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Markov Processes and Functional Analysis
Proceedings of the International Mathematical Conference L.H.Y. B e n . T.B. Ng,M.J. Wicks (edr.) 0 North-Holhndhblishing Company, 1982
187
MARKOV PROCESSES AND FUNCTIONAL ANALYSIS Masatoshi Fukushima College of General Education Osaka University Toyonaka, Osaka Japan
51 Introduction 1.1 From analysis on the state space toward Markov processes 1.2 Capacity related to Markov processes 1 2 Dirichlet forms 2.1 Transition functions and semigroups 2.2 Dirichlet forms and symmetric Markov processes 9 3 Capacity 3.1 Capacitary potentials and hitting probabilities 3.2 Tests of irreducibility and attainability 3.3 Examples 11 1.1 -
INTRODUCTION From analysis on the state space toward Markov processes
In dealing with stochastic processes, we usually try to understand their sample path behaviours from some fundamental analytical data. For instance Gaussian processes and processes with independent increments have been studied in terms of convariance functions and Lkvy-Khinchin exponents respectively. For the theory of Markov processes, the corresponding important analytical data are infiniEquivalent tesimal generators ( o f transition semigroups). roles are played by Dirichlet forms in a large class of Markov processes. These notions, being relevent to diverse spaces of functions defined on the state space, may well be the objects of independent interests without refering to the associated Markov processes on the state space. Indeed the Hille-Yosida theory of semigroups and the Beurling-Deny theory of Dirichlet spaces were of this character and contributed considerably to the development of potential theoretic aspects of functional analysis. Only afterward those theories were applied to the study of the associated Markov processes. Perhaps it is worthwhile to point out a difference between the formulation of a Markov process and that of other important stochastic process, e.g. a martingale : while the latter concerns a single probability measure P on a basic sample space $2, a Markov process refers to a collection of probabilty measures Px on $2 indexed by a parameter x ranging over the state space ( x indicates the This location from where the sample paths start at time 0 ) . means that the study of a Markov process would be more complicated but at the same time suggests its advantage that it can be carried
M. Fukushima
188
out in relation to the well developed mathematical analysis on the state space--functional analysis, potential theory and so on. Our position is that we make full use of the analysis on the state space and thereby make clearer not only the sample path structure of the associated Markov process but also the structure of the analysis being used as a tool in the beginning. When the state space is infinite dimensional, our approach would be even more adequate because then the analysis on the state space itself bears We refer to a recent paper by some probabilistic characters. Kusuoka [:lo] in this respect. One may ask : what are the links connecting the analysis on the Certainly state space with the associated Markov process on it ? the theory of martingales is one of the important links. An excessive function is a supermartingale along the sample path of a Markov process under the probability law Px for each fixed x. As we will explain in subsection 3.1 , this fact plays a key role in interpreting the analytical potential theory on the state space in terms of the associated Markov process. The theory of stochastic integrals based on martingales also made a great deal of contribution to the theory of Markov processes. The key observation here is that an additive functional with zero mean of a Markov process is a martingale under the probability law Px for each x . Since any function in the Dirichlet space naturally generates such an additive functional, one can relate functional calculus to stochastic calculus. The following diagram is a rough indication of the relations mentioned above especially for the symmetrizable Markov process.
martingale additive transition
We focus our attention on the left part of this diagram. notion o f capacity plays essential roles in this respect.
1.2 -
The
Capacity related to Markov processes
In this article we are particularly concerned with the notion of capacity and its roles in the study of Markov processes. A (Choquet) capacity on a topological space X is by definition a set function LapLA) defined for every subset A of X such that A C B 3 Cap" 5 Cap(B1
a
An + Cap(UAn) = SUP Cap(An) An compact + =$Cap(nAn) = inf Cap(An)
In the following , we only consider a non-negative and subadditive capacity. "Quasi-everywhere" or means "except on a set of A function u di%: q.e. on X is said to zero capacity". be quasi-continuous if, for any E > 0 , there exists an open set A
Markov processes and functional analysis
with Cap(A) continuous.
E
such that the restriction of u
189
to
X - A
is
In relation to a Markov process, the capacity arises in two rather different ways. One is a probabilistic way of defining Cap(A) as a mathematical expectation of a random variable related to the hitting time of A by the sample path, Another is an analytical way of defining Cap(A) using the associated Dirithlet form, Green function and so forth. The capacity in the first sense was successfully utilized by Hunt in proving the measurability of hitting times. Since the capacity in this sense is an intrinsic concept for a Markov process, so is the associated notion of quasicontinuity. A recent work of Lejan [ll]clarifies the significance of quasi-continuity in the theory of Markov processes. We now like to take the secondapproach by introducing a capacity using a Dirichlet form. Accordingly we restrict our attention to the class of symmetrizable Markov processes associated with regular Dirichlet forms. The three dimensional Brownian motion is a typical example in this class and then the associated Dirichlet form is given by 1/2*{VuVv dx, the Green function is the Newtonian kernel Clx - y1-l and consequently the associated capacity is the classical Newtonian capacity. The next section will give an elementary and brief introduction to Markov processes and Dirichlet forms. 13 concerns the capacity and in particular we give in subsection 3.2 some criteria of the irreducibility of the Markov process in terms of capacity. Subsection 3.3 will be devoted to examples. The point is that, using such concrete analytical data as Dirichlet forms and through the notion of capacity, we may have a good understanding of some important sample path behaviours - - attainability of a set, irreducibility and so on - - of the Markov processes. Most results in subsection 3 . 2 and 3.3 are presented here for the first time and may be considered as a generalization of a part of the joint paper by Albeverio, Fukushima, Karwowski and Streit [I]. DIRICHLET FORMS
12 2.1
Transition functions and semigroups
We use the following notations : " locally compact separable Hausdorff space the family of all Bore1 sets, i.e., the a-field generated by all open sets the space of continuous functions on X with compact U support the space of continuous functions on X vanishing at Cm ( X I infinity u c @J, u is a bounded @measurable function u e u is a non-negative @measurable function.
6
K = K(x,B) is called a a measure on (X,@X)) B-measurable function. x K(x,X) 6 1 for any function u on X , KU
kernel if, for a fixed x 6 X, K(x,.) is -or a fixed B 0 @(X), K(*,B) is a A kernel K is called Markovian if E X. For a kernel K and a measurable is defined by
M. Fukushima
190 (KU)(X)
=
jx K(x,dy)u(y),
whenever this makes sense.
A family {pt, t > 01 of Markovian kernels is said to be a Wtion function if (2.1.1) PtPSU = Pt+sU’ t, s > 0 , lJ 6 % (2.1.2) pO(x,B) = tix(B) &-measure x 6 X, u Co(X). (2.1.3) lim ptu(x) = u(x) t+O The equation (2.1.1) is called the Chapman-Kolmogorov equation. A transition function pt is called conservative if pt(x,X) = 1. t > 0 , x € X , namely, pt(x,.) is a probability measure. A transition function can always be made to be a conservative transition function on an enlarged space XA = X U A by setting x 4 X, (1 - ptl(x))~IA~(B), Pt(X,X) = P~(X, B - (A}) Pt(A,B) = &{A(B) Usually A is added as a point at infinity if X is non-compact and as an extra point if X is compact. We may think of a transition function pt(x,B) as the probability of finding a moving particle in the set B at time t under the condition that it starts from x at time 0. Then A may well be said to be a death point.
-
+
A transition function can be viewed as a collection of linear operators on the space o f bounded measurable functions. The property (2.1.1) is nothing but the semigroup property of this collection. If each p t makes the space Cw(X) invariant, then the transition function Ep,, t > 01 is said to be a Feller transition function o r Feller semigroup. In general, a collection of linear operators I Tt, t > 01 on a Banach space V is called a strongly continuous contraction semigroup if TtTs = Tt+s, t,s > 0, llTtlI 2 1, IITtu - ull -+ 0 , t f 0,II 1 I being the Banach norm. Then the infinitesimal generator A of {Tt, t > 0 ) is defined by Ttu - u (2.1.4) Au = lim t+O with domain D(A) = {u 6 V : the limit in (2.1.4) exists in the I 1. The semigroup property and (2.1.4) formally lead sense of II 1 us to the equation a (Tp) (2.1.5) - A(Tt4 at which may be solved symbolically as Ttu = exp(tA)u. The HilleYosida theorem concerns necessary and sufficient conditions for a linear (but unbounded generally) operator A on V to be a generator of a unique strongly continuous contraction semigroup on
V.
In particular, a Feller transition function {p , t > 01 is a strongly continuous contraction semigroup on th& space Cm(X) endowed with the supremum norm Ibll, = sup Iu(x)I . In fact, X€X
Markov processes and functional analysis
191
[btullm llulJ is trivial. (2.1.3) implies the weak continuity lim c v , ptu - u> = 0 for u 6 C m ( X ) and any bounded signed measure v , which in turn means the strong continuity of p t’ In this case, the Hille-Yosida theorem mentioned above can be stated as follows : a linear operator A on the space Cw(X) is the generator of a Feller transition function if and only if (A. 1) D(A) is dense in C m ( X ) , (A. 2) if u E Cm(X) takes a maximum at x = x0’ then Au(xo) 6 0, (A. 3) for any X > 0 and v 6 C m ( X ) , the equation ( A - A)u = v has a solution u e D(A). Example 1. Let X = Rd the d-dimensional Euclidean space. Suppose a transition function pt is conservative and spatially d homogeneous in the sense that p t (x,B) = pt(x+y, B+y), y e R . Then pt is Feller because of the expression ptu(x) = For instance {Rd Pt(O,dy)u(x+y)
-
@
(2.1.6) pt(x,dy) = (27~t)d/ 2 exp( )dy is such a transition function. Let A be the generator on Cm(Rd) of the transition function (2.1.6) and let L be a linear operator defined bv 2 ‘ d Lu = 1 Au ( = a D(L) = C O2 ( R d ) , 2 i=l 1 2 where Co is the space of twice continuously differentiable functions. Using the Taylor expansion of u(x+y) around x, it is then easy to see (2.1.7) L C A. In this sense, the equation (2.1.5) can be interpreted as the heat equation. A related transition function on ( 0 , ~ )defined by 2 2 - (2~t)-~/’exp(- *)I dy pt(x,dy) = {(Z~t)-~/’exp(- k$-is still Feller but non-conservative : pt(x, (0,m)) < 1, x > 0. In this case also Co((O,m)) 2 C D(A) and Au = 1 u ” , u E. C 2o ( ( O , m ) ) .
7 ’ 9
It is in general hard to check directly the Feller property for non-spatially homogeneous transition functions. Moreover, as is illustrated in the above example, an explicit expression of the generator A is available only on a subset of the domain D(A). What we practically know is that A is a closed extension of some concretely given linear operator L with domain D(L) consisting of smooth functions. Kolmogorov [9] first presented a typical expression o f the operator L as an elliptic differential operator: d 2 d a. .(x) a ‘I + xbi(x) + c(x)u (2.1.8) Lu = axiaxj i=l axl i,j=l 1J ~
(Caij(x)titj 0 , c(x) o 1. He actually proved that, underdsome regularity conditions for the transition function pt on R , the parabolic differential equation
M. Fukushima
192
must hold with
L
of ( 2 . 1 . 8 ) .
A fundamental question arises : under what conditions on the coefficients aij, bi, c of the operator L, does the equation (2.1.9) admits a transition function as a solution ? This question has been answered directly or indirectly in several ways. One is a probabilistic way of solving the associated stochastic differential equations ( [ 8 ] - , [ 1 2 ] ) which has been quite successiful especially in the multidimensional cases. But we do not go into this direction now. The second way is to rely upon the theory of partial differential equations ( [ 4 ] ) . The third is the way of functional analysis yo construct, by taking a closed extension of L, an operator A = L satisfying the Hille-Yosida conditions(A.l), (A.2) and ( A . 3 ) . This method was successful in one-dimensional case but rather hard to be used in the multidimensional situations. From now, we like to present another way of a functional analytical character, namely, an L2-theoretic method rather than the C-theory. Since the main concern of probabilists is not a transition function but a stochastic process behind it called a Markov process, the method of SDE is in a sense more favorable than the analytical methods. But LZ-theoretic methods provides us with many Markov processes which can not be covered by the SDE method. Furthermore, as the title of the historical paper of Kolmogorov suggested, the theory of Markov processes can not get rid of the influence of the analysis on the state space X. Instead we make full use of a relevent functional analysis, that is , the theory of Dirichlet form. An L'-theoretic approach toward transition functions is based on the following considrations. Denote by m a positive Radon measure on X such that m is positive on each non-empty open set. L 2 (X;m) stands for the L 2 -space of m-square integrable functions on X with inner product ( u , v) = {Xu(x)v(x)m(dx). Let pt be a transition function. m is said to be an excessive measure with respect to pt if
lx
m(dx) pt(x,B) 2 m(B),
B C @(XI.
Lemma 1 If m is excessive with respect to a transition function p,, then {pt, t > 0 ) determines uniquely a strongly continuous semigroup {Tt, t > 01 on L 2 (X;m). Tt is Markovian in the 1, u G L 2 (X;m). sense that 0 2 Ttu 6 1 m-a.e. follows from 0 6 u (ptu(x)) 2 Proof By Schwarz inequality, we have for u 6 C o ( X ) , 2 2 I ptl(x) ptu (XI 5 ptu' (XI; and consequently (ptu, ptu) 6 2 Hence pt extends from Co(X) to ptu (x)m(dx) 6 ( u , u ) . L (X;m) as a unique linear operator Tt with llTtllLZ 2 1.
In
Semigroup property
Tt are clear.
TtTS = Tt+s and the Markovian property of each The strong continuity of Tt follows from
Markov processes and functional analysis
(ptU - u,PtU - u) 2.2 -
2 {(u,u) - (
u , ptu))
-+
t+O
0, u B C,(X).
193
q.e.d.
Dirichlet forms and symmetric Markov processes
We say that a transition function
jX
pt
is m-symmetric if
jx
(2.2.1) P,U(X) v(x)m(dx) = u(x) ptv(x)m(dx), u , v E @+. Evidently m is then an excessive measure of pt and , according to the preceding lemma, an m-symmetric transition function gives rise to a unique strongly continuous contraction semigroup {Tt, t > 0 ) on L 2 (X;m). In this case, Tt is not only Markovian but also a symmetric operator on L 2 (X;m). Such a semigroup is completely characterized by the notion of the Dirichlet form. In this subsection, we quickly mention relevent basic notions and relations by refering the readers to [S] for further details. densely defined non-negative definite symmetric bilinear form E on L2(X;m) is simply said to be a symmetric form. Its domain is We let denoted by D [ E ] . E a ( u , v) = E(u, v) + a ( u , v), U, v 6 D[E]. If D[E] is comlete with respect to the El-metric, then a symmetric form E is called closed. We say that a symmetric form E on L2(X;m) is Markovian if, for any E > 0 , there exists a real func t ion $,It), t 6 R 1 , satifying
A
& $,(t) 0 2 $,(t) such that u 6 D[E] (2.2.2)
-E
& 1
$,(t) = t
+E,
& t
- $,(t')
implies v
- t' =
$,(u)
for t e [0,11, for t' < t, E D[E] and E(v,v) , I E(u,u).
We call a Markovian closed symmetric form on L 2 (x;m) a Dirichlet A Dirichlet form E on L2(X;m) is in one-to-one form. correspondence to a strongly continuous contraction semigroup The {Tt,t > 01 of Markovian symmetric operators on Lz(X,m). correspondence is given by D[E] (2.2.3)
=
E(u,v)
=
1
( u , u - T u) < m ) t t+O 1 u , v 6 D[E]. lim -t- ( u , v - Ttv),
{u C L'
: lim
t +o
We have seen that any m-symmetric transition function gives rise to such a semigroup. We call the associated Dirichlet form the A natural question Dirichlet form of the transition function. arises : conversely, given any Dirichlet form, is it the Dirichlet form of some m-symmetric transition function ? Exam le 2 The transition function (2.1.6) on Rd is symmetric tce*w to the Lebesgue measure. Its Dirichlet form E on L2(Rd) is given by D[E] = H 1 (Rd ) (2.2.4) Vu Vv dx E(u, v) =
IRd
M. Fukushima
194
where H 1 (Rd) is the Sobolev space of order 1 : H 1 (Rd) = {u 6 L 2 : au Indeed, by taking the Fourier transform axi C L 2 , 1 g i d l . 2 of u 6 L and using the formula ( 2 . 2 . 3 ) , we easily get ( 2 . 2 . 4 ) . The Markovian property of the form ( 2 . 2 . 4 ) can directly be seen as follows. Take u G H1(Rd). We may assume that u is absolutely continuous on almost all lines parallel to the axes. Take for any E > 0 a C1-function $,(t) satisfying ( 2 . 2 . 2 ) . Then
jRd \ v Q E ( U > l dx
=
'I,
7 d
\VU\
2
which means the Markovian property of
\@;(U)\2
dx & E ( u ,
U),
E.
Given a transition function p on X, one can construct a Markov rocess governed by pt, namel?, there exists a collection o f xA with state space M = { Q , Q, Xt, PX} b t i c processes X,
such that for each
x h X
PX -a.s. Px(Xt+s E A l a t ) = P,(X~,A) (2.2.6) Px(Xo = x) = 1 where A is any element of @(X) and Dt denotes a sub a-field of @ making Xs, s 6 t, measurable. Suppose that the sample path X.(W) is right continuous and has left limits PX -a.s. for each x X. We then say that M has the strong Markov property if, for some right continuous o-fields {C$)t>o, equation ( 2 . 2 . 5 ) holds with time t being replaced by any @}-stopping time. If in addition almost every sample path is left continuous along any increasing sequence of stoppint times, then M is said to be a h ; ; process: In particular a strong Markov process whose sample is continuous up to the death time ~ ( w ) = inf {t > 0 : X,(W) Px-a.s., x f X, is called a diffusion (process). A) (2.2.5)
In regard to the question raised before Example 2 , we introduce two A Dirichlet form E on L 2 ( X ; m ) notions about Dirichlet forms. is called regular if the space D[E]nC,(X) is dense both in D[E] and in Co(X) E is said to be local if E(u, v ) vanishes whenever u, V C D[E] have disjoint compact support. Now the answer ty the above question is as follows : if a Dirichlet form E on L (X;m) is regular, then E admits a Hunt process, namely, E is the Dirichlet form of the m-symmetric transition function of The associated Hunt process M is unique some Hunt process M. in a certain sense. In this case, M can be taken to be a diffusion if and only if E is local.
.
Those strong statements would become void if there were not practical means of constructing regular local Dirichlet forms from some concrete data. Fortunately we possess the following useful assertions : if a symmetric form E with D[E]_dense in Co is Markovian and closable (that is, E admits a closed extension), then the closure E is a regular Dirichlet form. If D[E] satisfies some additional mild condition and if E is local, then E as also local.
Markov processes and functional analysis
Consider a non-negative function p t LioC(Rd). 1 d) , E(u, v) = lRd VUVV p dx, D[E] = CO(R
Example 3 (2.2.7)
$
195
Then
defines a Markovian local symmetric form on L2(Rd;pdx). If either (2.2.8) inf p(x) > 0 for any compact set K C Rd, x K or 2 d) , (2.2.9) the distribution derivatives Dx P are in Lloc(R i then E is closable on L2 (Rd ;pdx). Then the closure E' becomes a local regular Diri hlet form and consequently admits a diffusion process Mp on R .
s
A recent paper [ 7 ] concerns the conditions for the function p that the associated diffusion Mp is mutually absolutely continuous with respect to the d-dimensional Brownian motion. Brownian motion is by definition a diffusion process with transition function (2.1.6). In other words,Brownian motion is associated with the Therefore it is a version o f the Dirichlet form (2.2.4) . diffusion MD for p = 1. so
53
CAPACITY
3.1 Capacitary potentials and hitting probabilities -
Given a regular Dirichlet form E on L 2 (X;m), we define the associated capacity Cap(A) for an open set A C X by 1 m-a.e.on A}. Cap(A) = inf E1(u,u), where LA = { u E D[E]: u u6 LA We then extend Cap(A) for any set A C X as the outer capacity yielding a Choquet capacity (see subsection 1.2). The notions !!q.e.'! and !'quasi-continuous''will be used with respect to this When E is transient (or equivalently the as ciated capacity. Hunt process is transient), then the 0-order capacity Cap( 69 (A) can be introduced in the same way as above but by replacing El and D[E] with E and the extended Dirichlet space respectively. This replacement does not change the notions I'q.el' and "quasicontinuous". Cap(O)(A) related to the Dirichlet form (2 2 . 4 ) for d = 3 coincides with classical Newtonian capacity of A C Rs up to a multiplicative constant. The set L is closed and convex for open A C X. Hence there exists a uhique element eA in LA minimizing E1(u,u), which is said to be the e uilibrium potential or the capacitary potential of It then h:lds that 0 6 eA & 1 and A. (3.1.1)
Cap(A)
=
El(eA, eA). be a Hunt process associated with the
regular Dirichlet form E. probability pA of A by (3.1.2) where Ex ure Px.
We defige the (1-order) hitting
-U.
pA(x)
=
Ex( e A ; uA <
m)
, uA(u)
=
inf {t > 0 : Xt(u)EA}
is the expectation with respect to the probability meas-
M. Fukushima
196
Theorem 1
For an open set A
with
LA 9
4 , pA
is a version of
eA' Proof Let ITt, t > 01 be the L 2 -semigroup corresponding to E and M. Then both eA and pA are 1-supermedian : e-tTteA 6 -t It suffices to show the inequality eA3 TtPA & PA' (3*1.3)
PA
6
eA,
because we can then apply formula (2.2.3) to get pA c L and E1(pA,pA) 9 El(eA,eA) (here we use the fact that eA 6 AL 2 ) , which implies PA
=
eA'
To see (3.1.3), take a Borel version eA of eA and let h be any non-negative function with h(x)m(dx) = 1. Let Yt(w) = e-teA(Xt ( w ) ) , then (yt, a t , Ph.m) tBo is easily seen to be a supermartingale : E~.,,(Y~~@) 6 y S phmm-a.s., s < t. Choose any finite set D C ( 0 , m ) with min D = a, max D = b and let a(D ;A) = min {t 6 D : Xt 6 A). The optional sampling theorem for the supermartingale yields -u(D;A) : a ( D : A ) < b, & Eh.m(Ya(D:A)) 6 Eh.m(Ya) 6 ( h , eA)* Eh .m(e By making D increase to a dense subset of (0, b) and b + m, we get (3.1.3). q.e.d.
Ix
When E i transient, we may consider the 0-order capacitary (Opotf, ntial e(if o f an open set A along with Cap(O)(A). e A 2 in the extended Dirichlet space but may not belong to L ( X ; m ) . But we still have the identity (3.1.4) pio) is a version of eA( 0 ) , In fact we get where p p ) = Px( aA < m) . same way as above and hence the inequality
pA('1
eA( O )
in the
p p ) I\ aRaf 6 eiO)fi aRaf , a > 0 , f continuous 2 0, where Ra is the resolvent of pt. Since both hand sides are a-supermedian, we obtain the equality by the same argument as in the above proof. It now suffices to let a m. -f
Theorem 1 enables us to identify diverse potential theoretic notions relevent to the Dirichlet form E with those relevent to the Hunt process M. Here are some of the ikentifications : (3.1.A) Cap(N) = 0, N C X for some Borel N 2 N, PhSm(ur N, m($ = 0 and for any x E x - N, pX( xt or xt- (i 5 for some t 2 0 ) = 0 . (3.1.B) An open J. Cap(An) J. 0 Px(lim aA < 5 ) = 0 q.e. x. n n L.
197
Markov processes and functional analysis
(3.1.C) A function u locally in D[E] is quasi-continuous if and only if u(Xt) is right continuous in t 2 0 and lim u(Xt,) t'+t = u(Xt-), t > 0, Px-a.s. for q.e. x 6 X. "Only if" part of (3.1.C) is a direct consequence of (3.1.B). "If" part can be shown by using a fact that any function u locally in D[E] (i.e., f o r any relatively compact open set G there exists v 6 D[E] sucJh tkat u = v m-a.e. on G ) admits a quasicontinuous version u : u is quasi-continuous and F = u m-a.e. 3.2 -
Tests of irreducibility and attainability
Let E and M be as in the preceding subsection. We further assume that E is local and consequently M is a diffusion. A Borel set B is called T t -invariant if TtIB = IBTt, namely, Tt(IBu) = IB - Ttu for any u 6 Lz, and t > 0. B is said to be M-invariant if Px( ux-B < m) = 0 for any x B. We say that an increasing sequence {FnI of closed sets is a nest if Cap(X-Fn) -+ 0. Following Lejan[ll],we call a Borel set B uasi-o en (resp. quasi-closed) if there exists a nest {F,) s u c W n F n is open (resp. closed) in Fn for each n. F o r Borel sets B , u and B, B is said to be a modification of B if m ( E e B) = 0. Theorem 2 Following conditions are equivalent for a Borel set B. B is Tt -invariant. (i) (ii) u E D[E] IB.u E D[E]. (iii) IB is locally in D[E]. (iv) IE is quasi-continuous f o r some modification B of B. (v) B is quasi-open and quasi-closed f o r some modification of B. (vi) X can be decomposed as X = B1 + BZ + N where Bl(resp. B2) is a modification o f B (resp. X-B), both B1 and B2 are M-invariant and m(N) = 0.
-
h.
Remark 1 Condition (v) is the same as saying that there exists a nest {Fn} such that B n F n and Fn - B are closed for each n. This simplifies corresponding statements in [l]. The set N in condition (vi) is o f zero capacity according to ( 3 . 1 . A ) . Proof
(i)=)(ii)
:
o f identity { E A , X > 01
D[E]
= {U
L2 : 2
jmA 0
Tt and D[E] on L2(X;m)
d(EXu, u) <
m}.
are expres2ible by a resolution as Tt = e-XtdEX and (i) then implies
EhIBu
=
IB'EXu, u E L , from which (ii) is immediate. (ii)*(iii) : Since E is regular, there exists for any compact K, a function u E D[E] n C o ( X ) such that u = 1 on K. (iii)+(iv) : IB admits a quasi-continuous version $ . Since
198
M. Fukushima
= @ m-a.e., @ = o or 1 q.e. Hence @ = 11. q.e. for B c, some modification B o f B. (iv)+ (v) : trivial. : Let -cn be the first leaving time of the sample path. (v)+(vi) Since the sample path is continuous almost surely, x € B n F n , (see Remark 1). Px( Xt E B O F n for any t < T ~ )= 1, The same property holds for the set Fn - B. We then arrive at (vi) by virtue of (3.1.B). (vi)+ (i) : trivial. q.e.d.
$‘
We call E irreducible if any Tt -invariant set B is trivial in the sense that either m(B) = 0 or m(X - B) = 0. In view of condition (v) of the above theorem, we can say that E is irreduOn account of cible if and only if X is quasi-connected. condition (iv), we get Corollary Let m and m be mutually absolutely continuous and let E(l) and E(‘) be regular local Dirichlet forms on L2(X;m(l)) and L2 (X;m(2)) respectively. Suppose that any quasi-continuous function with respect to E(‘) is also quasicontinuous with respect to E(l). Then the irredubibility of E(’) implies the same property of E”). The condition in this Corollary is satisfied if E ( 2 ) is locally dominating E(l). To make this statement more precise, let us consider a dense subalgebra D of Co(X) satisfying the following two properties : (i) for any E > 0, there exists a function % possesssing property (2.2.2) such that C $ ~ ( U )tZ D whenever u e D , (ii) for any compact set K and a relatively compact open set G with K C G , D contains a function u with u = 1 on K and u = 0 on X - G. We let DG = { u E D : u = 0 on X - G}. We say that D is a of a Dirichlet form E if D is El-dense in D[E]. Theorem 3 (i) Let E(l) and E ( 2 ) be two local regular 2 Dirichlet forms on L (X;m) possessing a set D as above as their common cores. Suppose that for any relatively compact open set G (3.2.1) E(’)(U, u ) 2 y K E(’)(u, u), u 6 DG, for some constant y K > 0. Then the irreducibility of implies the same property of E(‘). (ii) Let m (l) and m be related as (3.2.2) dm(‘) = p dm(’) with yG = inf p(x) > 0 x€G
for any relatively compact open set G . Let 2 be local regular Dirichlet forms on L (X;m(”) respectively. Suppose that E(l) and E(’) D of the above type and
E(l)
E(l) and E(’) and L2(X;m(2)) have a common core
199
Markov processes and functional analysis
(3.2.3) ~ ( l ) = E(~) on DXD. Then the irreducibility of E(l) implies that of
E(').
(i) The restriction of a local regular Dirichlet form E Proof to {u E D[E] : u = 0 q.e. on X - G I is denoted by EG, which is known to be a local regular Dirichlet form on L 2 (G;m). A function on G is EG-quasi-continuous if and only if it is E quasi-continuous ( [ 5 ; Th. 4.4.21). Moreover, a function which is quasi-continuous on any relatively compact open set is (globally) quasi-continuous. Under the assumption of (i), DG becomes a common core of E';) and Ei2) "5; Prob. 3.3.41). Inequality (3.2.1) then implies an analogous inequality for the relevent capacities. Hence a function is E';) -quasi-continuous whenever it is E(:) -quasi-continuous. Thus, if a function is quasi-continous with respect to E(2), s o it is with respect to E(l). Now Corollary applies. (ii) By (3.2.2) and (3.2.3), , u 6 DG. q.e.d. E(:)(u, u ) + (u, u ) 2 E(;)(u, u ) + yG.(u, u) m2 ml Theorem 3 can be proved using condition (iii) (instead of Remark 2 0 ) f Theorem 2. Y. Lejan pointed out the significance of condition (iv) in regard to the paper [l]. I also owe to him the present simple proof of "(iii)+(iv)" in Theorem 2 (private communication).
-
3.3 Examples -
Example 4 The Dirichlet form (2.2.4) on L 2 (Rd ) is irreducible because the associated diffusion is Brownian motion whose transition function is given by (2.1.6). Consider the symmetric form E o f Example 3 and assume that the function p E satisfies i d the condition (2.2.8). Theorem 3 then applies with D = CO(R ) and we see that the closure E is a irreducible local symmetric Dirichlet form on L2(Rd; dx).
L~A~(R~)
'
Consider Example 3 for the one dimensional case. Example 5 More specifically we are concerned with the symmetric form (3.3.1) E(u, V) = l~'(x)V'(x)p(x)dx, D[E] = CO(R 1 1) , on
2 1 L (R ,pdx)
and we assume that
whenever 0 4 (a.b). p closable ( [ S ; Th. 2.1.41). The closure
-
E
jb
p d
Lloc
and
may be degenerate at
inf p(x) a
> 0
is
is irreducible if and onlv if
dS -b < for b > 0. In view o f Theorem 2 and Remark 1, this assertion follows from the next lemma. (3.3.2)
M. Fukushima
200
Lemma
Let
I d = (0, d ) .
l i m Cap(Id) d+O
= 0
,
@
b
>o.
Proof " j " : Suppose t h a t t h e above i n t e g r a l i s f i n i t e . Let G = ( - b , b ) , t h e n i t s u f f i c e s t o show l i m C a p ( I d ) > 0 where dGO CapG i s t h e c a p a c i t y r e l a t e d t o t h e form EG ( s e e t h e p r o o f of Theorem 3 ) . By [S; P r o b . 3 . . 3 2 ] , CapG(K) f o r K = [ c , d ] , O< c < d , c a n be computed a s CapG(K) =
inf E1(u,u) u6C&, u = l on K U = O on R'-G
2
which i s n o t smaller t h a n Hence we have 'I
1
Cap(Id)
inf uc c; u (d) =1, u ( b ) =O by Schwarz i n e q u a l i t y . > 0
(
: Suppose t h a t
K = [c,d]
Let
uo
and d e f i n e
by
(I
uo(x) =
uniformly i n
d.
O < c ' < c < d < d '< b ,
C'
L e t t i n -g
c'
+
CaP(Id)
s 21
(
d
+
0
and
c' &
d'
and
0
+
0.
c
+
0, we Eet
)-l +
jd 0
I
p(c)d[
,
which t e n d s t o z e r o a s
~.e.d.
T h i s lemma means t h a t , i f f o r i n s t a n c e
= m and
I,
dS
Po
< =, t h e n R1 i s t h e sum of two i n v a r i a n t - s e t s ( - - , O ) and [O,m). 0 i s a t t a i n a b l e by t h e a s s o c i a t e d sample p a t h from t h e r i g h t b u t n o t from t h e l e f t i n t h i s c a s e .
'
C o n s i d e r t h e same example f o r t h e two d i m e n s i o n a l Example 6 case. Thus, 2 2 1 2 ( 3 . 3 . 3 ) E(u,v) = 2(ux + u y ) P ( x , y ) d x d Y , D [ E I = CO(R 1 . I R L e t C = Ix = 1 1 . We assume t h a t p LIAc(RZ) and i n f p ( x , y ) (x,Y)eK i s p o s i t i v e f o r any compact s e t K w i t h K n C - = $- . We f u r t h e r assume t h a t t h e form ( 3 . 3 . 3 ) i s c l o s a b l e on L'(RL; pdxdy) BY t h e same r e a s o n i n g a s i n t h e p r e c e d i n g e x a m p l e , we c a n c o n c l u d e t h a t E i s i r r e d u c i b l e i f f o r some a B and b > 0
.
Markov processes and functional analysis
201
sufficient condition for the reducibility can be stated as follows: For a n-interval J, we denote the integral jJp(c,n)dr, by m If I J k ) k=-.. is an open covering of R1 and if for each k ,
A
(3.3.5)
0
either
dc bJk0
-
cJ.
= for some j-bk bk > 0, then the left hklf plane {x > 01 and the right half plane {x > 0 ) are not attainable from each other. =
Or
IR3
Exam le 7 We still consider the same example but for three dimenh( 3 . 3 . 6s) E(u, e : V) = ( u i + u2 + uz)p(x,y,z)dxdydz 2 Y and we are concerned with the attainability of one point, say, the origin 0, of the associated sample path. In other words, our problme is whether Cap({O)) is zero or not. We assume that p is locally integrable and satisfies (2.2.8). P may be singular at 0. Our assertion is this :
Cap((O1)
Ib & 0 if
=
p(x,
Cap({O})
y, z)
=
p(r)
> 0
if
and
= m , b >O. r p(r) To verify the first assertion, we proceed as in the proof of Lemma: Letting G = {r < 1) and d < 1 and using the inequality
(3.3.8)
0
E1(u,u) 2 ~ ~ ~01 0 2 n0 o:r2p(r,e,$)sin$d=ded$,
we have
J'p +1-l
inf E~(U,U) 2 8n2(j 'sin drded u6C1, u=l on rl which is bounded away from zero uniformly in d. Cap({r
=
M.Fukushima
202
References
S. Albeverio, M. Fukushima, W. Karwowski and L. Streit, Capacity and quantum mechanical tunneling, to appear in Comm. Math. Phys. A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A.,.45(1959), 208-215. R.M. Blumenthal and R.K. Getoor, Markov processes and potential theory, Academic press , 1968. E.B. Dynkin, Markov processes, Springer, 1965. M. Fukushima, Dirichlet forms and Markov processes, Kodansha and North Holland, 1980. M. Fukuhsima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motions, Physics Reports 502 (New Stochastic Methods in Physics), North Holland, 1981. M. Fukushima, On absolute continuity of multidimensional symmetrizable diffusions, t o appear in Proc. Katata Conf. on "Functional Analysis in Markov Processes", Lecture Notes in Math., Springer. N. Ikeda and S . Watanabe, Stochastic differential equations and diffusion processes, Kodansha and North Holland, 1981. A.N. Kolmogorov, uber die analytischen Methoden in der W a h r s c h e i n l i c h k e i t s r e c h n u n g , Math. Ann. 104(1931), 415-458. S. Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, to appear. Y. LeJan, Quasi-continuous functions and Hunt processes, t o appear. D.W. Stroock and S.R.S. Varadhan,Multidimensional diffusion processes, Springer, 1979. K. Yosida, Functional analysis, Springer, 1968.
187
MARKOV PROCESSES AND FUNCTIONAL ANALYSIS Masatoshi Fukushima College of General Education Osaka University Toyonaka, Osaka Japan
51 Introduction 1.1 From analysis on the state space toward Markov processes 1.2 Capacity related to Markov processes 1 2 Dirichlet forms 2.1 Transition functions and semigroups 2.2 Dirichlet forms and symmetric Markov processes 9 3 Capacity 3.1 Capacitary potentials and hitting probabilities 3.2 Tests of irreducibility and attainability 3.3 Examples 11 1.1 -
INTRODUCTION From analysis on the state space toward Markov processes
In dealing with stochastic processes, we usually try to understand their sample path behaviours from some fundamental analytical data. For instance Gaussian processes and processes with independent increments have been studied in terms of convariance functions and Lkvy-Khinchin exponents respectively. For the theory of Markov processes, the corresponding important analytical data are infiniEquivalent tesimal generators ( o f transition semigroups). roles are played by Dirichlet forms in a large class of Markov processes. These notions, being relevent to diverse spaces of functions defined on the state space, may well be the objects of independent interests without refering to the associated Markov processes on the state space. Indeed the Hille-Yosida theory of semigroups and the Beurling-Deny theory of Dirichlet spaces were of this character and contributed considerably to the development of potential theoretic aspects of functional analysis. Only afterward those theories were applied to the study of the associated Markov processes. Perhaps it is worthwhile to point out a difference between the formulation of a Markov process and that of other important stochastic process, e.g. a martingale : while the latter concerns a single probability measure P on a basic sample space $2, a Markov process refers to a collection of probabilty measures Px on $2 indexed by a parameter x ranging over the state space ( x indicates the This location from where the sample paths start at time 0 ) . means that the study of a Markov process would be more complicated but at the same time suggests its advantage that it can be carried
M. Fukushima
188
out in relation to the well developed mathematical analysis on the state space--functional analysis, potential theory and so on. Our position is that we make full use of the analysis on the state space and thereby make clearer not only the sample path structure of the associated Markov process but also the structure of the analysis being used as a tool in the beginning. When the state space is infinite dimensional, our approach would be even more adequate because then the analysis on the state space itself bears We refer to a recent paper by some probabilistic characters. Kusuoka [:lo] in this respect. One may ask : what are the links connecting the analysis on the Certainly state space with the associated Markov process on it ? the theory of martingales is one of the important links. An excessive function is a supermartingale along the sample path of a Markov process under the probability law Px for each fixed x. As we will explain in subsection 3.1 , this fact plays a key role in interpreting the analytical potential theory on the state space in terms of the associated Markov process. The theory of stochastic integrals based on martingales also made a great deal of contribution to the theory of Markov processes. The key observation here is that an additive functional with zero mean of a Markov process is a martingale under the probability law Px for each x . Since any function in the Dirichlet space naturally generates such an additive functional, one can relate functional calculus to stochastic calculus. The following diagram is a rough indication of the relations mentioned above especially for the symmetrizable Markov process.
martingale additive transition
We focus our attention on the left part of this diagram. notion o f capacity plays essential roles in this respect.
1.2 -
The
Capacity related to Markov processes
In this article we are particularly concerned with the notion of capacity and its roles in the study of Markov processes. A (Choquet) capacity on a topological space X is by definition a set function LapLA) defined for every subset A of X such that A C B 3 Cap" 5 Cap(B1
a
An + Cap(UAn) = SUP Cap(An) An compact + =$Cap(nAn) = inf Cap(An)
In the following , we only consider a non-negative and subadditive capacity. "Quasi-everywhere" or means "except on a set of A function u di%: q.e. on X is said to zero capacity". be quasi-continuous if, for any E > 0 , there exists an open set A
Markov processes and functional analysis
with Cap(A) continuous.
E
such that the restriction of u
189
to
X - A
is
In relation to a Markov process, the capacity arises in two rather different ways. One is a probabilistic way of defining Cap(A) as a mathematical expectation of a random variable related to the hitting time of A by the sample path, Another is an analytical way of defining Cap(A) using the associated Dirithlet form, Green function and so forth. The capacity in the first sense was successfully utilized by Hunt in proving the measurability of hitting times. Since the capacity in this sense is an intrinsic concept for a Markov process, so is the associated notion of quasicontinuity. A recent work of Lejan [ll]clarifies the significance of quasi-continuity in the theory of Markov processes. We now like to take the secondapproach by introducing a capacity using a Dirichlet form. Accordingly we restrict our attention to the class of symmetrizable Markov processes associated with regular Dirichlet forms. The three dimensional Brownian motion is a typical example in this class and then the associated Dirichlet form is given by 1/2*{VuVv dx, the Green function is the Newtonian kernel Clx - y1-l and consequently the associated capacity is the classical Newtonian capacity. The next section will give an elementary and brief introduction to Markov processes and Dirichlet forms. 13 concerns the capacity and in particular we give in subsection 3.2 some criteria of the irreducibility of the Markov process in terms of capacity. Subsection 3.3 will be devoted to examples. The point is that, using such concrete analytical data as Dirichlet forms and through the notion of capacity, we may have a good understanding of some important sample path behaviours - - attainability of a set, irreducibility and so on - - of the Markov processes. Most results in subsection 3 . 2 and 3.3 are presented here for the first time and may be considered as a generalization of a part of the joint paper by Albeverio, Fukushima, Karwowski and Streit [I]. DIRICHLET FORMS
12 2.1
Transition functions and semigroups
We use the following notations : " locally compact separable Hausdorff space the family of all Bore1 sets, i.e., the a-field generated by all open sets the space of continuous functions on X with compact U support the space of continuous functions on X vanishing at Cm ( X I infinity u c @J, u is a bounded @measurable function u e u is a non-negative @measurable function.
6
K = K(x,B) is called a a measure on (X,@X)) B-measurable function. x K(x,X) 6 1 for any function u on X , KU
kernel if, for a fixed x 6 X, K(x,.) is -or a fixed B 0 @(X), K(*,B) is a A kernel K is called Markovian if E X. For a kernel K and a measurable is defined by
M. Fukushima
190 (KU)(X)
=
jx K(x,dy)u(y),
whenever this makes sense.
A family {pt, t > 01 of Markovian kernels is said to be a Wtion function if (2.1.1) PtPSU = Pt+sU’ t, s > 0 , lJ 6 % (2.1.2) pO(x,B) = tix(B) &-measure x 6 X, u Co(X). (2.1.3) lim ptu(x) = u(x) t+O The equation (2.1.1) is called the Chapman-Kolmogorov equation. A transition function pt is called conservative if pt(x,X) = 1. t > 0 , x € X , namely, pt(x,.) is a probability measure. A transition function can always be made to be a conservative transition function on an enlarged space XA = X U A by setting x 4 X, (1 - ptl(x))~IA~(B), Pt(X,X) = P~(X, B - (A}) Pt(A,B) = &{A(B) Usually A is added as a point at infinity if X is non-compact and as an extra point if X is compact. We may think of a transition function pt(x,B) as the probability of finding a moving particle in the set B at time t under the condition that it starts from x at time 0. Then A may well be said to be a death point.
-
+
A transition function can be viewed as a collection of linear operators on the space o f bounded measurable functions. The property (2.1.1) is nothing but the semigroup property of this collection. If each p t makes the space Cw(X) invariant, then the transition function Ep,, t > 01 is said to be a Feller transition function o r Feller semigroup. In general, a collection of linear operators I Tt, t > 01 on a Banach space V is called a strongly continuous contraction semigroup if TtTs = Tt+s, t,s > 0, llTtlI 2 1, IITtu - ull -+ 0 , t f 0,II 1 I being the Banach norm. Then the infinitesimal generator A of {Tt, t > 0 ) is defined by Ttu - u (2.1.4) Au = lim t+O with domain D(A) = {u 6 V : the limit in (2.1.4) exists in the I 1. The semigroup property and (2.1.4) formally lead sense of II 1 us to the equation a (Tp) (2.1.5) - A(Tt4 at which may be solved symbolically as Ttu = exp(tA)u. The HilleYosida theorem concerns necessary and sufficient conditions for a linear (but unbounded generally) operator A on V to be a generator of a unique strongly continuous contraction semigroup on
V.
In particular, a Feller transition function {p , t > 01 is a strongly continuous contraction semigroup on th& space Cm(X) endowed with the supremum norm Ibll, = sup Iu(x)I . In fact, X€X
Markov processes and functional analysis
191
[btullm llulJ is trivial. (2.1.3) implies the weak continuity lim c v , ptu - u> = 0 for u 6 C m ( X ) and any bounded signed measure v , which in turn means the strong continuity of p t’ In this case, the Hille-Yosida theorem mentioned above can be stated as follows : a linear operator A on the space Cw(X) is the generator of a Feller transition function if and only if (A. 1) D(A) is dense in C m ( X ) , (A. 2) if u E Cm(X) takes a maximum at x = x0’ then Au(xo) 6 0, (A. 3) for any X > 0 and v 6 C m ( X ) , the equation ( A - A)u = v has a solution u e D(A). Example 1. Let X = Rd the d-dimensional Euclidean space. Suppose a transition function pt is conservative and spatially d homogeneous in the sense that p t (x,B) = pt(x+y, B+y), y e R . Then pt is Feller because of the expression ptu(x) = For instance {Rd Pt(O,dy)u(x+y)
-
@
(2.1.6) pt(x,dy) = (27~t)d/ 2 exp( )dy is such a transition function. Let A be the generator on Cm(Rd) of the transition function (2.1.6) and let L be a linear operator defined bv 2 ‘ d Lu = 1 Au ( = a D(L) = C O2 ( R d ) , 2 i=l 1 2 where Co is the space of twice continuously differentiable functions. Using the Taylor expansion of u(x+y) around x, it is then easy to see (2.1.7) L C A. In this sense, the equation (2.1.5) can be interpreted as the heat equation. A related transition function on ( 0 , ~ )defined by 2 2 - (2~t)-~/’exp(- *)I dy pt(x,dy) = {(Z~t)-~/’exp(- k$-is still Feller but non-conservative : pt(x, (0,m)) < 1, x > 0. In this case also Co((O,m)) 2 C D(A) and Au = 1 u ” , u E. C 2o ( ( O , m ) ) .
7 ’ 9
It is in general hard to check directly the Feller property for non-spatially homogeneous transition functions. Moreover, as is illustrated in the above example, an explicit expression of the generator A is available only on a subset of the domain D(A). What we practically know is that A is a closed extension of some concretely given linear operator L with domain D(L) consisting of smooth functions. Kolmogorov [9] first presented a typical expression o f the operator L as an elliptic differential operator: d 2 d a. .(x) a ‘I + xbi(x) + c(x)u (2.1.8) Lu = axiaxj i=l axl i,j=l 1J ~
(Caij(x)titj 0 , c(x) o 1. He actually proved that, underdsome regularity conditions for the transition function pt on R , the parabolic differential equation
M. Fukushima
192
must hold with
L
of ( 2 . 1 . 8 ) .
A fundamental question arises : under what conditions on the coefficients aij, bi, c of the operator L, does the equation (2.1.9) admits a transition function as a solution ? This question has been answered directly or indirectly in several ways. One is a probabilistic way of solving the associated stochastic differential equations ( [ 8 ] - , [ 1 2 ] ) which has been quite successiful especially in the multidimensional cases. But we do not go into this direction now. The second way is to rely upon the theory of partial differential equations ( [ 4 ] ) . The third is the way of functional analysis yo construct, by taking a closed extension of L, an operator A = L satisfying the Hille-Yosida conditions(A.l), (A.2) and ( A . 3 ) . This method was successful in one-dimensional case but rather hard to be used in the multidimensional situations. From now, we like to present another way of a functional analytical character, namely, an L2-theoretic method rather than the C-theory. Since the main concern of probabilists is not a transition function but a stochastic process behind it called a Markov process, the method of SDE is in a sense more favorable than the analytical methods. But LZ-theoretic methods provides us with many Markov processes which can not be covered by the SDE method. Furthermore, as the title of the historical paper of Kolmogorov suggested, the theory of Markov processes can not get rid of the influence of the analysis on the state space X. Instead we make full use of a relevent functional analysis, that is , the theory of Dirichlet form. An L'-theoretic approach toward transition functions is based on the following considrations. Denote by m a positive Radon measure on X such that m is positive on each non-empty open set. L 2 (X;m) stands for the L 2 -space of m-square integrable functions on X with inner product ( u , v) = {Xu(x)v(x)m(dx). Let pt be a transition function. m is said to be an excessive measure with respect to pt if
lx
m(dx) pt(x,B) 2 m(B),
B C @(XI.
Lemma 1 If m is excessive with respect to a transition function p,, then {pt, t > 0 ) determines uniquely a strongly continuous semigroup {Tt, t > 01 on L 2 (X;m). Tt is Markovian in the 1, u G L 2 (X;m). sense that 0 2 Ttu 6 1 m-a.e. follows from 0 6 u (ptu(x)) 2 Proof By Schwarz inequality, we have for u 6 C o ( X ) , 2 2 I ptl(x) ptu (XI 5 ptu' (XI; and consequently (ptu, ptu) 6 2 Hence pt extends from Co(X) to ptu (x)m(dx) 6 ( u , u ) . L (X;m) as a unique linear operator Tt with llTtllLZ 2 1.
In
Semigroup property
Tt are clear.
TtTS = Tt+s and the Markovian property of each The strong continuity of Tt follows from
Markov processes and functional analysis
(ptU - u,PtU - u) 2.2 -
2 {(u,u) - (
u , ptu))
-+
t+O
0, u B C,(X).
193
q.e.d.
Dirichlet forms and symmetric Markov processes
We say that a transition function
jX
pt
is m-symmetric if
jx
(2.2.1) P,U(X) v(x)m(dx) = u(x) ptv(x)m(dx), u , v E @+. Evidently m is then an excessive measure of pt and , according to the preceding lemma, an m-symmetric transition function gives rise to a unique strongly continuous contraction semigroup {Tt, t > 0 ) on L 2 (X;m). In this case, Tt is not only Markovian but also a symmetric operator on L 2 (X;m). Such a semigroup is completely characterized by the notion of the Dirichlet form. In this subsection, we quickly mention relevent basic notions and relations by refering the readers to [S] for further details. densely defined non-negative definite symmetric bilinear form E on L2(X;m) is simply said to be a symmetric form. Its domain is We let denoted by D [ E ] . E a ( u , v) = E(u, v) + a ( u , v), U, v 6 D[E]. If D[E] is comlete with respect to the El-metric, then a symmetric form E is called closed. We say that a symmetric form E on L2(X;m) is Markovian if, for any E > 0 , there exists a real func t ion $,It), t 6 R 1 , satifying
A
& $,(t) 0 2 $,(t) such that u 6 D[E] (2.2.2)
-E
& 1
$,(t) = t
+E,
& t
- $,(t')
implies v
- t' =
$,(u)
for t e [0,11, for t' < t, E D[E] and E(v,v) , I E(u,u).
We call a Markovian closed symmetric form on L 2 (x;m) a Dirichlet A Dirichlet form E on L2(X;m) is in one-to-one form. correspondence to a strongly continuous contraction semigroup The {Tt,t > 01 of Markovian symmetric operators on Lz(X,m). correspondence is given by D[E] (2.2.3)
=
E(u,v)
=
1
( u , u - T u) < m ) t t+O 1 u , v 6 D[E]. lim -t- ( u , v - Ttv),
{u C L'
: lim
t +o
We have seen that any m-symmetric transition function gives rise to such a semigroup. We call the associated Dirichlet form the A natural question Dirichlet form of the transition function. arises : conversely, given any Dirichlet form, is it the Dirichlet form of some m-symmetric transition function ? Exam le 2 The transition function (2.1.6) on Rd is symmetric tce*w to the Lebesgue measure. Its Dirichlet form E on L2(Rd) is given by D[E] = H 1 (Rd ) (2.2.4) Vu Vv dx E(u, v) =
IRd
M. Fukushima
194
where H 1 (Rd) is the Sobolev space of order 1 : H 1 (Rd) = {u 6 L 2 : au Indeed, by taking the Fourier transform axi C L 2 , 1 g i d l . 2 of u 6 L and using the formula ( 2 . 2 . 3 ) , we easily get ( 2 . 2 . 4 ) . The Markovian property of the form ( 2 . 2 . 4 ) can directly be seen as follows. Take u G H1(Rd). We may assume that u is absolutely continuous on almost all lines parallel to the axes. Take for any E > 0 a C1-function $,(t) satisfying ( 2 . 2 . 2 ) . Then
jRd \ v Q E ( U > l dx
=
'I,
7 d
\VU\
2
which means the Markovian property of
\@;(U)\2
dx & E ( u ,
U),
E.
Given a transition function p on X, one can construct a Markov rocess governed by pt, namel?, there exists a collection o f xA with state space M = { Q , Q, Xt, PX} b t i c processes X,
such that for each
x h X
PX -a.s. Px(Xt+s E A l a t ) = P,(X~,A) (2.2.6) Px(Xo = x) = 1 where A is any element of @(X) and Dt denotes a sub a-field of @ making Xs, s 6 t, measurable. Suppose that the sample path X.(W) is right continuous and has left limits PX -a.s. for each x X. We then say that M has the strong Markov property if, for some right continuous o-fields {C$)t>o, equation ( 2 . 2 . 5 ) holds with time t being replaced by any @}-stopping time. If in addition almost every sample path is left continuous along any increasing sequence of stoppint times, then M is said to be a h ; ; process: In particular a strong Markov process whose sample is continuous up to the death time ~ ( w ) = inf {t > 0 : X,(W) Px-a.s., x f X, is called a diffusion (process). A) (2.2.5)
In regard to the question raised before Example 2 , we introduce two A Dirichlet form E on L 2 ( X ; m ) notions about Dirichlet forms. is called regular if the space D[E]nC,(X) is dense both in D[E] and in Co(X) E is said to be local if E(u, v ) vanishes whenever u, V C D[E] have disjoint compact support. Now the answer ty the above question is as follows : if a Dirichlet form E on L (X;m) is regular, then E admits a Hunt process, namely, E is the Dirichlet form of the m-symmetric transition function of The associated Hunt process M is unique some Hunt process M. in a certain sense. In this case, M can be taken to be a diffusion if and only if E is local.
.
Those strong statements would become void if there were not practical means of constructing regular local Dirichlet forms from some concrete data. Fortunately we possess the following useful assertions : if a symmetric form E with D[E]_dense in Co is Markovian and closable (that is, E admits a closed extension), then the closure E is a regular Dirichlet form. If D[E] satisfies some additional mild condition and if E is local, then E as also local.
Markov processes and functional analysis
Consider a non-negative function p t LioC(Rd). 1 d) , E(u, v) = lRd VUVV p dx, D[E] = CO(R
Example 3 (2.2.7)
$
195
Then
defines a Markovian local symmetric form on L2(Rd;pdx). If either (2.2.8) inf p(x) > 0 for any compact set K C Rd, x K or 2 d) , (2.2.9) the distribution derivatives Dx P are in Lloc(R i then E is closable on L2 (Rd ;pdx). Then the closure E' becomes a local regular Diri hlet form and consequently admits a diffusion process Mp on R .
s
A recent paper [ 7 ] concerns the conditions for the function p that the associated diffusion Mp is mutually absolutely continuous with respect to the d-dimensional Brownian motion. Brownian motion is by definition a diffusion process with transition function (2.1.6). In other words,Brownian motion is associated with the Therefore it is a version o f the Dirichlet form (2.2.4) . diffusion MD for p = 1. so
53
CAPACITY
3.1 Capacitary potentials and hitting probabilities -
Given a regular Dirichlet form E on L 2 (X;m), we define the associated capacity Cap(A) for an open set A C X by 1 m-a.e.on A}. Cap(A) = inf E1(u,u), where LA = { u E D[E]: u u6 LA We then extend Cap(A) for any set A C X as the outer capacity yielding a Choquet capacity (see subsection 1.2). The notions !!q.e.'! and !'quasi-continuous''will be used with respect to this When E is transient (or equivalently the as ciated capacity. Hunt process is transient), then the 0-order capacity Cap( 69 (A) can be introduced in the same way as above but by replacing El and D[E] with E and the extended Dirichlet space respectively. This replacement does not change the notions I'q.el' and "quasicontinuous". Cap(O)(A) related to the Dirichlet form (2 2 . 4 ) for d = 3 coincides with classical Newtonian capacity of A C Rs up to a multiplicative constant. The set L is closed and convex for open A C X. Hence there exists a uhique element eA in LA minimizing E1(u,u), which is said to be the e uilibrium potential or the capacitary potential of It then h:lds that 0 6 eA & 1 and A. (3.1.1)
Cap(A)
=
El(eA, eA). be a Hunt process associated with the
regular Dirichlet form E. probability pA of A by (3.1.2) where Ex ure Px.
We defige the (1-order) hitting
-U.
pA(x)
=
Ex( e A ; uA <
m)
, uA(u)
=
inf {t > 0 : Xt(u)EA}
is the expectation with respect to the probability meas-
M. Fukushima
196
Theorem 1
For an open set A
with
LA 9
4 , pA
is a version of
eA' Proof Let ITt, t > 01 be the L 2 -semigroup corresponding to E and M. Then both eA and pA are 1-supermedian : e-tTteA 6 -t It suffices to show the inequality eA3 TtPA & PA' (3*1.3)
PA
6
eA,
because we can then apply formula (2.2.3) to get pA c L and E1(pA,pA) 9 El(eA,eA) (here we use the fact that eA 6 AL 2 ) , which implies PA
=
eA'
To see (3.1.3), take a Borel version eA of eA and let h be any non-negative function with h(x)m(dx) = 1. Let Yt(w) = e-teA(Xt ( w ) ) , then (yt, a t , Ph.m) tBo is easily seen to be a supermartingale : E~.,,(Y~~@) 6 y S phmm-a.s., s < t. Choose any finite set D C ( 0 , m ) with min D = a, max D = b and let a(D ;A) = min {t 6 D : Xt 6 A). The optional sampling theorem for the supermartingale yields -u(D;A) : a ( D : A ) < b, & Eh.m(Ya(D:A)) 6 Eh.m(Ya) 6 ( h , eA)* Eh .m(e By making D increase to a dense subset of (0, b) and b + m, we get (3.1.3). q.e.d.
Ix
When E i transient, we may consider the 0-order capacitary (Opotf, ntial e(if o f an open set A along with Cap(O)(A). e A 2 in the extended Dirichlet space but may not belong to L ( X ; m ) . But we still have the identity (3.1.4) pio) is a version of eA( 0 ) , In fact we get where p p ) = Px( aA < m) . same way as above and hence the inequality
pA('1
eA( O )
in the
p p ) I\ aRaf 6 eiO)fi aRaf , a > 0 , f continuous 2 0, where Ra is the resolvent of pt. Since both hand sides are a-supermedian, we obtain the equality by the same argument as in the above proof. It now suffices to let a m. -f
Theorem 1 enables us to identify diverse potential theoretic notions relevent to the Dirichlet form E with those relevent to the Hunt process M. Here are some of the ikentifications : (3.1.A) Cap(N) = 0, N C X for some Borel N 2 N, PhSm(ur
197
Markov processes and functional analysis
(3.1.C) A function u locally in D[E] is quasi-continuous if and only if u(Xt) is right continuous in t 2 0 and lim u(Xt,) t'+t = u(Xt-), t > 0, Px-a.s. for q.e. x 6 X. "Only if" part of (3.1.C) is a direct consequence of (3.1.B). "If" part can be shown by using a fact that any function u locally in D[E] (i.e., f o r any relatively compact open set G there exists v 6 D[E] sucJh tkat u = v m-a.e. on G ) admits a quasicontinuous version u : u is quasi-continuous and F = u m-a.e. 3.2 -
Tests of irreducibility and attainability
Let E and M be as in the preceding subsection. We further assume that E is local and consequently M is a diffusion. A Borel set B is called T t -invariant if TtIB = IBTt, namely, Tt(IBu) = IB - Ttu for any u 6 Lz, and t > 0. B is said to be M-invariant if Px( ux-B < m) = 0 for any x B. We say that an increasing sequence {FnI of closed sets is a nest if Cap(X-Fn) -+ 0. Following Lejan[ll],we call a Borel set B uasi-o en (resp. quasi-closed) if there exists a nest {F,) s u c W n F n is open (resp. closed) in Fn for each n. F o r Borel sets B , u and B, B is said to be a modification of B if m ( E e B) = 0. Theorem 2 Following conditions are equivalent for a Borel set B. B is Tt -invariant. (i) (ii) u E D[E] IB.u E D[E]. (iii) IB is locally in D[E]. (iv) IE is quasi-continuous f o r some modification B of B. (v) B is quasi-open and quasi-closed f o r some modification of B. (vi) X can be decomposed as X = B1 + BZ + N where Bl(resp. B2) is a modification o f B (resp. X-B), both B1 and B2 are M-invariant and m(N) = 0.
-
h.
Remark 1 Condition (v) is the same as saying that there exists a nest {Fn} such that B n F n and Fn - B are closed for each n. This simplifies corresponding statements in [l]. The set N in condition (vi) is o f zero capacity according to ( 3 . 1 . A ) . Proof
(i)=)(ii)
:
o f identity { E A , X > 01
D[E]
= {U
L2 : 2
jmA 0
Tt and D[E] on L2(X;m)
d(EXu, u) <
m}.
are expres2ible by a resolution as Tt = e-XtdEX and (i) then implies
EhIBu
=
IB'EXu, u E L , from which (ii) is immediate. (ii)*(iii) : Since E is regular, there exists for any compact K, a function u E D[E] n C o ( X ) such that u = 1 on K. (iii)+(iv) : IB admits a quasi-continuous version $ . Since
198
M. Fukushima
= @ m-a.e., @ = o or 1 q.e. Hence @ = 11. q.e. for B c, some modification B o f B. (iv)+ (v) : trivial. : Let -cn be the first leaving time of the sample path. (v)+(vi) Since the sample path is continuous almost surely, x € B n F n , (see Remark 1). Px( Xt E B O F n for any t < T ~ )= 1, The same property holds for the set Fn - B. We then arrive at (vi) by virtue of (3.1.B). (vi)+ (i) : trivial. q.e.d.
$‘
We call E irreducible if any Tt -invariant set B is trivial in the sense that either m(B) = 0 or m(X - B) = 0. In view of condition (v) of the above theorem, we can say that E is irreduOn account of cible if and only if X is quasi-connected. condition (iv), we get Corollary Let m and m be mutually absolutely continuous and let E(l) and E(‘) be regular local Dirichlet forms on L2(X;m(l)) and L2 (X;m(2)) respectively. Suppose that any quasi-continuous function with respect to E(‘) is also quasicontinuous with respect to E(l). Then the irredubibility of E(’) implies the same property of E”). The condition in this Corollary is satisfied if E ( 2 ) is locally dominating E(l). To make this statement more precise, let us consider a dense subalgebra D of Co(X) satisfying the following two properties : (i) for any E > 0, there exists a function % possesssing property (2.2.2) such that C $ ~ ( U )tZ D whenever u e D , (ii) for any compact set K and a relatively compact open set G with K C G , D contains a function u with u = 1 on K and u = 0 on X - G. We let DG = { u E D : u = 0 on X - G}. We say that D is a of a Dirichlet form E if D is El-dense in D[E]. Theorem 3 (i) Let E(l) and E ( 2 ) be two local regular 2 Dirichlet forms on L (X;m) possessing a set D as above as their common cores. Suppose that for any relatively compact open set G (3.2.1) E(’)(U, u ) 2 y K E(’)(u, u), u 6 DG, for some constant y K > 0. Then the irreducibility of implies the same property of E(‘). (ii) Let m (l) and m be related as (3.2.2) dm(‘) = p dm(’) with yG = inf p(x) > 0 x€G
for any relatively compact open set G . Let 2 be local regular Dirichlet forms on L (X;m(”) respectively. Suppose that E(l) and E(’) D of the above type and
E(l)
E(l) and E(’) and L2(X;m(2)) have a common core
199
Markov processes and functional analysis
(3.2.3) ~ ( l ) = E(~) on DXD. Then the irreducibility of E(l) implies that of
E(').
(i) The restriction of a local regular Dirichlet form E Proof to {u E D[E] : u = 0 q.e. on X - G I is denoted by EG, which is known to be a local regular Dirichlet form on L 2 (G;m). A function on G is EG-quasi-continuous if and only if it is E quasi-continuous ( [ 5 ; Th. 4.4.21). Moreover, a function which is quasi-continuous on any relatively compact open set is (globally) quasi-continuous. Under the assumption of (i), DG becomes a common core of E';) and Ei2) "5; Prob. 3.3.41). Inequality (3.2.1) then implies an analogous inequality for the relevent capacities. Hence a function is E';) -quasi-continuous whenever it is E(:) -quasi-continuous. Thus, if a function is quasi-continous with respect to E(2), s o it is with respect to E(l). Now Corollary applies. (ii) By (3.2.2) and (3.2.3), , u 6 DG. q.e.d. E(:)(u, u ) + (u, u ) 2 E(;)(u, u ) + yG.(u, u) m2 ml Theorem 3 can be proved using condition (iii) (instead of Remark 2 0 ) f Theorem 2. Y. Lejan pointed out the significance of condition (iv) in regard to the paper [l]. I also owe to him the present simple proof of "(iii)+(iv)" in Theorem 2 (private communication).
-
3.3 Examples -
Example 4 The Dirichlet form (2.2.4) on L 2 (Rd ) is irreducible because the associated diffusion is Brownian motion whose transition function is given by (2.1.6). Consider the symmetric form E o f Example 3 and assume that the function p E satisfies i d the condition (2.2.8). Theorem 3 then applies with D = CO(R ) and we see that the closure E is a irreducible local symmetric Dirichlet form on L2(Rd; dx).
L~A~(R~)
'
Consider Example 3 for the one dimensional case. Example 5 More specifically we are concerned with the symmetric form (3.3.1) E(u, V) = l~'(x)V'(x)p(x)dx, D[E] = CO(R 1 1) , on
2 1 L (R ,pdx)
and we assume that
whenever 0 4 (a.b). p closable ( [ S ; Th. 2.1.41). The closure
-
E
jb
p d
Lloc
and
may be degenerate at
inf p(x) a
> 0
is
is irreducible if and onlv if
dS -b < for b > 0. In view o f Theorem 2 and Remark 1, this assertion follows from the next lemma. (3.3.2)
M. Fukushima
200
Lemma
Let
I d = (0, d ) .
l i m Cap(Id) d+O
= 0
,
@
b
>o.
Proof " j " : Suppose t h a t t h e above i n t e g r a l i s f i n i t e . Let G = ( - b , b ) , t h e n i t s u f f i c e s t o show l i m C a p ( I d ) > 0 where dGO CapG i s t h e c a p a c i t y r e l a t e d t o t h e form EG ( s e e t h e p r o o f of Theorem 3 ) . By [S; P r o b . 3 . . 3 2 ] , CapG(K) f o r K = [ c , d ] , O< c < d , c a n be computed a s CapG(K) =
inf E1(u,u) u6C&, u = l on K U = O on R'-G
2
which i s n o t smaller t h a n Hence we have 'I
1
Cap(Id)
inf uc c; u (d) =1, u ( b ) =O by Schwarz i n e q u a l i t y . > 0
(
: Suppose t h a t
K = [c,d]
Let
uo
and d e f i n e
by
(I
uo(x) =
uniformly i n
d.
O < c ' < c < d < d '< b ,
C'
L e t t i n -g
c'
+
CaP(Id)
s 21
(
d
+
0
and
c' &
d'
and
0
+
0.
c
+
0, we Eet
)-l +
jd 0
I
p(c)d[
,
which t e n d s t o z e r o a s
~.e.d.
T h i s lemma means t h a t , i f f o r i n s t a n c e
= m and
I,
dS
Po
< =, t h e n R1 i s t h e sum of two i n v a r i a n t - s e t s ( - - , O ) and [O,m). 0 i s a t t a i n a b l e by t h e a s s o c i a t e d sample p a t h from t h e r i g h t b u t n o t from t h e l e f t i n t h i s c a s e .
'
C o n s i d e r t h e same example f o r t h e two d i m e n s i o n a l Example 6 case. Thus, 2 2 1 2 ( 3 . 3 . 3 ) E(u,v) = 2(ux + u y ) P ( x , y ) d x d Y , D [ E I = CO(R 1 . I R L e t C = Ix = 1 1 . We assume t h a t p LIAc(RZ) and i n f p ( x , y ) (x,Y)eK i s p o s i t i v e f o r any compact s e t K w i t h K n C - = $- . We f u r t h e r assume t h a t t h e form ( 3 . 3 . 3 ) i s c l o s a b l e on L'(RL; pdxdy) BY t h e same r e a s o n i n g a s i n t h e p r e c e d i n g e x a m p l e , we c a n c o n c l u d e t h a t E i s i r r e d u c i b l e i f f o r some a B and b > 0
.
Markov processes and functional analysis
201
sufficient condition for the reducibility can be stated as follows: For a n-interval J, we denote the integral jJp(c,n)dr, by m If I J k ) k=-.. is an open covering of R1 and if for each k ,
A
(3.3.5)
0
either
dc bJk0
-
cJ.
= for some j-bk bk > 0, then the left hklf plane {x > 01 and the right half plane {x > 0 ) are not attainable from each other. =
Or
IR3
Exam le 7 We still consider the same example but for three dimenh( 3 . 3 . 6s) E(u, e : V) = ( u i + u2 + uz)p(x,y,z)dxdydz 2 Y and we are concerned with the attainability of one point, say, the origin 0, of the associated sample path. In other words, our problme is whether Cap({O)) is zero or not. We assume that p is locally integrable and satisfies (2.2.8). P may be singular at 0. Our assertion is this :
Cap((O1)
Ib & 0 if
=
p(x,
Cap({O})
y, z)
=
p(r)
> 0
if
and
= m , b >O. r p(r) To verify the first assertion, we proceed as in the proof of Lemma: Letting G = {r < 1) and d < 1 and using the inequality
(3.3.8)
0
E1(u,u) 2 ~ ~ ~01 0 2 n0 o:r2p(r,e,$)sin$d=ded$,
we have
J'p +1-l
inf E~(U,U) 2 8n2(j 'sin drded u6C1, u=l on r
=
M.Fukushima
202
References
S. Albeverio, M. Fukushima, W. Karwowski and L. Streit, Capacity and quantum mechanical tunneling, to appear in Comm. Math. Phys. A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A.,.45(1959), 208-215. R.M. Blumenthal and R.K. Getoor, Markov processes and potential theory, Academic press , 1968. E.B. Dynkin, Markov processes, Springer, 1965. M. Fukushima, Dirichlet forms and Markov processes, Kodansha and North Holland, 1980. M. Fukuhsima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motions, Physics Reports 502 (New Stochastic Methods in Physics), North Holland, 1981. M. Fukushima, On absolute continuity of multidimensional symmetrizable diffusions, t o appear in Proc. Katata Conf. on "Functional Analysis in Markov Processes", Lecture Notes in Math., Springer. N. Ikeda and S . Watanabe, Stochastic differential equations and diffusion processes, Kodansha and North Holland, 1981. A.N. Kolmogorov, uber die analytischen Methoden in der W a h r s c h e i n l i c h k e i t s r e c h n u n g , Math. Ann. 104(1931), 415-458. S. Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, to appear. Y. LeJan, Quasi-continuous functions and Hunt processes, t o appear. D.W. Stroock and S.R.S. Varadhan,Multidimensional diffusion processes, Springer, 1979. K. Yosida, Functional analysis, Springer, 1968.