A Holder condition for Brownian local time

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1 J. Math. Kyoto Univ. 1- (196) A Holder condition for Brownian local time By H. P. MCKEAN, JR.' (Communicated by Prof. K. Ito, November, 1961) Given a standard Brownian motion on l e beginning at, H. Trotter [ 3 ] proved the (simultaneous) existence of the local times: 1. t(t, a) = u m measure (s: a < x(s) <b, s <t) b+a b a t >, ae R ' and derived the law a. p r l i m I t(t, b) t(t, a)i 1 = 1. p [ u rn. \ / lg118 I give simple proofs leading to the sharper bound b. p r /. t(t, b) t(t, a)i < \/max t(t, )1 1 ji r = N/8 1g118 b is proved assuming t exists and is continuous in space ; afterwards, I go back and prove the latter statement. H. Tanaka's (unpublished) expression for the local time as a stochastic integral : l t ( t, a) = max [x(t) a, ] max [ a, O] x ( d s ) ] 1 3. P [ s x (s )> a and the bound' 4a. E [ea(t)]< 1 for the functional The support of the ONR, U.S. Govt. is gratefully acknowledged. See, for example, E. B. Dynkin

2 196 H. P. McKean, Jr. 4b. a(t) r o f [x(s)]x(ds) 1 r o f[x(s)]ds 4e.P [ s : J - T x ( s ) ] c ls < + D o i are the basic tools for this. I want to thank H. Tanaka for communicating his integral 3 and for a helpful conversation about the sample path f of 17 below. Tanaka's (unpublished) proof of 3 is as follows. Bringing in the indicator e a b o f th e interval (a, bea<b), an application of the formula for stochastic _differential gives 5. 1 measure (s : a < x (s )< b, s < t) = - r o e a b [x(s)]cls = ēa b [x (t)]- ēab[x()] ro ea b[x(s)]x(ds) with e <a 6a. e.a ea b c17) =.( a < < b n<1., b a > b 6b. -eab() Pa b dn --- n t < a ( a) a < < b (b a)( a + 11 ) > b, and, using 7a. and lim(b a) - 1 b() = max [ a, ] Ir j, 7b. E f t ( eab i \b a _ E rft( e a t, _ e a c., ) e c o )x (ds) a I- i \b a d s ]

3 A Holder condition for Brownian local time 197 Er e a b (b x(s)y d s ] b a )._ <E [ e a b ds] < constant (b a), 3 is immediate on letting b l a in 5, assuming, as I now do, the existence of the local time t(t, a). Given positive numbers a and /3, points a < b, an d putting b a=8, an application of 4 a gives 8a. P [ eabx(ds)> [a+13 max t(t, )] N/8 1g118] R i < P N e a b x (d s)>.\/ 8 ) C id i 9 /(/) lg118 = P [ e a b x(ds) e bd s > a -V8 / g ild J o a <E[e7s1,eabx(d.3) A e b d s]e -7 N/6 1g VS < e - cosig vs = and since the same bound applies to e a b x (ds) as well, 5t 8. p [ e a b x(ds) > [a+ re max t(t, )] N/8 lg118] < 8 6 g, leading at once to Pr 9. max - b= j " <i j< " lal < d < E (k - n)ai 3 < k < " lai <d < 4d- n [ - - e ) o f t e ] e a b x(ds) / 8 g la >ad-re max t(t, ) which is the general term of a convergent sum provided d=1,,

4 198 H. P. McKean, Jr. 3, etc. is fixed, ar >1, and &> is so small that (1-8)cei3-1-&<. Tanaka's integral ( =3), the Borel-Cantelli lemma, and the fact that max [ b - a, ] is piecewise smooth can now be combined with 9 to establish 1. P - li m a = i ", b= j " - - <k= j i<"! b a=& $ lal <d it(t, b)-t(t, a)i < a + R m a x t ( t, ) = 1 N/8 1g118 le, for each choice o f d>1, a i e> 1, and < 6 < a 4-1 ar +1 But now, taking into account the fact that t(t, a) is continuous in space, it is plain sailing over the course laid out by P. Lévy [] for the proof of 11. p1 I x (, ) _ x ( s), < i i 1 1 -t-s= s 1./8 1g 1/8 o<s< t< 1 to deduce from 1 1. p ri i m b ) - t ( t, a) I-lb-al-a o V 8 /g 1/8 la <d < a + R max t(t, )] = 1 for each d > 1 and ar >1, and b follows on letting d + (use near ± ), letting a g 1, and making a+ max t as small as possible subject to c = 1. I n o w g o back and prove that t exists and is continuous. Beginning with the stochastic integrals e.x(ds) e(a)(a E RI), the trick is to prove, as I now do, that e can be modified so as to be continuous in space. Because 13. P r max eabds > n - n] a = (k 1) - " b= k - " la <d <d"1 3 [ : e - d s > n - "] _<d(1e)"e [ex p(" e _ ds)] R i _

5 A Holder condition for Brownian local time _<d(1e)net c 8 ele[ex p ( " e ds)] +- = d(1e)net c e d [1 + + n i do, do, ndb, ndb, e e -- (b - 1, 1) 1 ( - 1 ) C ( b 1-1, 1-1 ) / ( ) V 71-1 N / 7 T ( 1) V 7r(,-,,) <d(1e)nef = d(1e)net c 'e l [1 + E 1 convoluted with 11\/7r 1 times] i=i = d(1e)net \ / V 1 is the general term of a convergent sum a n d ea b d s is monotone in a and h, one finds Ç e d s 14. P [ urn b lb al-51 8 lg 1/8 i a < + and so, using the obvious bound 15a. P h t t ea b x (ds)> ce+ 1, o f t ) o ea b x(ds) 16. P lim a=i - - ", b= j " 8 1g1 IS - lb-al=s io ea b ds] = P p 5:eabx(ds)_1 5t eabds > a d < e - `4 3 with cev 81g118 and 3 IV 8 (a19> 1 ) in place of a and IS to obtain 15b. P [ e a b x (ds) > c c 81g118+ e d s b V Jo < 8cdo a = lb al, it follows as in the proof of b above that

6 H. P. McKean, Jr. But this means that the modified sample path 17. f(a)u r n e ( b ) lim eb,.a(ds) a E Rl b=k " a b = k - - a o is continuous ; in addition, 18. P h f(c)dc e coo d c)x (d sd = 1 a < b a a because P[f(a)----e(a)] - =-1 (a E R '), and since measure (s : a < a b, x(s) s< t) e b d s, and fd c are all continuous in a and b, an application of 5 gives 19. P[-1 measure (s : a < x(s) b, s < t) = a b [ X ( t ) ] - - a b ( ) f leading at once to the fact that d c, a < b i= 1,. 1 t(t, a) max [x(t) a, ] max [ a, ] f(a) exists and is continuous, as was to be proved. A second application of the above method gives the bound' 1. P [ r e o bx (ds) > [a + m a x t(t, a)] \/ 1 g 118] O ct b <(1g 118), - leading at once to. P L 1 im 1t 8 (t, ) t(t o N / 8 1 g 1 ' )1 /8 < 4 ' )] 1. Given t > and a E, the conditional local time [t(t, b):b E R, 1 P( /x(t) a)] is a diffusion, and, expressing it in terms of a standard Brownian motion (via a change of scale and a time substitution), it is immediate that the bounds b and are best possible ; this beautiful result will appear in a forthcoming paper by D. B. Ray. Massachustetts Institute o f Technology November g c =lg ( lg c ).

7 A Holder condition for Brownian local time 1 REFERENCES E. B. Dynkin : Additive functionals of a Wiener process determined by stochastic integrals. Teor. Veroyatnost. i ee Primenen. 5, (196). P. Lévy : Théorie de l'addition des variables aléatoires. Paris H. Trotter : A property o f Brownian motion paths. Ill. J. Math., (1958).

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