498 V. E. Benes, L. A. Shepp Теорема 4. Пусть на гладком многообразии Е класса С2 задан невырожденный эллиптический оператор L, причем коэффициенты диффузии непрерывны, а коэффи­ циенты переноса и убивания измеримы и локально ограничены. Тогда существует диффузионный Т-процесс на Е с производящим оператором L. Поступила в редакцию 27.2.68 ЛИТЕРАТУРА [1] И. В. Г и р е а н о в , Сильно-феллеровские процессы, I, Общие свойства, Теория вероят. и ее примен., V, 3 (I960), 314—330. [2] Ph. C o u r r e g e , P. P r i o u r e t , Recollements de processus de Markov. C. R. Acad. Sci., Paris, 259 (1964), 4220—4222. [3] X. T a n а с a, Existance of diffusions with continious coefficients, Mem. Fac. Sci. Kyashu Univ., A( 18), 1964, 89—103. [4] H. В. К р ы л о в , О квазидиффузионных процессах, Теория вероят. и ее примен., XI, 3 (1966), 424-443. [5] Е. Б. Д ы н к и н , Марковские процессы, М., Физматгиз, 1963. [6] D. R a y , Resolvents, transitions functions and strongly markovian processes, Ann. Math., ser. 2, 70, 1 (1959), 43—72. [7] M. Г. Ш у р , Локализация понятия эксцессивной функции, связанной с марков­ ским процессом, Теория вероят. и ее примен., VII, 2 (1962), 191—196. [8] Б. В. Д ы н к и н , Основания теории марковских процессов, М., Физматгиз, 1959. [9] Д е Р а м , Дифференцируемые многообразия, М., ИЛ, 1956. [10] А. Л. Р о з е н т а л ь, Броуновское движение с отражением и задача с наклонной производной, Кандидатская диссертация, МГУ, М., 1967. STRONGLY-FELLER'S DIFFUSION PROCESSES ON SMOOTH MANIFOLDS S. A. MOLCANOV (MOSCOW) (Summary) We prove that the coalescence of two regular strongly-Feller's processes is a pro­ cess of the same type. Using this result, we construct non-degenerate strongly-Feller's diffusion processes on smooth manifolds. Some of the main results of this paper have been formulated by I. V. Girsanov in his article [1]. WIENER INTEGRALS ASSOCIATED WITH DIFFUSION PROCESSES V. E. BENES, X. A. SHEPP I. Statement of results. Let F be a functional on the space Q of real continuous functions x — x(t), 0 sg: t ^ T. Let EWF denote the Wiener integral of F, defined as J F(x)d\iw(x), with \iw the Wiener measure on Q determined by the condition: x(A 'a is Gaussian with mean zero and covariance min (s, t). Also let Y be the stochastic pro­ cess defined by the stochastic differential equation Y = a (Y) + W, 0 г£ t ^ T, with W the Wiener process, and let Цу be the measure induced on Q by the Y-process. Now Kac Wiener integrals associated with diffusion processes 499 Г [1] has given a formula for EWF in case p (x) = exp I — у \ V (x (t)) dt\, with F(-) о bounded, and Prokhorov [2] has shown that |x r ~ |iw with T T ^ — (ж) = exp | . о (x (t)) dx (t) — "2 \ «2 (» (0) <"} , (1) о о whenever a(-) e Lipi. (See also Skorokhod [3]). We show that Kac's formula can be obtained from Prokhorov's theorem. 2. Evaluating Wiener integrals by Newton's technique. A familiar technique for ь evaluating [ fdt is to find a function g with g' = f, so that £ fdt == g(b) — a a — g- (a). In our context, to calculate, J i'djiw we find a measure v on Q with - — = pt i.e. such that the functional F is the Radon — Nikodym derivative of v with respect to \iw T h e n E w . F = v(Q). So far only two classes of Wiener integrals have been calculated or at least reduced to more familiar problems: т (a) F{x) = e x p j — у \ f{t)x*(t)dt\ with / continuous [4]; о T (b) F (ж) = е х р | — у \ V(x(t))dt\ w i t h 7 ( - ) bounded [1] The above technique can be used to find EWF in these cases. Numerous proofs of the formulas thus obtained are known, but we feel that Newton's technique provides a simple unified approach which is far from being exhausted. Case (a) has been treated in pre­ vious work of Shepp [4]; the desired measure v corresponds to a scale change in the Wiener process. In case (b) the desired measures v correspond to diffusion processes. (See Dynkin [5], vol. I, p. 11.) Use of Newton's technique in describing stochastic proces­ ses has been made by Skorokhod [9]. 3. Kac's formula. For case (b) Kac [1] has given the following procedure for fin00 ding EWF: EWF = 00 \ q(T,y)dy where ф (я, у) = \ e~s'g (t, y) dt is determined by 0 —00 the Schrodinger equation Vtf|>», = (VsV + * ) * . 0 < | j / | < oo, s > 0; i M * , 0 - ) - * v ( « , 0+) = 2. (2) 4. Diffusion. We consider a diffusion process Y(-) defined as the solution of the stochastic differential equation [6] Y(t)=a(Y(t))+W(t), OsStsST, in which the function a(-) describes the local drift, and W(l) is the Wiener process. We restrict attention to the case Y(Q) = 0 , a(-) e Lipi. Here Picard's theorem implies that for any continuous function W there is a unique У satisfying т Y (t)=\a{Y{u))du + W (t), 0<£<r, 0 the process Y(-) is a Markov process, and p(t, y) = ¥{Y(t) = y} satisfies the forward equation [7] pf. + (w)v = ЧгРяу, with p(0, y) = 6(y). т 5. Proof of Kac's formula from Prokhorov's theorem. T h e integral J a(x(t))dx(t) о 8* 500 V. E. Benes, L. A. Shepp n occurring in (1) is the «left-handed» stochastic integral [7] defined as lim 2 e ( s j - i ) • {XJ — ж,-_0, with ж,- = x(t}), 0 ^ h < t2 < ... < tn ^ T. It can be expressed in terms of Lebesgue integrals by a method of ltd [8, Appendix 2]: г т [a(x (J)) efo (i) = A(x (J1)) — 7з С a' (x (t)) dt, (3) о о z where 4(z) = J a(a)du. We make the assumption (#) that F(-) is sucb that there is о a sulution a (•) of the Riccati equation a' + a2 = F (4) with the property: a ( - ) e L i p i . (This supposition will be justified presently.) Using this function a(-) to define the diffusion process Y(-), we find from (1), (3) and (4) that a(z)dz~1k\!V(x{t))dtY о ^-=exp{ J о d\iY F (*) = -Щ^ daw (*) exp [ - A (x (T))\. Integrating with respect to Wiener measure, we obtain EWF = J «"* <ж <T» du. y (*) = j e - A («) /> (Г, у) A,. To see that our formula (5) is equivalent to Kac's (2), set g(t, y) — p(t, Then '.'Va?*'» = lh(Pve~A—ape~A)v (5) —oo Я = l/ie~A{pvv y)e~A^>. — 2apy — a'p + a2p) = j< + Va^S. ' (6) by the forward equation and (4), Taking transforms gives ; *ftfyy = ,$ + */аУф, ;2/=^0, and the condition on t|),, follows from (6) by integration over (—8, e): £ , , £ gy(t,&) — gv(t,— 8 ) ' = 2 \j ? t ( i , z ) d z + jj F (z) 9 (t, z) dz, —e , —Is £ £ % (s, 6) — ^ y ( s , - 8) = 2 ^ [si|) (s, z) — 6 (г)] dz + § 7 (г) ф(*, z) dz, —г —e . iM»,.0-)-iM«,'0+) = 2 . The exact conditions on V(-) such that assumption (*) holds are apparently not simple. But for V(-) bounded and positive the existence of «(•) such that a + a2 = F and a ( ; ) e Lipi, is ensured by the following Lemma. / / 0 г^ F ^ к2, then the equation d-j-a2 = F has a solution «(•) with 6 < а (з) «S к. P r o o f . Standard existence and uniqueness theorems guarantee that through any point (x, a(x)) there passes exactly one trajectory a(-) satisfying a + a2 = F. Consider the trajectories a(-) with 0 ^ e(0) ^ к. It is easy to see that these are bounded for x > 0, never leaving the region R = [0, к]. Assume then that every trajectory a (•) with a(0) Е Й has either а (ж) < 0 for some x < 0 or a(y) > к for some г / ' < 0 ; it is not possible that both properties accrue to a (•), because а, г£ О at the boundary к, and a ^ ^ 0 at the boundary 0, so once a trajectory is in R, it stays there. We will derive a contradiction and so prove the lemma. Интегралы Винера, связанные с диффузионными процессами 501 Case (1): For some хй < 0; х <. х0 implies V(x) = 0. Then the solution a(-) with a(x) = 0 for x < Xo lies in Й. Case (2): For some x0 < 0, ж < ж0 implies V(x) == A2. Then the solution a(-) with a (x) = ft lies in Й. Case (3): F ( i „ ) < ft2 for some sequence ж„ < 0, ж„-^оо and F(j/„) > 0 for some sequence yn < 0, г/„ -*- — oo. For a trajectory a(-) with a(0) е Л , define t« = inf {t : a(i) е й } , X = sup {a(0) : a(ta) = 0, e(0) 6 « ) . Let a(-) be the1 solution with a(0) = X. Either a(ra) = 0 or a(t a ) = ft. In the latter case find x < xa with V(x) < ft2. The trajectory b(-) with 6 (ж) = ft satisfies 6(») < < a(u) and 6(0) < a(0) = X, so that a(-) crosses &(•)• This is impossible. In the for­ mer case find у < xa with V{y) > 0. The trajectory c(-) with c(y) = 0 satisfies c ( a ) > >a(u) and c(0) > a(0) = X, so that a(-) crosses c(-). This contradicts the definition of X and shows that the original supposition that every trajectory a(•) with о(0) е й be outside R for some x is false. Hence there is a trajectory lying entirely in R. It is obvious that if a(-), V(-) are bounded and d + a?- = F, then a(-) e Lipi. BeH Telephone Laboratories, Incorporated Murray Hill, New Jersey Поступила в редакцию 25.1.67 REFERENCES [1] M. K a c , On some connections between probability theory and differential and in­ tegral equations, Proc. 2nd Berkeley Sympos., Math. Statist, and Prob., 1951, 189— 215. (Русский перевод: М. Кац, О некоторых связях между теорией вероятностей, дифференциальными и интегральными уравнениями, Математика (сб. перево­ д о в ) , ! (2) 1957,93—124), [2] Ю. В. П р о х о р о в , Сходимость случайных процессов и предельные теоремы тео­ рии вероятностей, Теория вероят. и ее примен., I, 2 (1956), 177—238. [3] А. В. С к о р о х о д , О существовании и единственности решений стохастических дифференциальных уравнений, Сиб. матем. ж., 2, 1 (1961), 129—137. [4] L. A. S h e p p, Radon — Nikodym derivatives of Gaussian measures, Ann. Math. Stat., 37 (1966), 321-354. [5] E. В. Д ы н к и н . Марковские процессы, М., Фйзматгиз, 1963. [6] К. 11 б, Stationary random distributions, Mem. Fac. Sci. Kyoto Univ., Ser. A 28 (1953), 209—223. [7] Дж. Л. Д у б, Вероятностные процессы, М., ИЛ, 1956. [8] К. I t б, On stochastic differential equations, Mem. Amer. Math. Soc, 4 (1954). [9] А. В. С к о р о х о д . Конструктивные методы задания случайных процессов, Успе­ хи матем. наук, 20, 3 (1965), 67—87. ИНТЕГРАЛЫ ВИНЕРА, СВЯЗАННЫЕ С ДИФФУЗИОННЫМИ ПРОЦЕССАМИ В. Е. БЕНЕШ, Л. А. ШЕИ И (НЬЮ-ДЖЕРСИ, США) (Резюме) Для V(-) и «(•)> связанных уравнением Риккати а + а2 = V, из формулы Про­ хорова т т exp J a (x (t)) dx (t) — -. . a2(x(t))dt\ о о для производной Радона — Никодима (по мере Винера) меры, соответствующей диф­ фузионному процессу Y(Y = a{Y) + W, где W — винеровский процесс), выводится метод Каца вычисления интеграла Винера от функционала т expj—1J -. о V(x'(t))dt}. • • , . - • " . .