Potential Analysis 3: 339-357, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands'.

Properties at Infinity of Diffusion Semigroups and Stochastic Flows via Weak Uniform Covers

XUE-MEI LI Department of Mathematics, University of Warwick, Coventry CV4 7AL, U.K.

(Received: 5 August 1992; accepted: 19 September 1993)

Abstract. A unified treatment is given of some results of H. Donnelly, P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifieations, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.

Mathematics Subject Classifications (1991). 31B25, 60H30, 60J50, 60J60.

Key words. Brownian motion, diffusion, heat semigroup, Riemannian manifold, compactification, boundary, uniform cover.

1. Introduct ion

Let M be a manifold, {f~, Y , ~ , P} a filtered p robab i l i t y space. A diffusion process

on M is a s tochast ic process which is pa th con t inuous and s t rongly Markov . These

diffusions are often given by solut ions to s tochast ic differential equa t ions of the fol lowing type:

dx t = X(x,)o dB t + A(xt)dt (1)

Here B t is a R" valued Brownian mot ion , X is C 2 f rom M x R" to the tangent bundle

T M with X ( x ) : R " - - . T~M a l inear m a p for each x in M, A is a C 1 vector field on

M. Let u be an M-va lued r a n d o m var iable independen t of ~'~o- Deno te by F~(u) the

so lu t ion s ta r t ing f rom u, ~(u) the explos ion time. Then {Ft(x)} is a diffusion for each

x ~ M . Assoc ia ted with F t there are also the p robab i l i s t i c semigroup Pt and the co r re spond ing infini tesimal genera to r ~ .

I. MAIN RESULTS

The ma in a im [17] of this ar t ic le is to give unified t r ea tment to some of the results

f rom H. Donnel ly , P. Li and L. Schwartz. I t gives first a p robab i l i s t i c in te rp re ta t ion

3 4 0 X U E - M E I LI

and extends part of the latter. We first introduce weak uniform covers in an analogous way to uniform covers, which gives a nonexplosion test by using estimates on exit times of the diffusion considered. As a corollary this gives the known result on nonexptosion of a Brownian motion on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function [14].

One interesting example is that a solution to a stochastic differential equation on R" whose coefficients have linear growth has no explosion and has the C o property. Notice under this condition, its associated generator has quadratic growth. On the other hand let M = R n, and let L be an elliptic differential operator:

~2 b" ~ L = ~ al;' i c?x~---j + ~. ~ c~x i

where aij and bi are C a. Let (sij) be the positive square root of the matrix (a~j). Let X ~ = Zj s~j c3/c?xj, A = Zj bj ~?/~xj. Then the s.d.e, defined by:

i i (ItO) dx t = ~ X (x~)dB t + A(x~)dt

has generator L. Furthermore if t(aij)l has quadratic growth and b~ has linear growth, then both X and A in the s.d.e, above have linear growth. In this case any solution u, to the following partial differential equation:

~u~ = Lu t 8t

satisfies: ut e Co(M), if u o ~ Co(M) (see next part).

II. P R E L I M I N A R I E S

A diffusion process is said to be a C o diffusion if its semigroup leaves invariant C o (M), the space of continuous functions vanishing at infinity, in which case the semigroup is said to have the Co property. A Riemannian manifold is said to be stochastically complete if the Brownian motion on it is complete, it is also said to have the C o property if the Brownian motion on it does. A Brownian motion is, by definition, a path continuous strong Markov process with generator i z/k, where A denotes the Laplacian Beltrami operator. Equivalently a manifold is stochastically complete if the minimal heat semigroup satisfies: P,I = 1.

One example of a Riemannian manifold which is stochastically complete is a complete manifold with finite volume. See Gaffney [11]. More generally a complete Riemannian manifold with Ricci curvature bounded from below is stochastically complete and has the Co property as proved by Yau [20]. See also Ichihara [14], Dodziuk [6], L. Karp and P. Li [16], Bakry [2], Grigory/m [12], Hsu [13], Davies

PROPERTIES AT INFINITY 341

[5], and Takeda [19] for further discussions in terms of volume growth and bounds on Ricci curvature. For discussions on the behaviour at infinity of diffusion processes, and the C o property, we refer the reader to Azencott [1] and Elworthy [10].

Those papers above are on a Riemannian manifold except for the last reference, for a manifold without a Riemannian structure, Elworthy [9] following It6 [15] showed that the diffusion solution to (1) does not explode if there is a uniform cover for the coefficients of the equation. See also Clarke [4]. In particular this shows that the s.d.e. (1) does not explode on a compact manifold if the coefficients are reasonably smooth. See [3], [8]. To apply this method to check whether a Riemannian manifold is stochastically complete, we usually construct a stochastic differential equation whose solution is a Brownian motion.

HI. HEAT EQUATIONS, SEMIGROUPS, AND FLOWS

Let M be a compactification of M, i.e., a compact Hausdorff space. We assume M is first countable. Let h be a continuous function on M. Consider the following heat equation with initial boundary conditions:

O--j= zSf, x e M (2)

f (x ,O)=h(x) , x ~ I (3)

f (x , t) = h(x), x e ~?M (4)

It is known that there is a unique minimal solution satisfying the first two equations on a stochastically complete manifold, the solution is in fact given by the semigroup associated with Brownian motion on the manifold applied to h. So the above equation is not solvable in general. However, with a condition imposed on the boundary of the eompactification, Donnelly and Li [7] showed that the heat semigroup satisfies (4). Here is the condition and the theorem:

THE BALL CONVERGENCE CRITERION. Let {x,} be a sequence in M converging to a point 2 on the boundary, then the geodesic balls ~,(x,), centered at x, of radius r, converge to 0~ as n goes to infinity for each fixed r.

An example of a manifold which satisfies the ball convergence criterion is R" with sphere at infinity, but not the compactification of a cylinder with a circle at infinity added at each end. The one point compactification also satisfies the ball convergence criterion.

THEOREM 1.1 (H. Donnelly and P. Li). Let M be a complete Riemannian manifold with Ricci curvature bounded from below. The over determined system (2)-(4) is solvable

342 XUE-MEI LI

for any given continuous function h on M, if and only if the ball convergence criterion

holds for ~ .

Notice if the Brownian motion starting from x converges to some point on the boundary to which x converges, (2)-(4) is clearly solvable. See Section 5 for details. We would also like to consider the opposite question: Do we get any information on the diffusion processes if we know the behaviour at infinity of the associated semigroups? This is true for many cases. In particular for the one point compactification, Schwartz has the following theorem [18-1, which provides a partial converse to 1-10]:

T H E O R E M 1.2 (L. Schwartz). Let F t be the standard extension of F t to M = M w { ~ }, the one point compactification. Then the map (t, x ) ~ F~(x) is continuous from R + × ]9I to L°(f~, P, M), the space of measurable maps with topology induced from convergence in probability, if and only if the semigroup Pt has the Co property and the map t ~ P t f

is continuous from R + to C o (M).

2. Weak Uniform Cover and Nonexplosion

D E F I N I T I O N 2.t [9-1. A stochastic dynamical system (1) is said to admit a uniform cover (radius r > 0, bound k), if there are charts {~b,, U,} of diffeomorphisms from open sets U, of the manifold onto open sets ~b,(U,) of R", such that:

1. B3~ c qSi(Ui), each i. (B~ denotes the open ball about 0, radius e). 2. The open sets {q~Tl(Br)} cover the manifold. 3. If (qSi),(X) is given by:

(~i),(X)(v)(e) = (D~i)¢; ,(~,)X(dp~ lv)(e)

with (qS~),(A) simitary defined, then both (~b,),(X) and d(~bl) are bounded by k on B2r, here d is the generator for the dynamical system.

Ui

~i(Ui)

PROPERTIES AT INFINITY 343

Let {ej} be an orthonormal basis for R.,, and define X j to be X(x)(ej). Then for Stratonovich equation (1) on M,

d(e)~) = ~ xkx~(~,) + d4),(A). k,l = 1

Let M = R". If we replace the Stratonovich differential in (1) by the It6 differential, then

1 m

~(~i ) (X) -~- 2 k~l=l O2~i(x)(Xk(x)' x l (x)) + O~i(x)(a(x)),

which does not involve any of the derivatives of XJ's.

D E F I N I T I O N 2.2. A diffusion process {Ft, 3) is said to have a weak uniform cover if there are pairs of connected open sets {U °, U,}, and a non-increasing sequence {6,} with 6, > 0, such that:

1. U ° c U,, and the open sets {U °} cover the manifold. For x~ U ° denote by C(x) the first exit time of F,(x) from the open set U,. Assume z" < ~ unless C = co.

2. There exists {K,},~I , a family of increasing open subsets of M with wK, = M, such tha t each U, is contained in one of these sets and intersects at most one boundary from {~K,,}m~=l.

3. Let x s U ° and U. ~ K m, then for t < 6.,:

P{co:C(x) < t} <~ Ct 2 (5)

4. E,~ 1 6, = oe.

Km

Notice the introduction of {K,} is only for giving an order to the open sets {U.}.

This is quite natural when looking at concrete manifolds. In a sense the condition says the geometry of the manifold under consideration changes slowly as far as the diffusion process is concerned. In particular if 6, can be taken all equal, we take K. = M; e.g., when the number of open sets in the cover is finite. On a Riemannian manifold the open sets are often taken as geodesic balls.

344 XUE-MEI LI

LEMMA 2.1. Assume there is a uniform cover for the stochastic dynamic system (1), then the solution has a weak uniform cover with 6, = 1, all n.

Proof This comes directly from Lemma 5, p. 127 in [9].

REMARKS.

1. Let T be a stopping time, the inequality (5) gives the following from the strong

Markov property of the process:

let V c U °, and V c K,,, then when t < 6,,:

P{z"(FT(X)) < tlFT(X)a V} <~ Ct 2, (6)

since

l ' P {z"(Fr(x)) < t, Fr(x ) ~ V} = Jv P(z"(y) < t)Pr(x, dy) <<. Ct2p{Fr(x) V},

here PT(X, dy) denotes the distribution of Fr(x ). 2. Denote by pV. the heat solution on U, with Dirichlet boundary condition, then

(6) is equivalent to the following: when x ~ U °,

1 - PV"(1)(x) ~< Ct 2 (7)

3. The methods in this article work in infinite dimensions to give analogous results.

EXIT TIMES. Given such a cover, let x e U °. We define stopping times {Tk(x)} as follows: Let T o = 0. Let Tl(x ) = inf{t > 0:Ft(x, to)¢U-£, } be the first exit time of f t (x ) from the set U,. Then Frl(X, to) must be in one of the open sets {U°}. Let

nl = {o):Frl(x)(to)e U °, Tl(X, to) < oo}

n,~ to :F~(x )e u ° o = - - U i , T 1 < c o

j = l

Then {gl~} are disjoint sets such that ~ = {T 1 < oo}. In general we only need to consider the nonempty sets of such. Define further the following: Let T 2 = m, if T 1 = oo. Otherwise if to a gl~, let:

Tz(X , co) = TI(x , to) + zk(FTI(X, CO)) (8)

In a similar way, the whole sequence of stopping times {Tj(x)} and sets {II~}L1 are defined for j = 3, 4 . . . . . Clearly IY~ so defined is measurable with respect to the

sub-algebra ~-rj.

PROPERTIES AT INFINITY 345

L E M M A 2.2. Given a weak uniform cover as above. Let x ~ U ° and U, ~ Km. Let

t < 5m+ k. Then

P{co:Tk(x, co ) - Tk_l(X,(.O ) < t, Tk_ 1 < 00} ~ C t 2 . (9)

Proof Notice for such x, FTk(X)~Km+k_ P Therefore for t < 5m+ k we have

P{o~:r~(x, co) - r~_l(x, co) < t , ~ _ , < oo}

j = l

= ~ P{zJ(FT~_,(x))< t,~)~-1} j = l

.< ct2 ct2, j = l

as in Remark 1. Here ZA is the characterist ic function for a measurable set A, and E

denotes taking expectat ion. •

L E M M A 2,3. I f Z , t, = 0% t, > 0 non-increasing. Then there is a non-increasing

sequence {s,}, such that 0 < s, <~ t,:

(i) Y .s .= oo (ii) Z s 2 < oo.

Proof Assume t. ~< 1, all n. G r o u p the sequence {t.} in the following way:

t 1; t 2 , • • • , t k 2 ; t k 2 + 1 , " • • , t k a ; t k a + 1 • • •

such tha t 1 ~< t 2 + " " q'- tka ~ 2, 1 <. tk,+l + "'" + tk,+, ~< 2, i /> 2. Let s 1 = tl,

s z = tz/2 . . . . . Sk2 = tkJ2, Sgz + 1 ---- tk2 + 1/3, "'" , Sk~ = tga/3, Sk3 + 1 = tk3 + 1/4 . . . . . Clearly the s, 's so defined satisfy the requirements. •

So without losing generality, we m a y assume from now on that the constants {6,} for a weak uniform cover fulfill the two condit ions in the above lemma. With these established, we can now state the nonexplos ion result. The p roof is ana logous to the a rgumen t of Theo rem 6 on p. 129 in [9].

T H E O R E M 2.4. I f the solution F,(x) o f the equation (1) has a weak uniform cover,

then it is complete (nonexplosion),

Proo f Let x ~ K, , t > 0, 0 < e < 1. Pick a number p (may be depending on e and

n), such tha t ~i=,+l';'"+v ~6 i > t, which is possible from: Z ®i=l 6~ = Go. So

346

P{~(x) < t} ~<

~<

~<

P{Tp(x) < t, Tp_ 1 < or}

P { ~ I (Tk(x) - k Tk-I(X)) < t' TP-1 < ~ }

P

2 P{T~(x)- Tk_l(x ) <e6.+k, Tk_ 1 < o0} k = l

n+p

Z < 2 k=n k = l

Let e ~ 0, we get: P { ~(x) < t} = 0.

XUE-MEI LI

REMARK. The argument in the above proof is valid if the definition of a weak uniform cover is changed slightly, i.e. replacing the constant C by C. (with some slow growth condition) but keep all 6. equal.

To understand more of the weak uniform cover, we look into an example:

EXAMPLE 1. Let {U,} be a family of relatively compact open (proper) subsets of M such that U, c Un+ 1 and w,~ 1 U. = M. Assume there is a sequence of numbers {6.} with Y., 5. = o% such that the following inequality holds when t < 5,_ i and x e U . _ l : P { z V " ( x ) < t} <<. ct 2. Then the diffusion concerned does not explode by taking {U.+ 1 - U._I , Un -- U ._ I } to be a weak uniform cover and K. = t7..

3. Boundary Behaviour of Diffusion Processes

To consider the boundary behaviour of diffusion processes, we introduce the following concept:

DEFINITION 3.1. A weak uniform cover {U °, U,} is said to be regular (at infinity for M), if the following holds: let {xj} be a sequence in M converging to £~0M, and xj~U,,jet° suO~,j,=l, then the corresponding open sets {U,,j}j~=lc {U,}~ converges to :~ as well. A regular uniform cover can be defined in a similar way.

p Go For a point x in M, there are a succession of related open sets {W~ )p=l, which are defined as follows: Let W~ be the union of all open sets from {U,} such that U ° contains x, and W~ be the union of all open sets from { U. } such that U ° intersects one of the open sets U. ° defining W~. The {W f } are defined similarly. These sets are well defined and in fact form an increasing sequence.

PROPERTIES AT INFINITY 347

L E M M A 3.1. Assume F t has a regular weak uniform cover. Let {x,} be a sequence in M which converges to a point ~ ~ ~M. Then I4(. p converges to ~ as well for each fixed p.

Xn

Proof We only need to prove the following: let {Zk} be a sequence Zke W~Vk, then

Zk--O~, as n----~ o9. First, let p = 2.

By definition, for each xk, Zk, there are open sets U, ° and U°~ such that x k e U ° nk~

z k e U°~ and U, ° ~ U°~ ¢ ~ . Fur thermore U,~ ~ Y as k---. oe. Let {Yk } be a sequence

of points with Yk ~ U, ° c~ U°~. But Yk ~ "2 since U,~ does. So Um~---" "2 again from the

definition of a regular weak uniform cover. Therefore Zk converges to 2 as k---, 0%

which is what we want. The rest can be proved by induction. •

T H E O R E M 3.2. I f the diffusion F t admits a regular weak uniform cover for 3~, with 5, = 6, all n, then the map Ft(-- ) : M--~ M can be extended to the compactification ~i

continuously in probability with the restriction to the boundary to be the identity map,

uniformly in t in finite intervals. (We will say F t extends.) Proof Take )~e OM and a sequence {x,} in M converging to 2. Let U be a

ne ighbourhood of ~ in M. We want to prove for each t:

lim P{co:F~(x.,(o)q~ U, for some s < t} = 0. n ~ o o

Since x, converges to 2, there is a number N(p) for each p, such that if n > N = N(p), W. p c U. Let t > 0, choose p such that 2tip < 6. For a number n > N(p) fixed, we have:

Xn

P{oo:F~(x,,e))¢U, for some s < t} c P{(o:F~(x,,e))¢ W~,, for some s < t}

<~ P{o):Tp(x,)(o9) < t, Tp_a(xn) < oo}

<~ ~ P Tk(X.) -- T k_ l(x.) < p Tk- l(X") < k = l

Ct 2 <<.--

P

Here C is the constant in the definition of the weak uniform cover. Let p go to infinity

to complete the proof. •

R E M A R K . If5, can be taken all equal, Theorem 2.4, Theorem 3.2 hold if(5) is relaxed to:

P{z'(X ) < t} <~ f(t),

for some nonnegative function f satisfying limt~ o f ( t) / t = O.

The weak uniform cover condit ion is not very easy to apply for a general diffusion on a manifold. For a Brownian mot ion in a Riemannian manifold it is relatively

simple. We often start with a uniform cover, ref. Lemma 2.1.

348 XUE-MEI LI

E X A M P L E 2. Let M = R" with the one point compactif icat ion. Consider the

following s.d.e.:

(It6) dx, = X(x,)dB, + A(xt}dt

Then if bo th X and A have linear growth, the solution has the C O property.

Proof There is a well known uniform cover for this system. See [4], or [9]. A

slight change gives us the following regular uniform cover: T a k e a coun tab l e set of po in ts {P,},~o c M such tha t the open sets

O ° = { z : l z - p , l < l p , I / 3 } , n = l , 2 , . . , and Uo ° = { z : E z - p o l < 2 } cover R". Let

U o = {z:lz - Pol < 6; and U, = {z:lz - P,I < Ip, I/2}, for n ¢ 0. Let ~b, be the char t m a p on U,:

4 . ( z ) - z - p .

Ip . l

This certainly defines a uniform cover (for details see Example 3 below). Fu r the rmore

if z,---+ oo and z , e U °, then any y e U, satisfies the following:

lyl > > 5 t z . I - - - , o o ,

since ]p,] i> 3]z,I/4. Thus we have a regular uniform cover which gives the required

C o property .

E X A M P L E 3. Let M = R', compactif ied with the sphere at infinity: M" = R" vo S ' - 1.

Consider the same s.d.e, as in the example above. Suppose bo th X and A have

sublinear growth of power ~ < 1:

IX(x)[ ~< K(lxf ~ + 1)

IA(x)J <~ K({x[ ~ + 1)

for a cons tant K. Then there is no explosion. Moreove r the solut ion F, extends.

Proof. The proof is as in Example 2, we only need to construct a regular uniform

cover for the s.d.e.: Take points Po, Pl, P2, . . . in R" (with [Po] = 1, ]p,] > 1), such that the open sets.{ U ° }

defined by: U o = {z:lz - P0l < 2}, U ° = {z:lz - Pil < IP,[=/6} cover R' . Let U o = {z:lz - Po[ < 6}, U~ = {z:iz - Pil < IPil'/2), and let ~bi be the chart m a p

f rom U~ to R":

6(z - Pi) , ~ ( z ) - - -

tp~l ~

PROPERTIES AT INFINITY 349

Then {qS~, Ui} is a uniform cover for the stochastic dynamical system. In fact, for i ~ 0, and y e B 3 c R":

I(q~),(X)(y)t ~< K(I + [q~-a(y)l= ) g lP~l ~ ~< i i~(1 + 21p~[ ~) < 18K

Similarly [(q~),(A)(y)f ~< 18K, and D2q~i = 0.

Next we show this cover is regular. Take a sequence x k converging to 2 in OR".

Assume x k e U °. Let Zk e Uk. We want to prove {zk} converge to ~. First the norm of z k converges to infinity as k---~ 0% since [Pk] > 21Xk[/3 and [zkt > IP~[ - ½[PY.

z

Let 0 be the biggest angle between points in Uk, then

0 [ z -p .1 tp.I" Ip.U 1 tan~ ~< sup - - ~< ~< - - -

~v° Ip, I ~ 2 ~0.

Thus { U,, 4), } is a uniform cover satisfying the convergence criterion for the sphere compactification. The required result holds from the theorem.

This result is sharp in the sense there is a s.d.e, with coefficients having linear growth but the solution to it does not extend to the sphere at infinity to be identity:

EXAMPLE 4. Let B be a one dimensional Brownian motion. Consider the following s.d.e, on the complex plane ~:

dx z = ix tdB ~

The solution starting from x is in fact xe ~*+t/2, which does not continuously extend to be the identity on the sphere at infinity.

4. Boundary Behaviour Continued

A diffusion process is a C O diffusion if its semigroup has the C O property. This is equivalent to the following [1]: let K be a compact set, and TK(x ) the first entrance time to K of the diffusion starting from x, then limx_,oo P{TK(x ) < t} = 0 for each t > 0, and each compact set K.

350 XUE-MEI LI

The following theorem follows from Theorem 3.2 when 6, in the definition of the

weak uniform cover can be taken all equal:

T H E O R E M 4.1. Let ffl be the one point compactification. Then if the diffusion process Ft(x ) admits a regular weak uniform cover, it is a C O diffusion.

Proof Let K be a compact set with K c Kfi here {Kj} is as in Definition 2.2. Let

e > 0, t > 0, then there is a number N = N(e, t) such that:

t 6 j + 2 -~ 6 j + 4 -F " '" -F" 6 j + 2 N - 2 > - .

Take xCKj+2N. Assume xeKr, , some m > j + 2N. Let T O be the first entrance time of Ft(x) to Kj+zN_ 1, T1 be the first entrance time of F~(x) to Ki+zN_ 3 after To, (if T o < oo), and so on. But P{T i < t, T/_ 1 < oo} -%< Ct 2 for t < 6j+zN-2~, i > 0, since any open sets from the cover intersects at most one boundary of sets from {K, }. Thus

P{TK(x ) < t} <~ P Ti(x ) < t, TN_ 2 <

N - 1

E P{Ti(x) < e6~+2u-2i, Ti-,(x) < oo} i=1

N - 1 oo ~< Ce2 ~ 2 6j+~N-21 <<. C~ 2 ~ 6~.

1 1

The proof is complete by letting e---~O.

EXAMPLE 5. Let M be a complete Riemannian manifold, p a fixed point in M. Denote by p(x) the distance between x and p, B,(x) the geodesic ball centered at x

of radius r, and Ricci(x) the Ricci curvature at x.

ASSUMPTION A.

v / ~ d r = ~ . (10)

Here K is defined as follows:

K( r )= - I i n f Ricci(x)A 0}. I.Br(p)

Let Xt(x) be a Brownian motion on M with Xo(x ) = x. Consider the first exit time

of Xt(x ) from BI(X):

r = inf {t >~ O:p(x, Xt(x)) = 1}.

PROPERTIES AT INFINITY 351

Then we have the following estimate on T from [13]:

If L(x) > ,JK(p(x) + t), then

P {T(x) <.% C~(X)} <<. e-c~L~)

for all xeM. Here cl, c 2 are positive constants independent of L.

This can be rephrased into the form we are familiar with:

Cl/~ .g(p(x ) "}- 1), when 0 ~< t <

P{T(x) <<, t} ~ e -( .... /t)

But limt~ o e - cc,c2/o/r2 = 0. So there is a 6 o > 0, such that: e - tc,c2/o ~< t 2, when t < 60. Thus:

E S T I M A T I O N O N EXIT TIMES: when t < c l / ~ + 1)/x 60,

P{T(x) < t} ~< t 2. (11)

Let 6, = 1/~fK(3n + 1) ^ 60, then we also have the following:

~1 6, >~ 1 x/K(3n + 1) f; ' - - >/ d r = o o ~ K ( 3 r )

(12)

with this we may proceed to prove the following from Hsu:

C O R O L L A R Y [13]. A complete Riemannian manifold M with Ricci curvature satisfying Assumption A is stochastically complete and has the C O property.

Proof. There is a regular weak uniform cover as follows:

First take any pEM, and let K, = B3,(p ). Take points p~ such that U ° = Bl(p~) covers the manifold. Let Ui = B2(pi). Then {U °, Ui} is a regular weak uniform cover for MuZX.

REMARK. Grigoryfin has the following volume growth test on nonexplosion. The Brownian motion does not explode on a manifold if there is a function f on M satisfying:

foo r Ln (Vol (B R)) dr

=C~O.

Here Vol(BR) denotes the volume of a geodesic ball centered at a point p in M. This result is stronger than the corollary obtained above by the following comparison theorem on an n dimensional manifold: let o , _ 1 denote the volume of the n - 1 sphere of radius 1,

3 5 2 XUE-MEI LI

VoltB.). Sinht / 7__lir) dr

Notice K(R) is positive when R is sufficiently big provided the Ricci curvature is not

nonnegative everywhere. So Grigory/m's result is stronger than the one obtained above.

The definition of weak uniform cover is especially suitable for the one point compactification. For general compactifications the following definition explores more

of the geometry of the manifolds and gives a better result:

D E F I N I T I O N 4.1. Let !~ be a compactification of M, 2e~?M. A diffusion process

F, is said to have a uniform cover at the point 2, if there is a sequence A, of open

neighbourhoods of 2 in M and positive numbers 6, and a constant c > 0, such that:

1. The sequence of A, is strictly decreasing, with c~A, = 2, and A, D ~A,+I. 2. The sequence of numbers 6, is non-increasing with E 6, = oc and E 6~ < oe.

3. When t < 6,, and x e A, - A, + 1,

p{~A--~(x) < t} < ct ~.

Here ra"(x) denotes the first exit time of F,(x) from the set A,.

P R O P O S I T I O N 4.2. I f there is a uniform cover for ,Y ~ ~M, then Ft(x) converges to ,2 continuously in probability, uniformly in t in finite interval, as x----~ 2.

Proof The existence of {A,} will ensure F~A,,(x) c A,_ 1, which allows us to apply a similar argument as in the case of the one point compactification. Here we denote

by z A the first exit time of the process Ft(x) from a set A. Let U be a neighbourhood of 2. For this U, by compactness of M, there is a

2 1/2 number m such that A,. c U, since C~k=l°° A k = ~. Let 0 < e < 1, g = (e/e Z k 6 k) . We may assume g < 1. Choose p = p(e) > 0 such that:

t 6,. + 6,.+ 1 + "'" + 6,.+p- 1 > =.

Let xEA"+p+2. Denote by To(x) the first exit time of F,(x) from Am+p+ l, TI(x )

PROPERTIES AT INFINITY 353

the first exit time of Fro(x) from A,,+p where defined. Similarly T~, i > 1 are defined. Notice if Ti(x ) < o% then FT,(:,)~A,,,+p_ i - A,,,+p+l_ i, for i = 0, 1,2 . . . . . Thus for

i > 0 there is the following inequality from the definition and the Markov property:

e{r (x) < < •

Therefore we have:

P{~U(x) < t} <~ P{zA~(x) < t}

e{T +--- + 71 < t, < } P

i = 1

P

<~ cg2 y , 2 Om+p-i <~ 8. i = l

This finishes the proof.

5. Property at Infinity of Semigroups

Recall a semigroup is said to have the C O property, if it sends Co(M ), the space of continuous functions on M vanishing at infinity, to itself. Let M be a compactification of M. Denote by /~ the point at infinity for the one point compactification. Corresponding to the C o property of semigroups we consider the following C, property for M:

DEFINITION 5.1. A semigroup Pt is said to have the C. property for .Q, if for each continuous function f on M, the following holds: let {x,} be a sequence converging to ~ in ~M, then

lira Ptf(x, , ) = f(ff). (13) n--* ~9

To justify the definition, we notice if 2~ is the one point compactification, condition C. will imply the C o property of the semigroup. On the other hand if Pt has the C O property, it has the C , property for M w/X assuming nonexplosion. This is observed by subtracting a constant function from a continuous function f on M w A: let 9(x) = f ( x ) - f ( A ) , then g ~ Co(M). So Ptg(x) = P t f ( x ) - f ( A ) . Thus

lira P J ( x n ) = lim Ptg(x,) + f ( A ) = f ( A ) , tl-+ c13 n ~ o O

if lim,~ ~ x, = / ~ , In fact the C, property holds for the one point compactification if and only if there

is no explosion and the Co property holds. These properties are often possessed by

354 XUE-MEI Lt

processes, e.g. a Brownian motion on a Riemannian manifold with Ricci curvature

which satisfies (10) has this property. Before proving this claim, we observe first that:

LEMMA 5.1. I f Pt has the C. property for any compactification M, it must have the C. property for the one point compactification.

Proof Let f e C ( M ~ A). Define a map fi from M to M u A: N x ) = x on the interior of M, and fl(x) = A, if x belongs to the boundary. Then fl is a continuous map from b2 to M w A , since for any compact set K, the inverse set

/~ - K = fl- t(M w A - K) is open in _M. Let g be the composition map o f f with fi: g = f ° fl: .~t--+ R. Thus g(x) I M = f(x) I M,

and g(x)iou = f ( A ) . So for a sequence {x,} converging to 2~8M, lim, P J ( x , ) = lim, Ptg(x.) = g(2) = f(A). •

We are ready to prove the following theorem.

T H E O R E M 5.2. I f a semigroup Pt has ~he C, property, the associated diffusion process F t is complete,

Proof We may assume the compactification under consideration is the one point

compactification from the lemma above. Take f - 1 , Ptf(x)= P{t < ¢(x)}. But P{t < ¢(x)} ~ 1 as x--* A from the assumption. More precisely for any e > 0, there

is a compact set K, such that if x ~ K~, P{t < ~(x)} > 1 - e. Let K be a compact set containing K,. Denote by ~ the first exit time of F,(x) from

K. So F~(x)¢K, on the set {r < oe}. Thus:

P{t < ~(x)} >i P{~ < 0% t < ~(F,(x))} + P{r = oo}

= E{Z~< ~E{z,<{(v,(x))I W~}} + P{z = oo}.

Here Za denotes the characteristic function of set A. Applying the strong Markov property of the diffusion we have:

E{Z,< ~E{zt<e(v~(x))l ~}} = E{Z~< ~E{t < ~(y) l F~ -- y}}.

However

E{Z,<¢(r) I F~ = Y} > 1 -- z,

SO

= 1 - e P { r < o e } .

Therefore P{t < ¢(x)} = 1, since e is arbitrary. •

PROPERTIES AT INFINITY 355

In the following we examine the relation between the behaviour at oe of diffusion

processes and the diffusion semigroups.

T H E O R E M 5.3. The semigroup P~ has the C, property if and only if the diffusion process F t is complete and can be extended to 37I continuously in probability with

Ft(x) 1aM = x. Proof Assume F t is complete and extends. Take a point ff s c~M, and sequence {x, }

converging to 2. Thus

lim Ptf (x , ) = lim Ef(Ft(x,) ) = Ef(x) = f (x ) n ~ cg) n -+ oo

for any continuous function on M, by the dominated convergence theorem.

On the other hand Pt does not have the C, property if the assumption above is

not true. In fact, let x, be a sequence converging to 2, such that for some neighbourhood

U of 2, and a number 6 > 0:

limm P{F,(x,)¢ U} = (~. n--+ oo

There is therefore a subsequence {x,, } such that:

lim P{F,(x,,)~ U} = 6.

Thus there exists N > 0, such that if i > N:

(f P{Ft(xn,)e2ff l- U} > ~.

But since 2~r is a compact Hausdorff space, there is a continuous function f from M

to [0,1] such that f t ~ - v = 1, and f(x)[ o = 0, for any open set G in U. Therefore

Pt f (x . ) = Ef(Ft(x.) )

>1 ~{~:F~(x,)~pa- v} f(Ft(x"))P(d°°)

6 = P { f , ( x . ) e ~ - U} > ~.

So lim Ptf (x , ) ~ f (x) = O. []

C O R O L L A R Y 5.4. Assume the diffusion process F t admits a weak uniJbrm cover regular for M w A, then its diffusion semigroup P, has the C. property Jbr M • A . The same is true for a general eompactifieation if all 6, in the weak uniform cover can be taken equal.

356 XUE-MEI LI

EXAMPLE 6 [7]. Let M be a complete connected Riemannian manifold with Ricci curvature bounded from below. Let ~r be a compactification such that the ball convergence criterion holds (ref. Section 1). In particular the over determined equation (2)-(4) is solvable for any continuous function f on M if the ball convergence criterion holds.

Proof We keep the notation of Example 5 here. Let K = -{infxRicci(x ) ^ 0}, 6 = c~/k, where c 1 is the constant in Example 5. Let p ~ M be a fixed point, and K, = ~ be compact sets in m. Take points {Pi} in m such that {B~(pi)} cover the manifold. Then {Bl(pi),B2(pi)} is a weak uniform cover from (11) and (12). Moreover this is a regular cover if the ball convergence criterion holds for the compactification.

Acknowledgements

This paper is developed from my Warwick Msc. dissertation. It is a pleasure to thank my supervisor Professor D. Elworthy for pointing out the problem and for discussions on the proofs and for critical reading of the paper. I am grateful to Professor Zhankan Nie of Xi'an Jiaotong University and for the support of the Sino-British Friendship Scheme. I also benefited from the EC programme SCI-0062-C(EDB). I also like to thank Professor I. Chavel for helpful comments.

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