ARKIV FOR MATEMATIK Band 6 nr 14
1.65 02 1 Communicated 27 January 1965 by LARS GiRDIN~ and LENNART CARLESOlV
L o r e n t z - i n v a r i a n t M a r k o v p r o c e s s e s in relativistic p h a s e s p a c e
By R. M. DUDLEY
1. Introduction
This paper deals with certain " r a n d o m motions" permitted b y the special theory of relativity; t h a t is, with probability measures on sets of trajectories on which speeds are less t h a n or equal to the speed c of light (which we take equal to 1 throughout). We deal with classes of such measures indexed b y possible initial states, related to each other b y the Lorentz group (implying certain invariance conditions for individual measures).
The definition of " s t a t e " as mentioned above is governed b y the Markov pro- perty, i.e. t h a t given the present state, further knowledge of past states should be irrelevant to the prediction of future states. I t is known t h a t position is insufficient for this purpose (see e.g. Theorem 11.3 below), and it seems natural to include the velocity in specifying the state. Indeed, velocities m u s t exist at almost all times since position is a Lipschitzian function of time, and the existence of velocities is generally incompatible with the Markov p r o p e r t y of a position process.
We distinguish between speeds strictly less t h a n 1 and those equal to 1, and do not consider processes in which b o t h occur (except in Theorem 11.3). This corre- sponds to the physical distinction between particles of positive or zero rest mass.
I n v a r i a n t processes of speed 1 t u r n out to be essentially uninteresting (see w 11) in t h a t they cannot change direction.
We are left, then, with processes of speeds almost always strictly less t h a n 1.
F o r these processes, we can introduce the relativistic " p r o p e r t i m e " on each trajec- t o r y (see w 6 below). The possible "4-velocities", i.e. derivatives of space-time posi- tion with respect to proper time, then lie in a three-dimensional hyperboloid ~ / w i t h a symmetric, Lorentz-invariant Riemannian structure (see w 7). :Because of our in- variance assumptions, the velocity process is itself Markovian (Theorem 3 . 2 ) a n d defined b y a "convolution" semigroup on ~/.
Such convolution semigroups have fortunately been completely classified b y Tutu- balin [1 ]. We thus arrive at an explicit description of all the processes which concern us, in Theorem 8.2, the main theorem of the paper. There are two extreme possi- bilities: on the one hand, "Brownian motions" in ~ (a one-parameter family indexed b y a diffusion constant), which yield (the only) processes in which velocity is a continuous (but, of course, not differentiable) function of time; see w 10. On the other hand, there are "Poisson" processes in which the velocity changes only b y jumps.
Finally, there are (roughly speaking) mixtures of the two.
R. M. DUDLEY, Lorentz-invariant Markov processes
in relativistic phase
spaceI n order to make the change from coordinate time to proper time as parameter, it seems necessary to make an assumption of continuity in probability. One would like to infer this from the other assumptions through a sort of "infinite divisibility,"
but in our situation this seems difficult before the change to proper time, since one has a convolution semigroup on a symmetric space only after the change.
Sections 2 and 3 deal with generalities about Markov processes. Section 4 intro- duces Lorentz-invariant Markov processes of speeds less t h a n 1. Sections 5-10 carry through the characterization of these processes. I n section l l , we establish the triv- iality of processes with speeds equal to 1 or states given b y position only. I n sec- tion 12 some unsolved problems are mentioned.
For Markov processes I have used specializations of the definitions in D y n k i n ' s books [1] and [2], since several results are also quoted. I t is worth noting, however, that this leads to at least one unsatisfying situation. I n the usual approach to the strong Markov property one demands right continuity for sample functions. Here, where a sample function is (in part) a derivative, it becomes a right derivative after imposition of right continuity. I t m a y seem unnatural to consider the right deriva- tive of a function at time t as known at t h a t time, if the left derivative is different.
The way out of this difficulty, if it is one, will be left for other researches.
The "diffusion" or "Brownian motion" processes studied in w 10 below have been worked on previously b y G. Sehay [1] and H. Dinges. Both the latter and R. Her- mann advised me against beginning with the proper time (i.e., essentially starting in the middle of w 6 below), as I did in an earlier draft, and I am now glad to have followed their advice. An exchange of correspondence with H. Dinges on this sub- ject in general has been most helpful.
2. Starting probabilities and M a r k o v processes
First we review some measure-theoretic notation. I f (X, S, #) and ( Y, if, ~,) are a-finite measure spaces (which for our purposes m a y as well be finite), then $ • 9"
denotes the a-field of subsets of the Cartesian product X • Y generated by all sets A • B, A E 5, B E if, and # • v the product measure. If F maps X onto another set S, then
F(5)
denotes the class of allC c S
such thatF-I(C) = {x:F(x)
EC} E 5. We let(#o F -1) ( C ) = # ( F - I ( C ) ) ,
CEF(5).
Finally if / is #-integrable and :H is a sub-a-field of 5, then
E,(/]
~4) denotes the /~-conditional expectation of / given :H, and if / is the indicator function of a setA,
we letE.(/I ~ ) = ~ ( A I
~).
To clarify the argument of a function F = #(A ] :H) we m a y write F(x) = ~(x : x e A I :H),
where "x E A " will be replaced by a defining condition.
Throughout this paper, R will denote the real line (with its usual topology), R = its n-fold Cartesian product, and R + the nonnegative axis [0, ~ ) . If S is a n y topo- logical space,
B(S)
will denote the Borel a-field generated b y the open sets in S.We shall consider certain function spaces defined as follows:
ARKIV FOR MATEMATIK. Bd 6 nr 14 Definition. I f S is a set, a path space for S is a pair (A, ~) satisfying the following conditions:
(1) ~ is a function on A with O~<~(/) ~< + c~,/E~4.
(2) Each /E ~4 is a function from the interval [0, $(/)) to S.
(3) F o r each s ~> 0 and x E S there is an / E A w i t h / ( s ) = x.
If S is a n y set, the set A of all functions / from intervals [0, ~(/)) to S will be called the maximal p a t h space for S.
Given a p a t h space (A, ~) and s ~> 0, let
A s = { / e A : ;(/) > s}.
Suppose $ is a a-field of subsets of S. I f O ~ s ~ t, B I ( A , $) will denote the a-field of subsets of A ~ generated b y all sets of the form
{/E A ~ :/(r) E A }
for s ~< r ~ t and A E $ (here and t h r o u g h o u t / ( r ) E A implies ~(/) > r). Bs(A, $) will be the a-field generated b y all the B~(A, $) for t > s. Context permitting, BI(A, $) will be written B~(A) or B~, and likewise for B ~. We call (14, $, S, $) a measurable path space.
We shall need the following fact, proved in D y n k i n [1] L e m m a 5.9:
L e m m a 2.1. Suppose S is a metric space and (~4, ~) is a path space/or S such that each / E ,'4 is continuous /rom the right wherever it is defined. For 0 <~ r <~ s let Mrs be the product a-field
B~(A, B(S) ) x B ( R §
in A • R +. Let F(/, t) = / ( t ) / o r r <~ t ~ s. Then the domain D o / F belongs to Mrs and F is M~-measurable on D.
For h ~> O let Oh be the transformation of functions defined b y (0h 1) (t) = / ( t + h), t/> 0.
A p a t h space (A, ~) will be called stationary if for each h >/0, Oh takes A into itself.
I ) S T h 8
I f (A, ~, S, $) is a stationary measurable p a t h space, clearly Oh takes ~,t+h onto B~
for a n y h>~O, t>~s>~O.
A Markov process will be defined below b y a class of measures on a p a t h space corresponding to different initial states. The restriction to p a t h spaces is a speciali- zation of the definitions in Dynkin [1] and [2]. The definition in [2] requires station- a r y transition probabilities, which are sufficient for our purposes. I t will be repro- duced after our own is completed.
Definition. Suppose (~4, ~, S, $) is a measurable p a t h space in which $ contains all one-point sets. Given x E S, a probability measure P on B~ will be called a starting probability at x on (A, ~, S, S) if
P(/ : /(0) - x ) = 1.
R . M . DUDLEY, Lorentz-invariant Markov processes in relativistic phase space
Definition. Suppose (A, ~, S, $) is s t a t i o n a r y a n d for each x E S, P z is a s t a r t i n g p r o b a b i l i t y a t x on A. T h e n ({P~}, .,4, $, S, S) or, briefly, ({Px}, 74) or {Px}, is a Markov process if
(1) F o r a n y t >~ 0 a n d A E $, P(t, x, A) = P~(/:/(t) E A) is S - m e a s u r a b l e in x.
(2) W h e n e v e r 0 ~< t <~ u, x E S, a n d A E S, P~(/: [(u) E A I Bt ~ = P(u -- t,/(t), A) al- m o s t e v e r y w h e r e with r e s p e c t to P~ on ~4 s.
T h e following is k n o w n ( D y n k i n [1], L e m m a 2.2):
L e m m a 2.2. I / ({P~}, ,.4, ~, S, $) is a Markov process and A EB~ $), then Px(A) is S-measurable in x.
T h e definition of M a r k o v process in D y n k i n [2] can be r e f o r m u l a t e d in our t e r m s as follows:
Definition. Suppose given a s t a t i o n a r y m e a s u r a b l e p a t h space (~t, ~, S, $), a set f~, a m a p p i n g
X : ~ x ( , ~ o )
of fl onto A, for each t ~> 0 a a-field ~ t of subsets of
~ , = { ~ c ~ : ~(x( , ~)) > t},
a a-field ~ ~ of subsets of f2 including all t h e ~ t , a n d a set (Qx, x E S ) of p r o b a b i l i t y m e a s u r e s on ~ o . T h e n (x( , ), $, { ~ t } , {Qx}) is a fields i a r k o v process if:
(A) F o r O<~t<~u a n d A E ~ t , or A E X - I ( B ~ $)), A n ~ . ~ .
(B) I f Px is Qx o X 1 restricted to B~ S), t h e n ({P,}, A, ~, S, S) is a M a r k o v pro- cess (in t h e p a t h space sense), a n d (2) holds with Bt ~ replaced b y X ( ~ t ) . T h e w o r d "fields" has b e e n a d d e d to clarify the difference b e t w e e n the two de- finitions. I f the Q, are defined o n l y on the m i n i m a l a-field X l(B~ we h a v e a n i s o m o r p h i s m .
I t is clear t h a t a M a r k o v process on a p a t h space defines a fields M a r k o v process, a n d conversely a fields M a r k o v process defines a p a t h space M a r k o v process. H o w - ever, n o n - i s o m o r p h i c fields M a r k o v processes m a y define the s a m e p a t h space Mar- k o v process.
A M a r k o v process on a m a x i m a l p a t h space is called " c a n o n i c a l " b y D y n k i n [1, 2.11]. A n y p a t h space M a r k o v process extends n a t u r a l l y to a canonical one.
Conversely, a canonical M a r k o v process ({Px}, B) can be restricted to a p a t h space A ~ ~ if a n d o n l y if .,I has o u t e r m e a s u r e 1 for all the P : ( D y n k i n [1] T h e o r e m 2.5).
W e shall also use the strong M a r k o v p r o p e r t y , which we proceed t.o define. Sup- pose (,-i, ~, X, S) is a m e a s u r a b l e p a t h space. A stopping time on the p a t h space is a B ~ function ~ f r o m .~ to R + such t h a t -c(f) ~< ~(/) for all / E ,-1 and, f o r a n y t ) 0,
{l E A : T(I) <~ t < $(f)} E B~
I f T is a stopping time, let
A " = {l e A : ~(l) > ~(1)}
a n d let B~ be the a-field of sets A c A such t h a t for a n y t ~> 0,
ARKIV FOR MATEMATIK. Bd 6 nr 14
A n { / : < ;(/)}
EB~
Then a Markov process (P~} on .4 is called
strongly Markovian
if:SM(1) Given B E $,
(t, x} -+ Px{/ : /(t) e B}
is
B(R +) •
S-measurable,SM(2) for a n y stopping time v, B E S, and B:-measurable function ~ >~ T,
Px(/:/(U)
e B[ B~) = Pro) (g : g(u - ~) E B)almost everywhere on .4~ with respect to Pz, and {x :/(T) = x for some / E .4} E S.
3. M a r k o v p r o c e s s e s i n p r o d u c t spaces
Suppose
(.4, ~, X, S)
and(B, ~1, Y, ~)
are stationary measurable p a t h spaces with~ U ~ + c~, and for each
x E X (yE Y), Px (Q~,)
is a starting probability on .4 (B).Let
Z = X •
Y, ' U = S • C = . 4 x ~ ,and ~--- + c~. For
z = ( x , y ) e Z
let R~ be the measurePx•
on C. Regardh =
(/, g} e C as the functionh(t) = (/(t),
g(t)} EZ, t/>- 0.Theorem 3.1. ({Rz},
C, ~, Z, "if) is a Markov process i/ and only i/both
{Px}and
{Qy}are Markov processes.
Proo/.
Clearly (C, ~, Z, ~ ) is a measurable p a t h space and eachRz
is a starting probability at z. First suppose {P,} and {Qy} are both Markovian. The class of sets C E ~ for which both 1) and 2) in the definition of Markov process hold is closed under finite, disjoint unions and countable increasing unions and decreasing inter- sections. Thus we can p u t C = A • B, A E S, B E 9", and the conclusion follows easily.Now suppose {R~} is Markovian; let us show t h a t {Px} is Markovian. The meas- urability condition 1) is clear. Given 0 ~<t ~<u, x EX, and A ES, let z = (x, y} for some y E Y and note t h a t the conditional probability of a "rectangle" for a product measure is itself a product, so t h a t if lg = 1 for all g,
Px(/: /(u) EA ]B~(.4)). lg
= R~((/,
g}:/(u)E A ]B ~ (C))
= R<i(t),o(t)>((r ,
~p}: r - t) e A)= Pm)(r r - t) E A)" lg,
almost everywhere for
Rz.
Choosing g suitably, we obtain t h a t {P~} is Markovian, q.e.d.Now suppose (X, $) and ( Y, g) are measurable spaces, Z = X • Y, ~ = $ • g, and ((R~}, C, $, Z, ~ ) is a Markov process. Suppose G is a transitive group of automor- phisms of the measurable space (X, S), and each ~ E G acts on Z b y
~(x, y} = (~,(x), y}
and on C in the obvious way. L e t K be the natural projection of Z onto Y and
K ( ( / ,
g}) = g, (/, g} E C, where / and g are both defined on the interval 0 ~< t < ~((/, g});R.M. DUDLEY,
Lorentz-invariant Markov processes in relativistic phase space
we can let ~/(g) = ~(</, g>). L e t B be the set of all
K(h)
for h E C. Clearly (B, ~, Y, if) is a stationary, measurable p a t h space.Theorem 3.2.
Suppose
{R~}ks G-invariant, i.e./or any z E Z , A EB~ and 7EG,
R~( A ) = R~(~)(~ ( A ) ).T h e n / o r each y E Y the measure
R<~.~>o K -1 is independent o / x . Calling it Q~,
({Q~}, B, ~?, Y, if)is a Markov process.
Proo/.
Each Q~ is well-defined b y G-invariance and is clearly a starting probability on B. For the Markov property, first given A E ~" we haveQ~(g: g(t)
q A ) = R<x,y>(</, g>:g(t) EA)
for a n y fixed x E X, and this is if-measurable in y. Second, suppose y E Y, 0 < t < u, A e 9": F o r a n y x E X,
R~.~( </,
g>:g(u) E A I B~
(C))= Rr(~),~(t)(<r ~>: ~ ( u - t) e A )
almost everywhere for R~,~ in
C t.
Transforming both sides b y K we get Q~(g: g(u) e A. I B ~ (B)) = Qg(t)(~o: yJ(u - t) E A)almost everywhere for Qy on B t, q.e.d.
4. L o r e n t z - i n v a r i a n t r a n d o m m o t i o n s
L e t X be a three-dimensional Euclidean space R 3 of ordered triples x = ( % x2, x3)
of real numbers and let
ix I = (x~ + x~ + x~) "2 L e t M be the space R 4 of pairs z = <x, t>, x E X, t ~ R, with
I l z l l 2 = t ~ - I x l '
s will denote the group of linear transformations of M into itself which preserve ]] II 2, have determinant 1, and do not change the sign of t (the proper, orthochronous Lorentz group).
Mo
will denote the universal " t a n g e n t space" of M, or space of derivatives of functions from the real line to M at points ofM. Mo
is naturally isomorphic to M, b u t for some purposes the two spaces will be distinguished.L e t A be the space of functions / from R + to X such t h a t (1)
I/(s)-/(t)l<ls-tl
for s , t > ~ 0(2) I f / ' ( s ) is defined (as it is for almost all s by (1)),
I/'(s) l <
1A R K I V F O R M A T E M A T I K . B d 6 n r 1 4
L e t I(/)-=- + oo. T h e n (•, I, X , B ( X ) ) is a measurable p a t h space. W e shall need t h e following fact:
L e m m a 4.1. Suppose U is an open subset o/ R k, S = B ( U ) , and (~, ~, U, $) is a measurable path space consisting o~ continuous/unctions. Let F = ~ x U and
y = BO(~) x S.
Then the/unction
(/, t) --> d/(t)/dt is de/ined on a set in y and y-measurable there.
Proo/. B y L e m m a 2.1,
(/, t)--> /(t) a n d (/, t)--> (/(t + r ) - [ ( t ) ) / r
for a n y r 4 : 0 h a v e the desired p r o p e r t y . F o r a n y open set V c R k, let
Z ( V ) = {(/, t): (/(t + r) - / ( t ) ) / r E V for small enough rational r 4: 0}. Clearly Z(V) E y.
F o r each n = 1, 2 . . . let {Umn}m~l be a locally finite open cover of R k b y sets of diameter less t h a n 1In. T h e n
A = {(/, t ) : / ' ( t ) exists} = N n [3 m Z(Umn).
N o w a n y o p e n V c R k is t h e union of the / Umn with (finn c V, where the b a r denotes closure. T h u s
{(/, t) : It(t) ~. V} = [3 m,n{ Z(Umn): ~f mn C V} n A . This completes the proof.
N o w let V be the open unit ball {v:iv] < 1} in R 3. V is the space of admissible velocities. W e shall consider families {Px v} of measures, where for each x E X a n d v E V Px ~ is a starting p r o b a b i l i t y at x on 04, I, X , B ( X ) ) . T h e family {Px ~} will be a s s u m e d to satisfy conditions A ) - E) to be f o r m u l a t e d below.
(A) F o r a n y n = l , 2 . . . . , t , > ~ O , A ~ E B ( X ) , x E X , a n d v E V ,
P~x(/ : /(t,) E A,, i = 1 . . . n) = Pg (] :/(t,) E A~ - x, i = 1 . . . n).
(B) F o r a n y x E X a n d v E V,
P~z(/:/'(0 +) = v) = 1.
Since each / E A is differentiable a l m o s t e v e r y w h e r e a n d {(/, t ) : / ' ( t ) exists} is p r o d u c t measurable,
{s > 0 : P~(/'(s) exists) < 1}
has Lebesgue measure zero b y the F u b i n i t h e o r e m for each x a n d v. W e assume this set is a c t u a l l y e m p t y , a n d f o r m u l a t e a M a r k o v p r o p e r t y :
R.M. DUDLEY, Lorentz-invariant Markov processes in relativistic phase space (C) I f t > O , x E X , a n d v E V , then
P~(/'(t) e x i s t s ) = 1 and for a n y A E Bt(~4),
P ~ ( A IB~~ - P } ( t ) ( O t A ) r'(~)
almost everywhere for Pz" (where Ot is the translation b y t, defined in w above).
We n e x t formulate a Lorentz-invariance condition. Each L E ~: defines a trans- formation L* of functions /E A b y
L ( / ( t ) , t) = ( L(1)(/(t), t), L(2)(/(t), t) ), L(i)(/(t), t) = (L* /)(L(2)(/(t), t) ).
Clearly L* is measurable from (A, B~ to itself. L also defines a transformation VL of velocities b y
VL(/'(t) ) = (L* /)' (L(2)(/(t), t) ).
Our Lorentz-invariance assumption is
(D) Pg o (L*) -1 = p~L(v) for any L E I: and v E V.
L e t W be the "phase space" X • V. Let d be the metric on W defined b y d ( ( x , u ) , (y, v)) = I x - Y l + I u - v I.
The last assumption is t h a t derivatives are continuous in probability:
(E) l i E Pg (d((/(t),/'(t)), (0, v)) >~ e) = 0 t~o
uniformly in v, for a n y e > 0.
I d o n ' t know whether (E) follows from the preceding assumptions.
L e t Q be the set of all functions from R + to W. Then (Q, I, W, B ( W ) ) is a station- ary, measurable p a t h space. Each measure P~ on sets
{[ : (/(t~),/'(t~-)~ E B , i = 1 . . . n}, t~ >~ O, B~ E B ( W ) ,
extends uniquely b y Kolmogorov's theorem (Lo~ve [1] p. 93) to a probability meas- ure Qx,, on B~ F o r ti > 0, t + can be replaced by ti. Of course, there m a y exist with positive probability points t, depending o n / , at w h i c h / ' ( t ) does not exist.
5. P r o p e r t i e s o f the Q ....
I n this section we infer from (A) - (E) t h a t {Qx.v} is a "Fellerian" Markov process, and discuss other continuity properties.
We consider the transition measures
Q(h, x, v, A ) = Qx.v(/ : /(h) EA)
ARKIV FOR MATEMATIK. B d 6 n r 1 4 defined for <x, v) E W, A E B(W), h >~ 0. E a c h Q(h, x, v) is a p r o b a b i l i t y measure on W.
W e consider weak* convergence of such measures:
# n - - > # (weak*)
if for every b o u n d e d continuous real-valued f u n c t i o n / on W,
L e t B L ( W ) be t h e class of b o u n d e d Lipschitzian real-valued functions on W, i.e.
functions [ with
II
! JIlL =sup l/(w) I + sup I/(u) -/(w)l
< ~ .~.~ d(u, w)
T h e n (BL(W), ]] IIBL) is a B a n a c h space, a n d a n y p r o b a b i l i t y measure # on W defines an element of the dual space BL*(W), with the dual n o r m
We h a v e # n - ~ # (weak*) if a n d o n l y if
I I ~ n - ~ l l ~ - + o ;
see e . g . R . R . R a o [1] T h e o r e m 3.1. (For "if", n o n n e g a t i v i t y of the measures is essential.)
Theorem 5.1. Under assumptions (A) - (E) o/w 4, Q(h, x, v, ) is jointly weak* conti- nuous in h, x, and v.
Proo!. W e shall prove c o n t i n u i t y in h a n d x u n i f o r m l y on the triple product, t h e n simple c o n t i n u i t y in v.
We h a v e c o n t i n u i t y in h, u n i f o r m l y in all x, v a n d h, b y assumptions (A) a n d (E). F o r c o n t i n u i t y in x, let
Ty<x, v> = (x + y, v>, a n d note t h a t
Q ( h , x + y , v , ) = Q ( h , x , v , ) o T ; 1 b y (A). L e t t i n g # = Q(h, x, v, ), we h a v e
H~--]~~
IIflIBL=I SUpf/d~- f/oT, d#l
Since I ( 1 - ! oT~)(w)l ~<lyl if w ~ w, y ~ x , and II111~ = 1, we have continuity in x, u n i f o r m l y in h a n d v.
I t remains to prove c o n t i n u i t y in v for fixed x a n d h, a n d we can t a k e x = 0.
Suppose v~--->v in V. There are L~ E s such t h a t v = VL~(%) for all n, with L~ con- verging to the identity. W e also choose n u m b e r s s~ ~ 0 such t h a t
Ln<x , h + s~> = <x', t'> a n d (t') 2 - I x' 12 >~ 0 i m p l y t' > h.
R . M . D U D L E Y ,
Lorentz-invariant Markoy processes in relativistic phase space
For / E ~4,/(0) = 0, let t~(/) be the unique value of s such t h a t</(s), s> = L~<x, h + ~>
for some x. (Then
tn(/)> h
for all /E ~4 with/(0) = 0.) We need the following:Lemma 5.2. / - ~
(/(t~(/)), VL,,/'(t~(/))> is Bh-measurable on A.
Proo/.
Clearly[-->t~(])
is Bh-measurable, and so is [-->(/,tn(/)>,
(from B h toB h •
oo)). The map</, s>-+ </(8), v~J'(s)>, s > h,
is
Bn• B(R*)-measurable
by Lemmas 2.1 and 4.1 and continuity of VL,. Composing the last two mappings finishes the proof of the Lemma.Now we apply assumption (C) to obtain, given e > 0,
(*) e~ ([: d((/(t~([)), VL,/'(t~(/))>, (/(h), /'(h))) >~ e]B ~
- - / z ( h ) t
-P~f(h) (9 : d( </(h ), /' (h ) >, <g(s,~(g) ), VL, g (Sn(g) ) > ) ~ e)
almost everywhere for P~, wheresn(g)
is the unique value of s such t h a t<g(s), s + h> = Ln<x, h + ~>
for some x. The set of
<y, s> E M
such that<y, s + h> = L~<x, h + ~n>
for some x E X is the set of points of the form
L=<z,
an> for some fixed an > 0 and arbitrary z E X, since L ; 1 takes parallel hyperplanes into parallel hyperplanes.For y E X the translation
T~: <~, t> --> <~ - y, t>
of M onto itself takes points of the form
L~<a, a~>
onto points of the formLn<b, an (y)>
for some
an(y).
If I Y I < h, then a~(y) > 0, and as n--> ~ , a~(y)--> 0 uniformly for!yl<h.
For a n y y E X and u E V, let
A~.~.~ = P~ (g : d(<y, u>, <g(s~(9)), VL~g'(sn(9))> ) >~ e.
Applying first Ty and then L ; 1, letting un = V -1 u, we obtain by (A) and (D)
L n
An.y.~ = p~n (r : d(<0, un>, <r r ~> e).
Of course, 0~<An.~.~<l for all
n , y
and u. As n--> ~ , since a~(y)-->0, and the sequence un and its limit u form a compact set for a n y u, we have b y (E) that An.y.u--~0 for a n y u and y with lY[ ~<h. Integrating (*) with respect to P~ over all of A, we conclude by the bounded convergence theorem t h a tP~ (/: d(</(t~(/)), VL./'(t~([))>,
</(h),/'(h)> ~> e) tends to 0 as n - ~ oo. Now if B E B(W) letARKIV FOR MATEMATIK. B d 6 n r 1 4
P , ( B ) = P~ (/: </(t,(/)), VL,,/'(t,(/))> 6 B ) , defined b y L e m m a 5.2. W e h a v e shown t h a t
P,~--~ Q(h, O, v) weak* as n - + co. B y (D),
P . ( B ) = Po (g : <g(h + e.), g'(h + e.)> 6 B) = Q(h + e., O, v., B). vn
W e also k n o w t h a t
]]Q(h+en, O,v,, ) - Q ( h , O , vn, )II*L-+O
as n - ~ co since en-~ 0 a n d the v~ lie in a c o m p a c t set. H e n c e Q(h, O, v~, ) --> Q(h, O, v, ) weak*, a n d t h e proof is complete.
I t is n o w e a s y to p r o v e
L e m m a 5.3. Under assumptions (A) through (E), ({Qx,v), (2) is a Markov process.
Proo/. W e k n o w t h a t ( Q , I , W, B ( W ) ) is a measurable p a t h space. T h e Qx.v are p r o b a b i l i t y measures on B~ B(W)) a n d are starting probabilities b y a s s u m p t i o n (B).
F o r the M a r k o v p r o p e r t y , we m u s t first show t h a t given t ~> 0 a n d A E B ( W ) , Q(t, x, v, A ) is jointly measurable in x a n d v. If A is open, x~-->x, a n d Vn-->v,
Q(t, x, v, A ) <~ lim sup Q(t, x~, v,, A )
b y T h e o r e m 5.1. This implies joint measurability in x a n d v, a n d t h e n A can be replaced b y an a r b i t r a r y Borel set. T h e M a r k o v p r o p e r t y itself t h e n holds b y assump- tion (C), q.e.d.
(The following section, which presents L e m m a 5.4, was received as m a n u s c r i p t a d d e d to proof on 15 S e p t e m b e r 1965.--Editor)
W e n e x t verify D y n k i n ' s condition " L ( F ) " for eomlzact sets F, which says in
o u r c a s e :
L e m m a 5.4. F o r a n y c o m p a c t set K c X • V a n d u >~ 0, lira sup Q(t, x, v, K) = O.
<x,v>-->c~ O ~ t ~ u
(Here ;x, v> co means ix oo a n d / o r iv
Proo/. F o r I x l --> co the result is clear. T h u s it suffices to show t h a t for a n y com- p a c t set C c V,
lira sup Q(t, O, v , X • C) = O.
W e m a y assume t h a t C is a closed ball centered a t 0, a n d t h e n t h a t v = (z, 0, 0) with 0 ~< z ~ 1. F o r <x, s> E M let
L(v) <x,
s>
= <(x~ + zs) / f l - z ~, x2, %, (s + zx~)/V1 ~-z~>.
T h e n L(v) 6 C a n d VL(v) (0) = v. Also for a n y t ~> 0, b y (D), Q(t, O, v, X • C) = P ( / : VL(~)/'(s(v)) 6 C), where P = P0 ~ a n d s(v) = s(v, t,/) is the unique n u m b e r s such t h a t
](s) = </(s), s) E H = L(v) -1 { <y, t> : y 6 X ) .
R. M. D U D L E Y , Lorentz-invariant Markov processes in relativistic phase space
L e t C~ he the cone Ix]--<Ss. B y a s s u m p t i o n (B), for a n y 5 > 0 f ( s ) w i l l P - a l m o s t surely lie in C~ for s small enough. I n H N C~, s ~> ~(v) > 0, w h e r e
a(v) = ~(v, t, 5) = t 1/1 - z2/(1 + z6),
so t h a t 0r --> 0 as z ~' 1. T h u s if D(5, v) is the set of all <x, a(v)} w i t h I x l ~< 5~(v), a n d S(v) = {/: ](~(v)) e D(5, v)},
t h e n P(S(v))--> 1 as z ~ 1. Also, b y (E), for a n y s > 0 lim P ( / : I l'(~(v)) I > ~) = 0.
z~'l
L e t u(/, v) = VL(v)/'(a(v)). W e can a p p l y t h e M a r k o v p r o p e r t y a t t i m e ~(v) to o b t a i n (*) P ( / : / ~ S ( v ) a n d ] VL(~)/'(s(v))--u(/,v)] <<.eIB~
= YI(I, v, a) = .r(~(~)) ~(~(~)) (g:l V~(v)g (~(v)) - u(l, v) l < ~), / c S(v)
= 0 otherwise,
P - a l m o s t everywhere; for / C S(v) we h a v e let a ( v ) = (~(v, g, 5, t) be t h e unique s such t h a t
<g(s), s + a(v)> ~ H . T r a n s l a t i n g b y - ] ( ~ ( v ) ) , using (A), we o b t a i n
H ( / , v, a) =
Pro(~(v)) (h :1 V~(~ h'(~(v)) - u(/,
v) I < ~), where ~r = x(v, h , / , 5, t) is t h e unique x such t h a t<h(x) + / ( a ( v ) ) , x + ~(v)> E H . N o w we t r a n s f o r m b y L(v), using (D), to get
l-I(/, v, 5) = p~,(r,~)(j: ] ?"(2(v)) - u(/, v) l ~< e),
where ]~(v) = t - L(v)(~) f(zc(v) ).
N o t e t h a t 2(v) does not d e p e n d on j, o n l y on v, t, 5, a n d / ( a ( v ) ) . N o w
L(v) <x, g(v)> = <(x~ + za(v))/]/1 - z 2, x2, x3, (zt(v) + z x i ) / V ~ - - z 2 > = < , x2, x3, ~>,
t zx 1
where T = i + z~ + 1 / ~ - z ~"
F o r <x, a(v)} E D(5, v), T is smallest w h e n x 1 = - (~(v).
T h e n T = t(1 - z5)/(1 + zS).
T h u s ~(v) <~ 2zSt/(1 § z5) ~< 25t.
H e n c e 1-[(/, v, 6) ~> inf p~(I.v) (] : I f ( s ) - u(/, v)[ ~< s).
O<~s<~26t
ARKIV FOR MATEMATIK. B d 6 n r 1 4 T h u s b y (E), lim l-I(/, v, (~) = 1,
6~,0
u n i f o r m l y in / a n d v. Thus, integrating o u r M a r k o v e q u a t i o n (*) with respect to P , we h a v e
lim lim P ( [ : / E S(v) a n d I VL(,)/'(s(v)) - u(/, v) I <~ e) = 1.
o$o Ivl~l F o r (a, b, c) E V we h a v e
VL(~) (a, b, c) = (a + z, b Vi-~-z i, cV1 - z2)/(1 + za), I VL(v)(a, b, c) - v] 2 ~ (ae(1 - z2) ~ § b e § c2)/(1 § za) e
< (a e + b e + ce)/(1 - [a ])3.
T h u s for 7 <~ 1, t(a, b, c)l ~< 7 implies
I VL(,) (a, b, c) - v l < 2r.
H e n c e for a n y e > 0 a n d 0 < 6 < 1,
lim P ( / : I VL(,)/'(c~(v)) -- v I ~< e) = 1.
IriS1 Thus, letting ~-->0, we get
lira P ( f : I VL~)/'(s(v)) -- v I <~ e) = 1.
Iv11"1
N o w for a n y c o m p a c t C c V, there is a n e > 0 such t h a t Iv 1> 1 - e a n d I w - v l > e i m p l y w ~ C. T h e n
lim P(/:
VL(,)/'(8(v)) E
C) = 0 Iv[ t'1This limit is u n i f o r m in 0 < t ~< u since
tim e(v) = lim 2(v) = 0
lvlr ~r
u n i f o r m l y for 0 --~ t ~< u. T h e proof is complete.
U n d e r assumptions ( A ) - ( E ) , each measure Qx.v gives outer measure 1 t o t h e class 0 of functions in Q h a v i n g limits f r o m t h e left a n d continuous from the right for each t >~ 0 ( D y n k i n [2], T h e o r e m 3.6).
The set of ( x ( ) , v ( ) ) E 6 such t h a t Ix(s) - x(t) I <~ I s - t I for all rational s, t ~ 0 is B~ a n d has Qy.~-measure 1 for each (y, u ) E W since P~ is concent- r a t e d in ,,t. T h u s b y definition of 6 , Ix(s) - x(t) l <~1 s - t l for all s, t >~ 0 with Q~.u- probability 1 on 6. T h e m a p
(t, (x( ), v( ) ) ) - , x ' ( t )
is B ( R +) • B0(6).measurable b y L e m m a 4.1. F o r each t, x ' ( t ) = v(t) with Q~.~-proba- bility one. Hence b y the Fubini theorem, for Qy.u-almost all ( x ( ) , v( )} E 6 we h a v e x'(t) = v(t) for almost all t, so t h a t since v( ) is locally b o u n d e d a n d x( ) is Lipschit- zian, x( ) is an indefinite integral of v( ) a n d x'(t +) =v(t) for all t>~0. L e t "/~ be t h e set of ( x ( ) , v( )} E 6 with x'(t +) = v(t) for all t ~> 0. T h e n we h a v e a M a r k o v process
({Q~,v},
lO, L w, B(W)).
R.M. DUDLEY, Lorentz-invariant Markov processes in relativistic phase space
L e t ls be the set of functions t---> <c + t, ](t) > , / E "llq , c, t E R. L e t W be the p r o d u c t space R • W, a n d let R~ be t h e u n i t m a s s c o n c e n t r a t e d in the f u n c t i o n t--> c + t. L e t Qr be t h e p r o d u c t m e a s u r e R~x Q .... T h e n b y T h e o r e m 3.1,
({Q ... }, W , I, W , B ( W ) )
is a M a r k o v process. This process is " F e l l e r i a n " b y T h e o r e m 5.1 a n d right contin- uous, hence s t r o n g l y M a r k o v i a n ( D y n k i n [1], T h e o r e m 5.10).
6. T h e proper t i m e
F o r a n y / E ~4 (defined early in w 4) the p r o p e r t i m e v(s,/) is defined (cf. Moller [1] w167 36, 37) b y
where the i n t e g r a n d is clearly b o u n d e d a n d measurable. B y a s s u m p t i o n on ~4, it is strictly positive a l m o s t e v e r y w h e r e (Lebesgue measure). F o r F E "lO, F(t) = </(t), v{t) >, or for F e ~ , F(t) = </(t), c + t, v(t)>, we let T(s, F) = v(s,/). T h e n let
~ ( f ) = ~(t, f) - ~ ( s , / )
for 0 ~ s ~< t, / E ]0. Clearly ~v~(/) = 0 for a n y s >~ 0, a n d ~0~(/) > 0 if 0 ~< s < t. W e h a v e
~(1) + ~.(1) = ~ ( I )
for 0 ~< s ~< t ~< u, / E ~0, a n d ~0~(/) is continuous in s a n d t. T h e functional ~0 is station- a r y in the sense t h a t
= ~ +
4(/)
for h>~O, O<~s<~t,[E~O.T h e strong M a r k o v process ({Q ... }, ~ , I , W, B(W)) is " s t r o n g l y " m e a s u r a b l e as defined b y D y n k i n [2], 3.17, i.e. the conclusion of L e m m a 2.1 a b o v e holds, since t h e h y p o t h e s i s (right continuity) does. W e can e x t e n d each of the (~-fields B~(~L 0) b y the sets which are subsets of sets of m e a s u r e zero for all t h e Q ... t h u s o b t a i n - ing a " c o m p l e t e " fields M a r k o v process as defined b y D y n k i n [2], 3.6.
T h u s we h a v e established all t h e h y p o t h e s e s of a t h e o r e m on " r a n d o m change of t i m e " : D y n k i n [2], T h e o r e m 10.10. T o s t a t e the t h e o r e m in our case, we let ~-->t(v,/) be t h e inverse of t h e increasing function t - + T(t,/) for a n y / E]L 0. T h e n for a n y f i x e d T > 0, t(T, ) is a s t o p p i n g time. F o r / E ~o let
(A/) (~) =/(t(z,/)), ~'(A/) = sup (z(t,/) : t/> 0), x(z,/) = (A/) (3).
T h e o r e m 6.1. (x(T, ), ~'o A , Bt(~), Q ... ) is a ]ields Markov process.
Corollary. ({Q ... o A - l } , A(~9), ~', W, B ( W ) ) is a (path space) Markov process.
W e h a v e t h e inclusion
A(Bt(~. )) ~ B~
I d o n ' t k n o w w h e t h e r the converse inclusion holds (given right continuity, in general)
ARKIV FOR MATEMATIK. B d 6 n r 1 4
T h e " v e l o c i t y c o m p o n e n t " of a f u n c t i o n in A(~9) is n o t t h e d e r i v a t i v e of t h e
" s p a c e c o m p o n e n t " . T o r e m e d y this situation, let U be the t r a n s f o r m a t i o n of V into M 0 defined b y
i ( v ) = Cv, ~ > / V ~ - I~
~.
T h e n ]] U(v)II 2 = 1 for a n y v E V, a n d t h e last c o m p o n e n t of U(v) is positive. U(v) is the four-velocity associated with t h e " v e l o c i t y " v (cf. Moller [1] w 37). L e t l / b e t h e set of all U(v), v E V. ~ is one n a p p e of a three-dimensional hyperboloid. F o r
~ = ( x , c , v ) E M x V ,
let U2(~ ) (x, c, U(v)) E M • ~ . T h e n U 2 is a h o m e o m o r p h i s m of M • V onto M x ~ , a n d it defines a t r a n s f o r m a t i o n U 2 of A{~9) o n t o a n o t h e r function space If. F o r (x, c, U) e M • z = (x, c), U = U(v), let
[Iz.~ = Qc,~,~ o A -1o U~ 1, r = ~'(/)
(U 2 is one-to-one). T h e n b y a s t a n d a r d result on t r a n s f o r m a t i o n of phase spaces ( D y n k i n [2] T h e o r e m 10.13) we h a v e
T h e o r e m 6.2. ((1-Iz. u}, If, ~, M • 1l, B ( M x ll)) is a Markov process.
W e n o w h a v e t h a t for ] = ( 4 , Y)} E lq, where q~ h a s values in M a n d y3 in 1/, r = v2(T ) for all v < ~(/), since the d e r i v a t i v e w i t h respect to v is the 4-velocity (Moller [1] w 37, e q u a t i o n (38)). Also ~fl has a limit f r o m the left for all ~ > 0. T h u s y~
is b o u n d e d on a n y b o u n d e d closed s u b i n t e r v a l of [0, ~(/)), a n d r is locally Lipschit- zian a n d a n indefinite integral of yJ.
A s s u m p t i o n (A) of w 4 (spatial h o m o g e n e i t y ) a n d the s a m e condition for t h e meas- ures R~ of w 5 i m p l y t h e corresponding condition for t h e l-L. v:
(I) I f y, z E M , U E I I , v>~O, a n d A E B ( M ) ,
l-L.v((r ~o) : r EA) = YIz+y.v((r ~ ) : r E A + y).
Also, a s s u m p t i o n (D) of w 4 yields
(II) F o r a n y L E s 1 7 6 a n d U E I / ,
1-[o. v(A) = ]-Io.L(~) (L(A)).
A n L E s acts on t h e t u n g e n t space Mo of M t h r o u g h t h e n a t u r a l i s o m o r p h i s m of M a n d Mo, or, e q u i v a l e n t l y here, t h r o u g h its differential dL, so t h a t if U E 1 / a n d r = U, t h e n
L ( U ) = dL(r
N o w let P be the n a t u r a l projection of M • 1 / o n t o l/. B y (I) a n d T h e o r e m 3.2, ((i-L.u o p 1}, p ( ~ ) , ~, 1/, B(I/))
is a M a r k o v process i n d e p e n d e n t of z, where ~(P(/)) = ~(/) for a n y / E If. P is one-to- one on If, so t h a t this process d e t e r m i n e s {1-Lv} (by integration).
F o r ~ ~> 0, U E I / , a n d A E B ( ~ ) , let
P,,(U, A) = YIz.v(<r ~o) E If : y~(a) E A)
R. l~I. DUDLEY, Lorentz-invariant Markov processes in relativistic phase space
(which is i n d e p e n d e n t of z). B y t h e M a r k o v p r o p e r t y , we h a v e t h e C h a p m a n - K o l - m o g o r o v e q u a t i o n (semigrou p p r o p e r t y )
P ~ ( U , B) = J~Po(U, dV) P~(V, B) for a n y a, ~ ~> 0, U E ~ , B E B ( ~ ) . B y (II), we h a v e
P , ( U, B) = Po( L( U), L( B) ), L E s
I n w 8, we shall reproduce t h e k n o w n classification of all semigroups {P,} satisfying our conditions.
L e m m a 6.3. For any {YIz. ~} satis/ying our conditions and ~ > O, [Ix. u {(~, yJ): ~'(3) exists} = l-Iz. u {/: ~(/) > T}.
Proo/. L e t ~ be the set of all 3 > 0 for which the conclusion does n o t hold. F o r / = (~, ~p) E ~, r exists for almost all 3 in the interval (0, ~(/)). T h u s b y L e m m a 4.1 a n d the Fubini theorem, ~ has Lebesgue mexsure 0. T h u s for a n y 3 > 0 there exist t > 0 a n d u > 0, neither of t h e m in ~ , such t h a t t + u = 3. F o r a n y a > 0,
[Io, v { ( r ~o): r exists}
is i n d e p e n d e n t of U b y condition (II): L e t t i n g a = u a n d applying the M a r k o v prop- e r t y at time t, we conclude t h a t T ~ ~ , q.e.d.
Of course, for (~, ~o) E ~, r is equal to ~p(v) if the f o r m e r is defined.
7. The space "U
is well-known as " h y p e r b o l i c space" or " L o b a c h e v s k y space" (Gelfand a n d Berezin [1]). I t is acted on transitively b y the proper L o r e n t z group ~. The subgroup K , of ~ leaving a point p of ~ fixed is isomorphic to the o r t h o g o n a l group K on three-dimensional E u c l i d e a n space, a n d is a m a x i m a l c o m p a c t subgroup of i:. ~ can be regarded as t h e homogeneous space ~ / K p of right cosets of Kp in t:. (Of course, K~ is n o t a n o r m a l subgroup of IZ.)
~/ has a n a t u r a l l~-invariant R i e m a n n i a n structure, inherited f r o m t h e "pseudo- R i e m a n n i a n " or L o r e n t z f o r m ]III 3 on Mo b y restriction to the t a n g e n t spaces of ~/.
W e p u t geodesic polar (or spherical) coordinates on ~/ (see Helgason [1] p. 401) as follows: we take p as (0, 0, 0, 1) for a given L o r e n t z coordinate s y s t e m (x, y, z, t) on Mo. Given a n y p o i n t U in ~ / w e let e(U) be the R i e m a n n i a n distance f r o m p to U in ~/. N o w each surface ~ - Q0 > 0 in ~/ is isometric to a Euclidean sphere of some radius /(Q0)- (Kp acts transitively on these surfaces, i.e. ~ / i s a " t w o - p o i n t homo- geneous" R i e m a n n i a n manifold as defined b y Helgason [1] p. 3 3 5 . ) i n each such sphere we choose s t a n d a r d spherical coordinates 0 and r 0 ~< 0 < 27~, - 7~/2 ~ ~ ~< ~ / 2 , c o n s t a n t on each geodesic e m a n a t i n g from p. The functions (Q, 0, 4)) m a k e up a regu- lar local coordinate s y s t e m except where Q or 0 is zero or ~ = _+ ~ / 2 .
:Now let r = (x ~ + y2 + z2)lm. T h e n ~ is a function of r in ~ , and/(~(r)) = r, 0 < r < ~ , since the induced R i e m a n n metric on a sphere D = c o n s t a n t can also be induced f r o m a 3-plane t = constant, which has the s t a n d a r d Euclidean structure.
To find the function Q(r), a n d hence its i n v e r s e / , it suffices to find the R i e m a n n s t r u c t u r e induced on t h e h y p e r b o l a H: t 2 - x 2 = 1, y = z = 0. H e r e r = Ix ]. x is a coor-
ARKIV FOR MATEMATIK. B d 6 nr 14 dinate function on
H,
a n d the t a n g e n t v e c t o rd/dx
"is (1, 0,O, x / ~ l +
x 2) at the p o i n t (x, 0, 0, ]/i + x 2) of H. This v e c t o r has " m a g n i t u d e "1 / V I + x ~
for the R i e m a n n met- ric induced on H . Thus a unit v e c t o rd/d~
for t h e R i e m a n n metric is [/1 + x 2d/dx
for x > 0, so t h a t
dx/dQ = V1 + x ~, Q =
log (x + ]/1~+ x 2) (since Q = 0 when x = 0), Ix[ = sinh ~ on H, a n d r = sinh Q t h r o u g h o u t ~ .N o w let g be the induced R i e m a n n f o r m on ~/:
g~ = g(~/~x,, ~/~x;),
where x~ = e, x~ = 0, x 3 = rT h e n g~j = 0 for i =~ ], gll = 1, g22 = sinh~- ~), a n d gaa = sinh2 0 sec2 ~ (except at t h e ex- ceptional points 0 = 0 etc.). N o w t h e " L a p l a c e - B e l t r a m i " o p e r a t o r A on ~ is (see Helgason [1] p. 386)
1 ~
(~ = d e t e r m i n a n t of (g~), g~ = inverse matrix)
= ~ / ~ + 2 coth ~ ~ / ~ + csch ~ ~ ~2/~r _ t a n r csch 2 ~ ~ / ~ + csch ~ ~ sec ~ r ~2/~0~
(wherever ~), 0, r form a coordinate system).
8 . S e m i g r o u p s o f m e a s u r e s o n ~/
A t the end of w 6 we h a d obtained a collection of nonnegative, K~-invariant measures
Pt(P, )
on B(~/) for p E ~/, t ~> 0, of total mass at m o s t 1. F o r a fixed :p, we call finite, Kp-invariant, n o n n e g a t i v e measures on ~/ " r a d i a l " . The "convolu- t i o n " of two radial measures # a n d r on ~ / i s given b y(/~ ~ v) (A) = j~ v(A~) #(dx)
for a n y A E B ( ~ ) , where A~ is a set
T~(A)
a n d T~ is a n y element of ~: t a k i n g x into p. B y Kp-invariance, it is irrelevant which such element is chosen, a n d we cantake Tx continuous in x to insure measurability.
Since Pt(U, B) =
Pt(L(U), L(B))
for all L E C, B E B(~/), we h a v eP ~ P t = P s + t
for alls,t>~o.
T h u s the P~ f o r m a semigroup u n d e r convolution. N o w each radial measure on has a " F o u r i e r t r a n s f o r m " , defined in terms of t h e eigenflmctions of A which are functions of ~ alone ("spherical f u n c t i o n s " : see Helgason [1] Chap. X w p. 398).
As given b y Tutubalin, Karpelevich a n d Shur [1] (hereafter called " T K S " ) t h e non- c o n s t a n t spherical functions on ~ , normalized to the value 1 at
p,
areF~(0) = sin ~ / / ~ sinh if,
where 2 is a n y complex n u m b e r a n d A 2'~ = ( - 1 - ~ ) _F;. F o r ~ or Q equal t o 0, F~(~) is defined b y continuity. These functions are b o u n d e d for
I l m ~ l
~ 1. T h e F o u r i e r t r a n ~ o r m of a radial measure ,u is given b yf~(~) = I F~(x) d#(x),
J u
R.M. DUDLEY, Lorentz-invariant M a r k o v processes in relativistic phase space
defined at least for I I m X I ~< 1. F o r radial measures convolution is c o m m u t a t i v e , a n d (~ ~ ~)^ (~) = ~(~) ~(~)
(see T K S [1]).
A radial measure # on ~ such t h a t for all n = 1, 2 . . . tt is the n-fold convolution (#n) n of a radial measure #n with itself, is called "infinitely divisible". Clearly the measures P t have this p r o p e r t y . Such measures h a v e been characterized b y Tutu- balin [1] and Gangolli [1]: # is infinitely divisible of total mass ~ > 0 if a n d only if
fi()~) =/~ exp (( - 1 - ~t 2) ~ - f u (1 - F~(u)) dL(u)) (*)
where a ~ 0 a n d L is a nonnegative, K~-invariant measure on ~ satisfying
~ @(u)~ d L ( u ) <
J o 1 + @(u) 2
(so t h a t the integral of 1 - F~ is absolutely convergent), a = ~(#) a n d L = L(tt ) are uniquely determined. Conversely, for a n y such measure L a n d n u m b e r s ~ a n d fl >~ 0, there is a radial, infinitely divisible measure # satisfying (*).
T u t u b a l i n [1] p r o v e d the representation f o r m u l a (*) u n d e r a definition of infinite divisibility requiring t h a t for e v e r y c > 0, # is a convolution of possibly distinct measures each c o n c e n t r a t e d within R i e m a n n distance ~ of p, except for mass c. A n a r g u m e n t in his proof, p a r t l y r e p r o d u c e d below, shows t h a t this is equivalent to the definition given above. A proof u n d e r our definition was also given b y R. Gangolli [1] for general s y m m e t r i c spaces of n o n - c o m p a c t type.
W e s a y a radial measure # divides a n o t h e r one, v, if #-)e @ = v for some radial @.
Clearly this imp]ies
~(#) § ~(@) = ~(v) a n d L(#) § L(@) = L(v), so t h a t # divides v if a n d only if a(#) ~< ~(~) a n d L(#) ~ L(v).
N o w given a convolution semigroup Pt, we h a v e L(P~t) = rL(Pt) a n d ~(P~t) = ro~(Pt) for all rational r > 0 b y uniqueness, a n d hence for all r > 0 b y the ordering just mentioned. Thus t h e semigroup is completely d e t e r m i n e d b y L(P1) a n d a(P1).
P t (P , ) converges weak* to t h e u n i t mass concentrated at p as t $ 0; s i n c e / 5 t converges to I pointwise and each I F~I for ~t real is b o u n d e d below 1 outside a n y n e i g h b o r h o o d of p. N o t e t h a t the original c o n t i n u i t y a s s u m p t i o n (E) was used only to o b t a i n a M a r k o v process with proper time as p a r a m e t e r ; for the latter, t h e conti- n u i t y properties follow f r o m t h e invariance conditions, a n d the first p a r t of assump- tion (C) is t a k e n care of b y L e m m a 6.3.
W e n o w show t h a t ~ is a c t u a l l y infinite:
L e m m a 8.1. F o r a n y z E M a n d U E "ll, I~z. ~(~ = + ~ ) = 1.
Proo/. The M a r k o v process
({YIz. v o P-~}, P ( ~ ) , ~/, ~ , B ( ~ ) )
is right continuous a n d Fellerian, hence strongly Markovian. Given / E P ( ~ ) , let to(/) = O. Given to(/) . . . tn(/), let t~+l(/) be t h e least t, if any, such t h a t t h e R i e m a n n distance f r o m
ARKIV FSR MATEMATIK. Bd 6 nr 14
/(to(/)+...+t~(/)) to / ( t o ( / ) + . . . + 4 ( / ) + t )
is g r e a t e r t h a n or equM to 1 (if such a t exists, tn+l(/) is well-defined b y r i g h t conti- nuity). Clearly tl(/) + ... + Q(f) is a stopping time for each n.
I f U(/) is finite, t h e n all the 4(/) are defined a n d their s u m converges. I n a n y case, each t~ is defined with I-L. u o p - x p r o b a b i l i t y one. The r a n d o m variables t~ for n ) 1 are i n d e p e n d e n t b y the strong M a r k o v p r o p e r t y and h a v e t h e same p r o b a b i l i t y distribution on the n o n n e g a t i v e real axis. E a c h is almost surely strictly positive b y r i g h t continuity. Thus t h e p r o b a b i l i t y of their s u m converging is zero, q.e.d.
I t follows t h a t u n d e r our assumptions the measures Pt(P, ) are all p r o b a b i l i t y measures. W e summarize our results as follows:
Theorem 8.2. There is a natural 1 - 1 correspondence between sets {P~} o/ measures on ,.4 satis/ying (A)-(E) o / w 4, and in/initely divisible radial probability measures P1 on ~ , and hence with pairs (L, ~) as described earlier in this section. Each P~ gives outer measure 1 to the set o//unctions / E ,,4 having left and right derivatives at all points (except le/t derivatives at 0)~ equal" except at countably many points, with right deriva- tives continuous/rom the right and le/t derivatives/rom the le/t.
9. F u r t h e r d i s c u s s i o n o f radial s e m i g r o u p s o n ~/
I n this section we discuss f u r t h e r facts which were n o t needed for the proof of T h e o r e m 8.2, b u t m a y well be of interest.
Suppose /~ is an infinitely divisible radial p r o b a b i l i t y measure on ~ , a n d {YIz. v}
t h e corresponding M a r k o v process on ~0. I t follows f r o m results of Gangolli [3] t h a t if n ( M , / , t) for M, t >~ 0 a n d f = <r ~fl> E ~0 is t h e n u m b e r of values of ~ < t such t h a t the R i e m a n n distance f r o m r ) to r + ) is at least M, t h e n
/ n( M , /, t) d~-[z, u(/) = tL(/u) ( U: ~(U) ~> M).
Thus if L = 0, ~ > 0, ~ is almost surely continuous. These processes ( " B r o w n i a n m o t i o n s " ) will be considered in w 10.
On the o t h e r hand, we have so-cal]ed " P o i s s o n " processes defined as follows: let /~, be normalized Kp-invariant "surface a r e a " measure on the sphere ~ = (r in 7~, for a n y a > 0. T h e n the "Poisson m e a s u r e " 7e((~, c) is defined b y
~((~, c) = e - ~ ~ ~ (ct~.) ~ / n ! ,
n = O
where t h e powers represent convolutions. I t is easy to v e r i f y t h a t Pt ~ ~(a, ct) gives a convolution semigroup. The associated process in ~ will r e m a i n at one p o i n t for some time, t h e n j u m p to one of the points at R i e m a n n distance a, with uniform probability distribution over the sphere of such points. T h e p r o b a b i l i t y of such a j u m p during an interval t o < t < t o + s is 1 - e -~c (see L o i r e [1] p. 548), which is as.
y m p t o t i c to ce as e ~ 0.
A Poisson process seems v e r y n a t u r a l as a velocity process, where a particle un.
dergoes collisions at various times which cause discontinuities in its velocity, c o n s t a n t between collisions.
R. M. D U D L E Y ,
Lorentz-invariant Markov processes in relativistic phase space
I f I is t h e i d e n t i t y o p e r a t o r on b o u n d e d c o n t i n u o u s f u n c t i o n s on ~/, t h e n l i m
(Pt
-I ) / t
tr
will be, in general, a n u n b o u n d e d o p e r a t o r d e f i n e d on a subspace, called t h e " g e n e - r a t o r " of t h e s e m i g r o u p
Pt.
T r a n s f e r r i n g o u r s e m i g r o u p s f r o m " t / t o s in a r a t h e r o b v i o u s way, we o b t a i n i n f o r m a t i o n on t h e i r g e n e r a t o r s from H u n t [1] (see also W e h n [1]).Since a n y r a d i a l m e a s u r e # on "U is a n a v e r a g e
tt = f#.dv((~)
for s o m e finite m e a s u r e v o n R +, we c a n e x p l i c i t l y c a l c u l a t e a n y c o n v o l u t i o n of r a d i a l m e a s u r e s if we can f i n d #a-)6 ~/b for a, b > 0. I n d o i n g this, we c a n a s s u m e b < a . S u p p o s e g i v e n c~> 0; l e t us f i n d
Clearly
/(a, b, c) = (tta ~e lib) (o <~ c).
/(a,b,c)=O
ifc < a - b ,
= 1 if
c > a + b .
F o r a n y u C ~ / a n d t > 0 l e t
S(u, t)
be t h e set of p o i n t s of ~ / a t R i e m a n n d i s t a n c e t f r o m u. L e tq = (sinh
a, O,
O, cosh a);t h e n 0 ( q ) = a. Also La E I~ t a k e s p = (0, 0, 0, 1) i n t o q, w h e r e
La(x, y, z, t) = (x
cosh a + t sinha, y, z, t
cosh a + x sinh a).S(p, b)
is t h e set of(X, y, z, T) E "tl
w i t h T = cosh b a n d X , + y2 + z 2 = sinh ~ b.T h u s
S(q, b)
is t h e set of p o i n t s(X cosh a + cosh b sinh a, y, z, cosh b cosh a + X sinh a)
w i t h X~ + y2 + z 2 = sinh2b. T h e n
S(p, c) 0 S(q, b)
is t h e set of (x, y, z, t) w i t h t = eosh c, x = X cosh a + eosh b sinh a w h e r e X = (cosh c - cosh b cosh a ) / s i n h a, a n d ya + z 2 = sinh2b - X 2. A p p l y i n g L - ~ we o b t a i n a circle of radius sinh~b - X 2 inS(p,
b), a sphere of r a d i u s sinh b, in t h e g i v e n c o o r d i n a t e s y s t e m for M o. W e n o w w a n t to f i n d t h e a r e a s of t h e t w o zones d e m a r c a t e d b y t h i s circle.F o r c small e n o u g h so t h a t cosh c ~< cosh b cosh a, t h e a r e a of t h e z o n e w h i c h con- cerns us is
2z~ sinh b (sinh b - C~ b c~ a--- e~ c)
aF o r cosh c > cosh b cosh a we w a n t t h e a r e a of t h e l a r g e r of t h e t w o zones m a r k e d off b y t h e circle, w h i c h is g i v e n b y t h e s a m e f o r m u l a . T h e t o t a l a r e a of t h e sphere b e i n g 4 z sinh2b, we o b t a i n
ARKIV FOR MATEMATIK. B d 6 n r 1 4
cosh c - cosh b cosh a/
/(a, b, c) = 89 1 + s i ~ bsisinhaa ] ' d/(a, b, c)/dc = sinh c/2 sinh a sinh b.
Proposition 9.1. F o r a n y a, b, ~ 0,
(a+b sinh x
~la-)~- /~b = ira ~1 2 sinh-as]sinh 5 ttxdx"
W e now t u r n to p r o v i n g " t r a n s i e n c e " or " n o n - r e c u r r e n c e " of our i n v a r i a n t pro- cesses in ~ . Proposition 9.1 will n o t be used.
Theorem 9.2. For any invariant Markov process {I/z. 9} as described in w 6, with l~z, v not concentrated in one/unction, and any compact subset K of "H,
I~z,v{ <r ~p> : y~(v) ~ K / o r all su//iciently large v} = 1.
W e first prove t h e following
L e m m a 9.3. For any radial probability measure tt on ~ not concentrated at p, 0 </2(0) < 1 a n d / o r any R >~ 0 there is a K > 0 such that
# " ( e < R) < I;[/2(0)]".
Proo/. Since tt is n o t c o n c e n t r a t e d at p,
o</2(o)= fu2hed~ < l.
T h e n
fsin Qh- ----e "= (o)= [/2(0)]".
T h u s given R ~> O,
sinh R
# " ( e ~< R) < ~ R ~ [/2(0)]", q.e.d.
N o w to prove T h e o r e m 9.2, suppose for some z, U, a n d c o m p a c t K c ~ , PK = l'-[z,U( <r y~} E "~ : y~(t) e K for t arbitrarily large) > 0.
F o r k = l, 2 . . . / = <~, y~> E ~q, let nk(/) be t h e kth integer n such t h a t yJ(t) E K f o r some t in t h e interval [n, n + 1), or + ~ if there is no such n; let tk(/) be the least such t, or + ~ if nk = + ~ . T h e n each tk is a stopping time.
L e t d be t h e R i e m a n n distance in ~ . There is a n R > 0 such t h a t I~,u(<r y~> : d(yJ(t), y~(0)) ~< R, 0 ~< t ~< 1) = e > 0.
e is clearly i n d e p e n d e n t of z a n d U. L e t Ak be t h e set of all / E ~ such t h a t nk(/) a n d tk(/) are finite, a n d let C~ be the set of <r yJ} E Ak such t h a t
d(~p(tk), ~p(tk + t)) <~R, 0 < t <~ 1.
m M. DUDLEY, Lorentz-invariant Markov processes in relativistic phase space Then b y the the strong Markov p r o p e r t y of {l~.v}
1-L. ~(GI B~) = almost everywhere on Ak. Thus
1-[~.v(Ck)=sl-Iz.v(Ak)>~ePs,>O for k = l , 2 . . . Thus 1I~. v{/: / E C~ for infinitely m a n y k} >~ SPK.
Hence if K a is the set of U E ~ such t h a t d(U, V) <~R for some V E K , H , . v { ( r v2): v/(n~ + 1) E Ka for infinitely m a n y k} > 0.
This contradicts L e m m a 9.3, so the proof of Theorem 9.2 is complete.
10. Diffusion processes in ~ and M x ~/
A convolution semigroup Pt in a symmetric space, e.g. in ~/, is called a diffusion semigroup if
lim Pt(~ >~ t)/t = O,
t~O
where 0 is the Riemannian distance from the fixed point used in defining convolu- tion. The theory of such semigroups tells us t h a t the generator A is defined at least on all functions which, together with their first- and second-order partial derivatives, are bounded and uniformly continuous on ~ , and is equal on these functions to a second-order differential operator (Yosida [1]).
On ~ , or other "two-point homogeneous" symmetric spaces G / K , G-invariance then implies t h a t A = mA + b where A is the L a p l a c e - B e l t r a m i operator. B y defini- tion of A it is zero on constants and A/~< 0 where / has a relative m a x i m u m , so t h a t b = 0 and m~>0.
Thus we have the "Brownian m o t i o n " semigroup with p a r a m e t e r m as defined b y Yosida [1]. On ~/, the Pt for this semigroup are given explicitly b y Tutubalin [1] as
Pt
= ( 4 3 z m t ) - 3 / 2 Q exp -- mt -- 9 N ,sinh ~ 4mr
where N is the s measure on ~ given by 4~(sinh2~) d~ d• and ~ is normal- ized orthogon~lly invariant surface area m e a s u r e on a sphere (ds = cos dpdOdr I n this case the measures P~ are concentrated in the set of functions / h a v i n g continuous first derivatives tiff), since the Pt are a diffusion semigroup (Yosida [1], D y n k i n [1] Theorem 6.5).
Now, the space M • ~ / i s a homogeneous space under the action of the "Poincar6 group" s generated b y s (acting on M and ~ / t o g e t h e r ) and b y translations of M which leave ~ / p o i n t w i s e fixed. The subgroup of s leaving the point (0, p) of M • fixed remains equal to K~, so t h a t M • can be regarded as the space s of right cosets of K v i n / : .
ARKIV FOR MATEMATIK. B d 6 n r 14 One can define a convolution for finite, (left) K f i n v a r i a n t measures on M • "U, as before, b y
(# ~ev) ( A ) = l" M
I
A E B ( M • "U), where Tz E F~ takes z into (0, p) a n d / ~ a n d v are i n v a r i a n t u n d e r Kp.
Our conditions (I) a n d (II) of w 6 can be r e g a r d e d as conditions of ~ - i n v a r i a n c e of transition probabilities
Qt((x, p) ; B) = YIx.,(/ : /(t) E B), t >~ 0, x E M , p E "1l, B E B(M • ~ ) . Our assumptions imply, letting
Q, = R,((0, p) ; ),
t h a t Qs % Qt = Qs+t on M • ~ . If Pt is a diffusion semigroup on ~ w i t h g e n e r a t o r mA, the g e n e r a t o r of Qt is the differential o p e r a t o r whose value at the point (x, U) of M • ~ is - U + M A , where mA acts on ~ a n d U acts as a first-order differential o p e r a t o r on M a t x.
G. S c h a y [1] also studied ~ - i n v a r i a n t diffusion processes in M • M0, a n d arrived at essentially the same g e n e r a t o r or "diffusion e q u a t i o n " (Schay [1] T h e o r e m 4 p. 39, E q u a t i o n 3.33). Our use of t h e p r o p e r time permits a considerable simplification of t h e result.
1 1 . N o n - e x i s t e n c e r e s u l t s
I n w 8 a b o v e we p r o v e d the existence of a class of L o r e n t z - i n v a r i a n t processes with speeds less t h a n 1 in M • V. I n this section we show t h a t such processes with speeds equal to 1 (in M • S 2, where S 2 is t h e sphere ] x ] = 1 in R a) are deterministic (trivial), a n d t h a t this remains essentially true if we allow stutes to be specified b y m o m e n t a r a t h e r t h a n velocities only. W e also prove the non-existence of i n v a r i a n t processes in M itself. The results in this section do n o t require the M a r k o v p r o p e r t y or c o n t i n u i t y in probability.
On a t r a j e c t o r y with speed almost everywhere equal to 1 the proper time ~ is a c o n s t a n t a n d hence n o t suitable as a p a r a m e t e r .
L e t ~ he the set of all functions / f r o m R + to X such t h a t for 0 <~ s <~ t,
I/(t) -/(s)
I < t - s, a n d i f / ' ( t ) is defined, I]'(t)] = 1.N o w suppose given starting probabilities Px" on the stationary, measurable p a t h space ( ~ , 1, X , B ( X ) ) , satisfying (A), (B) a n d (D) of w 4, with V replaced b y S ~.
A collection {Px'} will be called deterministic if each p v gives mass 1 to a set con- taining o n l y one function.
F o r a n y / E ~ w i t h / ' n o n - c o n s t a n t there is a u n i q u e t = to(]) > 0 such t h a t t 2
-/(t)
2 = 1(i.e., (/(t), t) E ~ ) . Clearly to( ) is B0(O)-measurable.
