smooth proper positive function fromDefinition 2.4. Consider the function u(t,x)=Ev(ξ_t,x(T)). By (iv) ofDefinition 2.4it is smooth.

Sincevis proper, the sets ˜Wn=v⁻¹([0,n]) are compact. Denote by ˜τnthe first exit time ofξt,x(s) from ˜Wn. Consider

Eu(t+∆t)∧τ˜_n,ξ_t,x(t+∆t)∧τ˜_n=Euη(t,x)

(t+∆t)∧τ˜_n. (3.7) Since the sample trajectories ofξ(t,x)are considered up to the first exit time from compact ˜Wn, we can apply It ˆo formula. Hence, since the expectation of It ˆo integral here is equal to zero,

Eu(t+∆t)∧τ˜n,ξt,x

(t+∆t)∧τ˜n

=u(t,x) +E

₍_t_+∆t₎_∧τ˜n

t Ꮽ^Tuη(t,x)(s)ds.

(3.8) Notice thatEu((t+∆t)∧τ˜n,ξt,x((t+∆t)∧τ˜n))=u(t,x). Indeed,

Eu(t+∆t)∧τ˜_n,ξ_t,x(t+∆t)∧τ˜_n=EEvξ(t+∆t)∧τ˜n,ξt,x((t+∆t)∧τ˜n)(T)

=EEvξ_t,x(T)=Evξ_t,x(T)

=u(t,x).

(3.9)

Then, taking into account (3.8), we obtain

0=Eu(t+∆t)∧τ˜_n,ξ_t,x(t+∆t)∧τ˜_n−u(t,x)

=E

₍_t_+∆t₎_∧_τ_˜_n

t Ꮽ^Tuη(t,x)(s)ds. (3.10)

Multiply the equality (3.10) by 1/∆tand pass to the limit as∆t→0. Then, taking into account (3.1), we get

0=lim

∆t→0

1

∆tE

(t+∆t)∧τ˜n

t Ꮽ^Tuη(t,x)(s)ds

=lim

∆t→0

1

∆tE

(t+∆t)_∧τ˜n

t

∂us,ξ_t,x(s)

∂s +Ꮽus,ξ_t,x(s)ds.

(3.11)

Recall again that in the case under consideration the processξt,x(s) belongs to the compact ˜Wn, hence the functionuis considered on compact [0,T]×W˜nand so its values are bounded. This means that we can apply Lebesgue’s theorem on passing to limit under integral (i.e., expectation) and that there exists a random

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variables∈[t,(t+∆t)∧τ˜n] such that (t+∆t)∧τ˜n

t

∂us,ξt,x(s)

∂s +Ꮽus,ξt,x(s) ds

=∂us,ξ_t,x(s)

∂s +Ꮽus,ξ_t,x(s)(t+∆t)∧τ˜_n−t.

(3.12)

One can easily see that (t+∆t)∧τ˜n−t=((t+∆t)−t)∧(˜τn−t)=∆t∧(˜τn−t).

Thus

0=lim

∆t→0

1

∆tE

∂us,ξ_t,x(s)

∂s +Ꮽus,ξ_t,x(s) ∆t∧ τ˜_n−t

=Elim

∆t→0

∂us,ξt,x(s)

∂s +Ꮽus,ξt,x(s) ∆t∧ τ˜n−t

∆t .

(3.13)

Notice that ˜τn−t >0 a.s., since attthe processξt,xtakes the valuexand sot cannot be the first exit time from ˜Wn, and that ˜τn−tis bounded and does not depend on∆t. Hence

∆tlim→0

τ˜_n−t

∆t ^{= ∞}, (3.14)

and so

∆tlim→0

∆t∧ τ˜n−t

∆t ⁼1∧ lim

∆t→∞

τ˜_n−t

∆t ⁼1. (3.15)

Froms∈[t,(t+∆t)∧τ˜n] and from Lebesgue’s theorem it follows thats→tas

∆t→0. So, equality (3.13) takes the form

∂ut,ξt,x(t)

∂t +Ꮽut,ξt,x(t)=0. (3.16) This means that

Ꮽ^Tu(t,x)=0. (3.17)

Lemma3.2. The functionuonM^Tis proper.

Proof. Suppose thatuis not proper. Then there exists a sequence of points (tk,xk)

∈M^T such that 0< u(t_k,x_k)< K for allk, whereK >0 is a certain real number, whilev(xk)→ ∞ask→ ∞. Sincevis proper this means thatxkleaves any specified compact inM.

However, if 0< u(t_k,x_k)< K, thenEv(ξ_tk,xk(T))< K and since the flowξ(s) isL¹-complete, it follows from (iii) ofDefinition 2.4thatxkbelong to a certain compactCK,T.

Introduce compact setsW_n=u⁻¹([0,n]) and denote byτ_nthe first exit time ofη(t,x)(s) fromW_n. Consider the random variablesu(η(t,x)(T∧τ_n)). Taking into

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account the It ˆo formula, the properties of mathematical expectation and the equalityᏭ^Tu(t,x)=0 proved above, we obtain

Euη(t,x)

T∧τn

=u(t,x) (3.18)

for eachn.

From the construction of functionuand flowηwe also get

Euη(t,x)(T)=Evξt,x(T)=u(t,x). (3.19) As well as in the proof of suﬃciency one can easily see that the sequence u(η(t,x)(T∧τn)) tends a.s. tou(η(t,x)(T)). Thus, from (3.18) and (3.19) it follows that

Elim

n→∞uη(t,x)

T∧τ_n=Euη(t,x)(T)=u(t,x) (3.20)

and this means that the random variablesu(η(t,x)(T∧τn)) are uniformly integrable. This completes the proof of necessity. The theorem follows.

Corollary3.3. The flowξ(s)onM isL¹-complete if and only if for any finite interval[0,T]there exists a smooth proper positive functionu_TonM^Tsuch that for any(t,x)∈M^T, the equalityᏭ^T(t,x)uT=C^T holds, whereC^T is a certain constant that can depend onT and for eachT the random variablesuT(η(t,x)(T∧τn))are uniformaly integrable.

Recall the following definition.

Definition 3.4. A Riemannian manifoldMis called stochastically complete if the flow of SDS with generator (1/2)∆(Laplace-Beltrami operator) is complete.

Denote the above-mentioned flow byw(s). For anyx∈M, the corresponding w0,x(s) is a Wiener process onMstarting atx.

Definition 3.5. A Riemannian manifoldMis calledL¹-stochastically complete if the floww(s) isL¹-complete.

Introduce the operator∆^T=∆+∂/∂tonM^T=[0,T]×M.

Theorem3.6. A Riemannian manifoldMisL¹-stochastically complete if and only if for anyT >0on the manifoldM^T =[0,T]×M, there exists a smooth proper positive functionuTsuch that for all(t,x)∈M^T, the values∆^T(t,x)uT are equal to a certain constant that may depend onT, and such that the random variablesu(T∧ τ_n,w_t,x(T∧τ_n))are uniformly integrable.

Theorem 3.6follows fromCorollary 3.3.

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Acknowledgment

This work was partially supported by grant 99-00559 from INTAS, by grant UR.04.01.008 of the Program “Universities of Russia” and by U.S. CRDF-RF Ministry of Education Award VZ-010-0. The authors are indebted to K. D. El- worthy for very useful discussions.

References

[1] Ya. I. Belopol’skaya and Yu. L. Dalecky,Stochastic Equations and Diﬀerential Geom- etry, Mathematics and Its Applications (Soviet Series), vol. 30, Kluwer Academic Publishers, Dordrecht, 1990.

[2] R. L. Bishop and R. J. Crittenden,Geometry of Manifolds, Pure and Applied Mathe- matics, vol. 15, Academic Press, New York, 1964.

[3] K. D. Elworthy,Stochastic Diﬀerential Equations on Manifolds, London Mathemati- cal Society Lecture Note Series, vol. 70, Cambridge University Press, Cambridge, 1982.

[4] I. I. Gihman and A. V. Skorokhod,Introduction to the Theory of Random Processes, 2nd ed., Izdat. Nauka, Moscow, 1977 (Russian).

[5] Ju. E. Gliklih,Conditions for nonlocal continuability of the integral curves of vector fields, Diﬀerencial’nye Uravnenija13(1977), no. 4, 743–744 (Russian).

[6] Yu. Gliklikh,Global Analysis in Mathematical Physics. Geometric and Stochastic Meth- ods, Applied Mathematical Sciences, vol. 122, Springer-Verlag, New York, 1997.

Yu. E. Gliklikh: Mathematics Faculty, Voronezh State University, 394006 Voronezh, Russia

E-mail address:yeg2000@pisem.net, yuri@yeg.vsu.ru

L. A. Morozova: Mathematics Faculty, Voronezh State University, 394006 Voronezh, Russia