Viability property on Riemannian manifolds

Probability Theory

Viability property on Riemannian manifolds

Shige Peng

, Xuehong Zhu

aInstitute of Mathematics, Shandong University, Jinan, 250100, China

bSchool of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China Received 16 October 2008; accepted after revision 2 October 2009

Available online 10 November 2009 Presented by Marc Yor

Abstract

This Note studies a sufficient and necessary condition for the viability property of a state system in a closed subsetKof a finite-dimensional compact Riemannian manifold without boundary. Our result is: the system enjoys the viability property inKif and only if the square of the distance function ofKis a viscosity supersolution of a second-order partial differential equation in some neighborhood ofK.To cite this article: S. Peng, X. Zhu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Propriété de viabilité sur une variété riemannienne.Dans cette Note on donne une condition nécessaire et suffisante pour que soit satisfaite la propriété de viabilité d’un système sur un sous-ensembleKd’une variété riemannienne, de dimension finie, sans bord. Le résultat s’énonce ainsi : le système surKpossède la propriété de viabilité si et seulement si le carré de la fonction distance àKest une sursolution de viscosité d’une équation aux dérivées partielles du second ordre définie sur un voisinage deK.

Pour citer cet article : S. Peng, X. Zhu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version française abrégée

Soient(Ω,F, P )un espace de probabilité complet et(W (t ), t 0)un mouvement brownien standard sur cet espace à valeurs dansR^d. Soit(Ft)_t0la filtration naturelle complète engendrée par(W (t ), t0)et augmentée des ensembles deP-mesure nulle deF.

Soit M un variété riemannienne complète et sans bord. SurM on considère dans l’intervalle[t, T], l’équation différentielle stochastique suivante :

dX_s^t,x=V0(s, X_s^t,x) ds+Vα(s, X_s^t,x)◦dW_s^α,

X^t,x_t =x∈M, (1)

oùV₀, V₁, . . . , V_dsontd+1 champs vectoriels réguliers surMparamétrés pars∈ [0, T].

D’après [3], il existe un unique processus, à valeurs dansM, qui résout l’équation (1).

E-mail addresses:[email protected] (S. Peng), [email protected] (X. Zhu).

1631-073X/$ – see front matter ©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2009.10.007

Soiti_M(x)le rayon d’injectivité deMenx. On a : i_M:=inf

i_M(x): x∈M

>0.

La distance dexà un ensembleK⊂Mest notée pardK(x):=infy∈Kd(x, y), oùd(·,·)est la distance riemannienne deM. Soit,

Γ_ε=

y∈M|d_K(y) < ε , et

τ =inf

st: dK

X^t,x_s

=ε , oùε=^iM₆.

SoitV˜_αle champ vectoriel surM×Mdéfini par :

∀(t, x, y)∈ [0,+∞)×M×M, V˜_α(t, x, y)=

V_α(t, x), V_α(t, y) . Nos hypothèses sont les suivantes :

pour tout couplex, y∈Γ_ε, on ad(x, y) <min{i_M(x), i_M(y)},t∈ [0,+∞), (H1) LxyV0(t, x)−V0(t, y)μd(x, y),

(H2) _d

α=1D²(d²(x, y))V˜_α(t, x, y),V˜_α(t, x, y) 2C0d²(x, y), (H3) _d

α=1L_xy∇V_α(t,x)V_α(t, x)− ∇V_α(t,y)V_α(t, y)C^∗₁d(x, y), (H4) _d

α=1L_xyV_α(t, x)−V_α(t, y)C₂^∗d(x, y),

oùC0,μ,C₁^∗etC^∗₂sont des constantes positives telles queC:=2μ+C0+C₁^∗+1>0.

Etant donnéKun ensemble fermé deM. Nous considérons la fonction suivante : U (t, x)=E

T∧τ t

e^{−C(s−t )}d_K² X^t,x_s

ds+e^{−C(T∧τ−t )}d_K² X^t,x_T∧τ

, (t, x)∈ [0, T] ×overlineΓ_ε. D’après [4],U (t, x)est l’unique solution de viscosité de l’équation de Hamilton–Jacobi :

⎧⎪

⎨

⎪⎩

u_t(t, x)+(V₀u)(t, x)+¹_2d

α=1(V_αV_αu)(t, x)+d_K²(x)−Cu(t, x)=0, (t, x)∈(0, T )×Γ_ε, u(t, x)=d_K²(x), (t, x)∈(0, T] ×∂Γ_ε,

u(T , x)=d_K²(x), x∈Γε.

(2)

Théorème 0.1.Sous les hypothèses(H1)–(H4), les assertions suivantes sont équivalentes: (i) Kest viable pour(1)¹;

(ii) d_K²(x)est une sursolution de viscosité de(2).² 1. Introduction

Let(W (t ), t0)be ad-dimensional standard Brownian motion on some complete probability space(Ω,F, P ).

We denote by(Ft)_t0the natural filtration generated byWand augmented by theP-null sets ofF.

In the sequel, we assume thatMis a compact Riemannian manifold without boundary. We consider the following stochastic differential equation onMin a fixed time interval[t, T]:

dX^t,x_s =V₀(s, X^t,x_s ) ds+V_α(s, X^t,x_s )◦dW_s^α,

X^t,x_t =x∈M, (3)

whereV₀, V₁, . . . , V_dared+1 one-parameter smooth vector fields onM.

1 On dit queKest viable si, quitte à changer d’espace de probabilié, pour tout(t, x)∈ [0, T] ×K, alors,P-p.s.X^t,x_s ∈K,∀s∈ [t, T]. 2 Pour la définition d’une sursolution de viscosité, voir [1].

SinceMis compact and without boundary, according to [3], there exists a uniqueM-valued continuous process which solves Eq. (3). Moreover, this solution does not blow up.

LetK be a given closed subset ofM. The aim of this paper is to give a necessary and sufficient condition that keeps the state(X^t,x_s , tsT )inKwheneverx∈K.

Definition 1.1.We say thatKenjoys the viability property with respect to (1) if, for all(t, x)∈ [0, T] ×K, there exist a probability space(Ω,F, P ), ad-dimensional Brownian motionW, such that,P-a.s.,X^t,x_s ∈K,∀s∈ [t, T].

The caseM=Rⁿhas been studied in [2] for time independent coefficients. However, general manifolds have not been considered in [2]. Our approach is similar to that in [2].

Leti_M(x)denote the injectivity radius ofMatx. SinceMis compact, we know iM :=inf

iM(x): x∈M

>0.

We denote byd_K(x):=infy∈Kd(x, y)the distance fromxtoK, whered(·,·)is the Riemannian distance inM. Set, Γε=

y∈MdK(y) < ε and

τ=inf

st: d_K X_s^t,x

=ε ,

whereε=^iM₆. Then our problem relates with the following function:

U (t, x)=E T∧τ

t

e^{−C(s−t )}d_K² X_s^t,x

ds+e^{−C(T∧τ−t )}d_K² X_T^t,x_∧τ

, (t, x)∈ [0, T] ×Γ_ε. (4) The constantCin (4) introduced later depends on thed+1 one-parameter tangent vector fields in (3).

We denote byL_xythe parallel transport along the unique minimizing geodesic connectingx toy. LetV˜_α denote the one-parameter tangent vector field onM×Mwhich is defined as follows:

∀(t, x, y)∈ [0,+∞)×M×M, V˜_α(t, x, y)=

V_α(t, x), V_α(t, y)

. (5)

We assume that, for allx, y∈Γεs.t.d(x, y) <min{iM(x), iM(y)},t∈ [0,+∞), (H1) LxyV0(t, x)−V0(t, y)μd(x, y),

(H2) _d

α=1D²(d²(x, y))V˜_α(t, x, y),V˜_α(t, x, y) 2C0d²(x, y), (H3) _d

α=1L_xy∇Vα(t,x)V_α(t, x)− ∇Vα(t,y)V_α(t, y)C₁^∗d(x, y), (H4) _d

α=1L_xyV_α(t, x)−V_α(t, y)C₂^∗d(x, y),

whereC₀,μ,C₁^∗andC₂^∗are positive constants, andC:=2μ+C₀+C₁^∗+1>0.

Consider now the following Hamilton–Jacobi equation:

⎧⎪

⎨

⎪⎩

ut(t, x)+(V0u)(t, x)+¹₂d

α=1(VαVαu)(t, x)+d_K²(x)−Cu(t, x)=0, (t, x)∈(0, T )×Γε,

u(t, x)=d_K²(x), (t, x)∈(0, T] ×∂Γ_ε,

u(T , x)=d_K²(x), x∈Γ_ε.

(6)

Definition 1.2. (See [1].) A continuous function u: [0, T] ×Γε→R is a viscosity supersolution (subsolution) of (6) if u(t, x)()d_K²(x)for all (t, x)∈(0, T] ×∂Γ_ε and u(T , x)()d_K²(x), for all x ∈Γ_ε, and for any ϕ∈C^1,2((0, T )×Γ_ε)and any(t, x)∈(0, T )×Γ_εat whichu−ϕattains a local minimum (maximum),

ϕ_t(t, x)+(V₀ϕ)(t, x)+1 2

d α=1

(V_αV_αϕ)(t, x)+d_K²(x)−Cu(t, x)()0.

Ifuis both a viscosity subsolution and a viscosity supersolution of (6), we say thatuis a viscosity solution of (6).

We will see in the next section thatU (t, x)is the unique viscosity solution of (6). The last section is devoted to the study of a sufficient and necessary condition for the viability property ofKwith respect to (3).

2. The unique viscosity solution to the Hamilton–Jacobi equations

For completeness, we first recall the comparison theorem of viscosity solution to the following parabolic PDE on Riemannian manifolds:⎧

⎪⎨

⎪⎩

u_t+F (t, x, u, du, d²u)=0 in(0, T )×Γ_ε, u(t, x)=f (t, x), (t, x)∈ [0, T )×∂Γ_ε, u(0, x)=ψ (x), x∈Γ_ε,

(7)

wheredu,d²umeand_xu(t, x)andd_x²u(t, x)(see [1]).T >0,ψ∈C(Γ_ε)andf∈C([0, T )×∂Γ_ε)are given.

We denote by T M_x^∗ the cotangent space ofM at a pointx∈M. LetT Mx stand for the tangent space atx and L²s(T Mx)denote the symmetric bilinear forms onT Mx. Set,

χ:=

(t, x, r, ζ, A): t∈ [0, T], x∈M, r∈R, ζ∈T M_x^∗, A∈L²s(T M_x) .

Proposition 2.1.(See [4].) LetΓ_ε be an open subset of a compact finite-dimensional Riemannian manifoldM, and F ∈C(χ , R) be continuous, proper for each fixed t ∈(0, T )and satisfy: there exists a function ω: [0,+∞] → [0,+∞]withω(0+)=0and such that

F

t, y, r, αexp⁻_y¹(x), Q

−F

t, x, r,−αexp⁻_x¹(y), P

ω

αd²(x, y)+d(x, y)

, (8)

for all fixedt ∈(0, T )and for allx, y∈Γ_ε which satisfyd(x, y) <min{i_M(x), i_M(y)},r∈R,P ∈L²_s(T M_x),Q∈ L²_s(T M_y)with

− 1

εα + A_α I 0 0 I

P 0

0 −Q

A_α+ε_αA²_α, (9)

whereA_α is the second derivative of the functionϕ_α(x, y)=^α₂d²(x, y) (α >0)at the point(x, y)∈M×M, and

ε_α= 1

2(1+ A_α).

Ifu(resp.,v) is an upper (resp., lower) semicontinuous function on[0, T )×Γ_εand a subsolution (resp., superso- lution) of (7) thenuvon[0, T )×Γ_ε.

In particular PDE (7) has at most one viscosity solution.

Proposition 2.2.The functionU (t, x)introduced in(4)is bounded and continuous in[0, T] ×Γ_ε. Moreover, it’s the unique viscosity solution of (6).

Proof. The functionU (t, x) is bounded since M is compact. For the proof of the continuity of U (t, x), see our forthcoming paper.

If we note that U (t, x)=E

(t+δ)∧τ t

e^{−C(s−t )}d_K² X_s^t,x

ds+e^{−C[(t+δ)∧τ−t]}U

(t+δ)∧τ, X^t,x_(t+δ)∧τ ,

we can use Definition 1.2 to prove, with the help of Itô’s formula thatU (t, x)is a viscosity solution of (6).

Then we will use Proposition 2.1 to get the uniqueness result. After transforming our PDE (6) to the standard form in (7), our functionF :χ→Ris as follows:

F (t, x, r, ζ, P )= −d_K²(x)+Cr−

ζ, V₀(t, x)

−1 2

d α=1

ζ,∇V_α(t,x)V_α(t, x)

−1 2

d α=1

P V_α(t, x), V_α(t, x) . (10) SinceMis compact, there exists a constantk0>0 s.t. the sectional curvature is bounded below by−k0onM. Then, according to [1], (9) implies:

P −L_yx(Q)3

2k₀αd²(x, y)I.

With the help of this relation and (H1)–(H4), we can easily prove that there exists a functionω: [0,+∞] → [0,+∞]

withω(0+)=0 such that (8) holds true for our functionF introduced in (10). So by Proposition 2.1, we can get the uniqueness result. 2

3. Characterization of the viability property

Theorem 3.1.Under the assumptions(H1)–(H4), the following conditions are equivalent:

(i) Kenjoys viability with respect to(3);

(ii) d_K²(x)is a viscosity supersolution of (6).

Proof. (ii)⇒(i): Ifd_K²(x)is a viscosity supersolution of (6), then, by Propositions 2.1 and 2.2, one hasU (t, x) d_K²(x),∀(t, x)∈(0, T] ×Γε. Thus, in particular, for all(t, x)∈(0, T] ×K,U (t, x)=0, what is equivalent to: for all t∈ [0, T],

X_s^t,x∈K, ∀s∈ [t, T], P-a.s.

(i)⇒(ii): For any given(t, x)∈(0, T )×Γ_ε, let x¯∈K such thatd_K²(x)=d²(x,x)¯ (ifx∈K, x= ¯x). Suppose thatKenjoys the viability property for (1). Then we have that, a.s.,X^t,_s^x¯∈K,∀s∈ [t, T]. Letϕ∈C^1,2((0, T )×Γ_ε) satisfies:

d_K²(x)−ϕ(t, x)=0d_K² x

−ϕ t, x

, for t, x

in a neighborhood of(t, x). (11)

For all 0< η < ε, we define:

τ_η=η∧inf

s0, d

x, X_t^t,x_+s

> η

∧inf

s0, d

¯ x, X_t^t,₊^x¯_s

> η . By (9), forη >0 small enough, we have:

ϕ

t+τ_η, X^t,x_t+τ

η

−ϕ(t, x)d_K² X^t,x_t+τ

η

−d_K²(x). (12) Applying Itô’s formula to the left-hand side of this inequality, we get

E ϕ

t+τ_η, X_t^t,x_+τ

η

−ϕ(t, x)

=E

t+τ_η

t

ϕ_s

s, X_s^t,x

+(V₀ϕ) s, X^t,x_s

+1 2

d α=1

(V_αV_αϕ) s, X_s^x

ds.

With the help of this relation we can prove easily that

ηlim→0

E[ϕ(t+τ_η, X^t,x_t+τ

η)−ϕ(t, x)]

Eτ_η =ϕt(t, x)+(V0ϕ)(t, x)+1 2

d α=1

(VαVαϕ)(t, x). (13)

Let us now consider the right-hand side of (12). By the viability assumption, we have X^t,_t+^x¯_τ

η ∈ K. Thus d_K²(X^t,x_t+τη)d²(X^t,x_t+τη, X_t^t,₊^x¯_τη). On the other hand, for alls∈ [0, τη],

d

X^t,x_t+s, X_t^t,₊^x¯_s

d

X^t,x_t+s, x

+d(x,x)¯ +d

¯ x, X^t,_t+^x¯_s

<3ε=i_M 2 .

Since d²(x, y) is C^∞ smooth on the set {(x, y)∈ M ×M: d(x, y) < ^iM₂}, we can apply Itô’s formula to d²(X^t,xs , X^t,s^x¯), s∈ [t, t+τη],

E d²

X^t,x_t+τη, X_t^t,₊^x¯_τη

−d²(x,x)¯

=E

t+τη

t

V˜0

s, X_s^t,x, X_s^t,x¯ d²

X^t,x_s , X^t,_s^x¯ +1

2 d α=1

(V˜αV˜α)

s, X^t,x_s , X^t,_s^x¯ d²

X_s^t,x, X_s^t,x¯

ds, (14)

where the tangent vector fieldV˜_α onM×Mhas been defined in (5).

According to [1],

∂

∂xd²(x, y)= −2 exp⁻_x¹(y), ∂

∂yd²(x, y)= −2 exp⁻_y¹(x), ∂

∂xd²(x, y)+L_yx ∂

∂yd²(x, y)=0.

Thus from (H1), we have:

V˜0

s, X^t,x_s , X_s^t,x¯ d²

X_s^t,x, X_s^t,x¯

= V₀

s, X_s^t,x

,−2 exp⁻_X¹t,x s

X_s^t,x¯ +

V₀ s, X_s^t,x¯

,−2 exp⁻¹

X^t,_s^x¯

X_s^t,x

= V₀

s, X_s^t,x ,2L_X^t,x¯

s X^t,x_s exp⁻¹

X_s^t,x¯

X^t,x_s +

V₀ s, X^t,_s^x¯

,−2 exp⁻¹

X_s^t,x¯

X^t,x_s

=2 L_Xt,x

s X_s^t,x¯V0

s, X^t,x_s

−V0

s, X^t,_s^x¯ ,exp⁻¹

X_s^t,x¯

X^t,x_s 2μd²

X_s^t,x, X^t,_s^x¯ ,

and combining this estimate with (H2), (H3) and (14), we get:

E d²

X_t^t,x_+τ

η, X^t,_t+^x¯_τ

η

−d²(x,x)¯

2μ+C₀+C₁^∗ E

t+τ_η

t

d²

X^t,x_s , X^t,_s^x¯ ds

(C−1)E

t+τη

t

1+1

η

d² X^t,x_s , x

+

1+1 η

(1+η)d² X_s^t,x¯,x¯

+(1+η)²d²(x,x)¯

ds

(C−1)(1+η)²Eτ_ηd²(x,x)¯ +O(η)Eτ_η. Substitution of the latter estimate and of (13) and (12) yields:

ϕ_t(t, x)+(V₀ϕ)(t, x)+1 2

d α=1

(V_αV_αϕ)(t, x)lim

η→0

E[d_K²(X_t^t,x_+τη)−d_K²(x)]

Eτη (C−1)d²(x,x).¯ Finally, fromd²(x,x)¯ =d_K²(x)=ϕ(t, x)we deduce that

ϕ_t(t, x)+(V₀ϕ)(t, x)+1 2

d α=1

(V_αV_αϕ)(t, x)+d_K²(x)−Cϕ(t, x)0. 2

Remark.Our assumptions are just the analogue to the Lipschitz conditions for the coefficients of SDEs inRⁿ. Our result extends that of [2] to SDEs on Riemannian manifolds.

References

[1] D. Azagra, J. Ferrera, B. Sanz, Viscosity solutions to second order partial differential equations on Riemannian manifolds, J. Differential Equations 245 (2008) 307–336.

[2] R. Buckdahn, S. Peng, M. Quincampoix, C. Rainer, Existence of stochastic control under state constraints, C. R. Acad. Sci. Paris 327 (1998) 17–22.

[3] Elton P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, 2002.

[4] X. Zhu, Viscosity solutions to second order parabolic PDEs on Riemannian manifolds, preprint.