some recent trends in the mathematical theory of diffusion

7/28/2019 SOME RECENT TRENDS IN THE MATHEMATICAL THEORY OF DIFFUSION

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SOME RECENT TRENDS IN THE MATHEMATICAL THEORYOF DIFFUSIONWILLIAM FELLER , ,

1. Introduction. This lecture may seem to sail under a false flag since itappears under the general heading of Applied Mathematics. Only a small partof the mathematical diffusion theory is connected with immediate applications,and much of the recent literature on the subject has an abstract character.Th e situation is here as in the theory of harmonic functions which likewise stemsfrom practical problems and has many applications, but which has spread intoand influenced many fields: complex variable theory, boundary value problemsfor general differential equations, subharmonic functions, generalized potentials,etc.Similarly, the diffusion theory has outgrown its origin as a special topic inpartial differential equations. Nevertheless, not even this particular chapter isclosed: recent applications to the theory of genetic evolution lead us to singulardiffusion equations and confront us with a new type of boundary problems.We find in this connection unsolved problems, but of even greater interest isthe fact that in diffusion theory we find for the first time an intimate interplaybetween differential equations and m easure theory in function spaces. The latte rthrows new light even on the classical parts of diffusion theory; it leads to newtypes of solutions, and opens a new avenue of attack on problems connectedwith boundary conditions which were left open for a long time. It turns out thatthe adjoint of a differential equation is not necessarily itself a differential equation, and this new information is of value for the general theory of semigroupsof operators. This is underlined by the fact that the probability approach leadsus to consider diffusion equations which are not of local characte r and whichcan be treated by the methods developed, under different forms, by Bochner,M. Riesz, and L. Schwartz. In this way the theory of differential equations istied in with other theories and we shall find unexpected connections betweenharmonic functions and diffusion processes.It is the purpose of the present address to outline some of the new results andopen problems. It is believed that the new methods promise to be fruitful alsoin other fields, in particular for other types of differential equations. [Added inproof: Since this was written , new results were obtained and some of the problemsare no longer open; cf. [8].]

2. Classical diffusion and the Wiener space. For a first orientation considerth e classical diffusion (or heat-conduction) equation(2.1) ut = uxxover the interval oo < x < oo. Its fundamental solution (or Green function)is given by the normal density function

322


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MATHEMATICAL THEORY OF DIFFUSION 32 3(2.2) ^ u(y; t, x) = ^ p exp {-(x - yf/tt].Under very general conditions on f(x) the function

/-foo f(y)u(y; t, x) dy- o o

represents the unique solution of (2.1) which, for fixed t > 0, is integra ble w ithrespect to x and which tends to f(x) as t > 0 .Now equation (2.1) is supposed to be connected with Brownian movement,the description of which involves a function space. In fact, the position of aBrow nian particle is a function X(t), and "qbserving a par t icle" means observingX(t). Thus each X(t) represents a sample point in our experiment, and theproblem is to describe the properties of the possible paths X(t) in probabilisticterms. This was first done by N. Wiener [25]. He .starts from the assumptionthat (2.2) defines the transition probability density of our process, that is, theconditional probability density of the relation X(U + t) = x if it is known thatX(tQ) = y; it is furthermore assum ed tha t the increm ents AX(t) = X(t + h) X(t) over nonoverlapping time intervals are statistically independent (or thatthe corresponding probabilities multiply). Wiener (and by other methods P.Levy [20]) shows that with probability one the path function is continuous butof unbounded variation in every ^-interval; the set of ^-values for which X(t) = 0has the structure of a Cantor set ; with probabil i ty one X(t) satisfies a Lipschitzcon dition of order 1/2 bu t no Lipschitz condition of order 1/2 + e (more pr ecise results are given by the local law of the iterated logarithm).

Obviously statements of this type describe the hypothetic diffusion processmore directly than does the differential equation (2.1). I t is less obvious thateven the formal theory of (2.1) can profit from the function space approach.Consider, for example, the bo un dar y value prob lem for (2.1) for a region bou nde dby the par t x > a of the -axis and a curve x =


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3 2 4 ' WILLIAM FELLERcan be imposed. Perhaps the most famil iar boundary condit ion is that of anabsorbing barrier, which is usually defined formally by the condition u(t, XQ) = 0.I t must be emphasized that th is condit ion is obtained only heurist ical ly by apassage to the limit from a discrete random walk and is justified by the results.This could be considered satisfactory were it not that the heuristic methodcompletely breaks down in other cases. In fact, a perusal of the literature showsthat even in comparat ively simple physical problems the appropriate boundaryconditions remain unknown^ We shall see later on that in a certain case the absorbing barrier is completely described by the condition u(t, x0) < oo f but againthis was found only from the properties of the solutions to which this conditionleads. In other cases we shall see that no boundary conditions can be imposed.In short , the appropriate analyt ical formulat ion of var ious boundary condit ionsis an unsolved problem. Apparently it is unavoidable to revert to the originaldefinition of the various conditions, and this definition usually refers directlyto the path functions. Thus, diffusion with absorbing barriers can be describedby saying that the laws of free diffusion prevail in the interior but that the process stops when the particle for the first time reaches a boundary, that is, whenX(t) equals x0 of xi .

3 . The adjoint diffusion equations. Consider now a more general diffusionprocess in the infinite line with transition probability u(y; t, x). By th is we meanthat i f a t any t ime fa the position of the particle is X(tQ) = y, t h en(3.1) Pr {a < X(t + to) < b} =* f u(y; t, x)Ja dx.If the initial position X(0) of the particle is a random variable with probabilitydensi ty f(x), then the probabil i ty densi ty of X(t) is given by the integral (2.3).'A general stochastic process cannot be described solely in trmo f the ini t ia ldistribution and the transition probabilities. A diffusion process, however, is ft h e Markov type which means, very roughly, tha t the fu tu re depends s ta t i s t i cal ly on the present state , but hot on the past h istory which led t it .1 Thisimplies in par t icular the fundamental identi ty

Z +OO u(y) s, > ( ; t, x) d,ooknown as the Chapman-Koltnogrov equation. By v i r tue of the M arkov propertyt h e transition'probability u(y; t, x) determines all probability relation in thspace of path functions X(t). In particular, the joint probability densi ty of(X(ti), X(h), , X(Q) for 0 < h < h < < tn is given by ""(3.3) u(h , xi)u(xi ; h h , x2) u(xn-i ; tn tn-x , xn)3where u(t, x) is defined by (2.3). Thus the properties of the path functions X(t)are implicitly given by u(y; t, x), and could be derived exactly as Wiener ob-

1 Cf. the discussion in 8.


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MATHEMA TICAL THEORY OF DIFFUSIO N 325tained the properties of the paths in Brownian motion from (2.2). Actually,Wiener's results have been only partially generalized; for example, general conditions which guarantee continuity of almost all path functions seem to beunknown.Among all stochastic processes for which th e Chapman-Kolmogorov equation(3.2) holds, the diffusion processes are characterized by the requirement thatthe probability of a change exceeding e during a short time interval At is smallas compared to At; more precisely, we assume2 that(3.4) ~[ u(y;At,x)dx-*0lit J|a;-|/|>efor every fixed y.If at some time t the position is X(t) = y, then the mean and the variance ofthe displacement AX(t) during the following time interval of duration At are

/+0 0 -+00

(x y)u(y; At, x) dx and / (x y)2u(y; At, x) dx.00 J O OActually these integrals may diverge,3 and we introduce therefore the truncatedmoments(3.6) / (x - y)u(y; At, x) dx = a(At; y),

J | say | < e

(3.7) [ (x - yfu{y; At, x) dx = 2b(Ai; y) .J I2/| 0 isindependent of e, and the physical significance of these quantities is essentiallythe same as that of the moments (3.5), except that less weight is attributed tolarge displacements. It is therefore natural to suppose that the limits(3.8) a(,) = l i m a - ^ p > , b(y) = lim ^exist.2 Th is definition w as given in [5] where diffusion processes a re called Ma rko v p rocesse sof the "purely con tinu ous " typ e. It seems to be an open problem whether (3.4) implies th atthe path functions X(t) are continuous with probability one.3 An (unpublished) example is given hy the densitjr u(y; i, x) = I2wlf2lll2}~1'(x)e x p { - [(x) - 4> G/)]2/4*) where (x) = log1/4 (100 + x/(l -f e"*)). This is a solution of theequations (3.12)-(3.13) with b(x) = \'{x)}-\ a{x) = (1/2) b'{x).4 It will be seen from the following derivation (which follows [5]) that our assumptionsactually imply the existence of the partial derivative ut{y; t, x). Conversely, if it is assumedt h a t u is differentiable and that (3.4) holds, then it suffices to assume that on e of the ratiosoccurring in (3.8) has a finite point of accumulation as At 0. Lett ing then At approachzero throug h an ap prop riate sequence of values it follows from (3.11) th at for this pa rticu larsequence also the other limit in (3.8) exists, and this leads to (3.12). The first derivationof (3.12) is due to Kolmogorov [17] who, however, uses strong uniformity conditions andassumes the existence of the moments (3.5).


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326 WILLIAM FEL LERIt is then easy to< derive a differential equation for u(y; tj x) as a function of

t and y. From (3.2) and (3.4) we have < ''/

+0O u(y; At, Qu{\ *, a) c?v > . y ; < = /" i*(y; A, tt(ft t, x) d + o(A*), J.-I 0. If we now assumethat for fixed t, x the function u(; t, x) has two continuous derivatives withrespect to , then the Taylor formula leads to

u(y; t + At, x) = u(y; t, x) / u(y; At, ) dJ|?-2/l


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MATHEMATICAL THEORY OF DIFFUSION 3274. Singular equations. The theory of the preceding section applies also if . thpinfinite -axis is replaced by a finite or semi-infinite interval, except that no;wboundary conditions have to be specified. As has been mentioned before, iliederivation of the familiar boundary conditions (such as absorbing, reflecting,and e lastic barrier s) is essentially of a heuristic naturse, and it would be desirableto der ive them frpm the basic assumptipns in the same way a the, diffusionequation itself has been obtained. We want now to , disouss some, nqw aspectsof the bou nda ry problems for a typ e of equations which is of both theo retica l a ndpractica^ ( jnterest.In,physical applications the coefficient b(y) is essentially posit ive. Now modem diffusion theory has found a new and interesting field of applications in

biology, in, pa rticu lar in the m athe m atica l the ory of evolution (cf. [6]). He rethe particle uncier, diffusion is to be interpreted symbolically and stands for apopulation size, gene frequency, etc. For example, the frequency of a particulargene in a population is a random variable X(t) which by definition is restrictedto the interval 0 < x < 1, and Sewall Wright 's theory of evolution assumesessentially that X(t) is subject to a diffusion process with(4.1) b(x) = x(l - x), a(x) = -yx + 8(1 - x)where , y, a re positive constants. Similarly, in the simplest growth processX(t) stands for a population size and can assume all positive values; the underlying stochastic process is described by our diffusion equations with(4.2) b(x) = x, a(x) = ax + y.

The fact that b(x) vanishes on the boundaries represents a singularity andhas important and curious consequences.Consider, for example, the diffusion equations with the coefficient^ (4.2),which are discussed in [7]. If 7 __ 0, then there exists a unique solution so thjatno boundary conditions can be imposed. For this solution the probability ,

(4.3) / u(y; t,x) dx = ir(t,y)J o

decreases steadily in time. The difference 1 ir(t, y) represents the probabil i tythat a population of initial size y dies out before time t. (This, is th e abso rptionor extinction probability.)The situation changes radically when 0 < 7 < , and resembles the morefamiliar circumstances of ordinary diffusion. There exist infinitely many solutions, among which there is one for which ir(t, y) = 1; this corresponds to reflecting barriers. For all other solutions ir(t, y) decreases, and the solution forwhich the ra te of decrease is greatest obviously corresponds to absorbing barr iers .It is the only solution for which u(y; t, 0) < 00, so that in th is par t icular casean absorbing barrier is described by the boundary condition u < 00.For 7 > the situation changes once more. In passing to the limit 7 ,the absorbing and the reflecting barrier solutions become identical, and for


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328 WILLIAM FELL ER7 > io we are once more in the situation that no boundary conditions can beimposed and that our problem is completely determined by the differentialequation itself.7

Accordingly, we can say that for y _ 0 and y > the origin acts as naturalboundary where no artificial conditions can be imposed, while for 0 < y < we are confronted w ith cond itions similar to tho se in physical diffusion theo ry.It is possible (cf. [8]) to give criteria to determine whether for a particularsingular equation the boundary acts as a natural one or whether there existinfinitely many solutions, but we are still unable to formulate the appropriateboundary conditions in all cases. Moreover, the biological applications lead usto a new type of boundary problems.The classical concept of an elastic barrier at x = 0 m ay be interpreted bysaying that whenever the particle reaches x = 0 it has a prob ability p to beabsorbed and probabil i ty q = 1 p to be reflected. The limiting cases p = 0and p = 1 represen t reflecting an d absorbing bar riers , respectively. Now b iological problems compel us to consider the more general case where the particleis only temporarily absorbed; that is, if the particle is absorbed, it remainsfixed at the boundary for a finite time, and reverts then to the diffusion process.The sojourn t ime at the boundary is a random variable with an exponentialdistribution. In the familiar terminology of a probability mass spread over thex-axis th is means that in addit ion to the solut ion u(t, x) we have to consider a

finite mass m(t) concentrated at the origin. This mass flows at a rate proportional to m(t) back into the interval x > 0, bu t it is pa rtl y replaced by new massbeing absorbed at the origin. If the diffusion starts with all the initial massdistributed over x > 0, the n m(0) = 0, and one would expect that m(t) willincrease to a certain saturation value.Unfortunately the appropriate boundary condit ions have not been formulatedand the precise conditions on a(y) and b(y) under which such a process can takeplace are not kno wn. Fo r an interest ing special case which is of some im porta ncein mathematical genetics cf. [6].5, Existence problems. Pathological solutions. We now return to the case ofan infinite interval and th e tw o diffusion eq ua tion s (3.12) and (3.13). As wasmentioned a t th e end of 3, th e whole setup as well as th e analogy w ith t heclassical diffusion equation uy uxx leads one to assume that the transi t ionprobabil i ty u(y; t, x) will satisfy th two equations (3.12) and (3.13) and, conversely, that each of these equations should have a fundamental solution whichis essentially uniquely determined and can serve as u(y; t, x). However , a lreadythe f irst a t tempt to prove this conjecture showed that i t cannot be true under7 However, for y > there exist solutions for which ir(t, y) increases. These solutions

have no probability significance and correspond to the solutions of the theory of heat withheat flowing into the medium. The existence of analogous solutions for the whole axis wasdiscovered by Hille [13]; his " exp losive " solutions are of th is kind w ith ic(t, y) > > ast - > * o > 0 .


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MATHEMATICAL THEORY OF DIFFUSION 329all circumstances. In fact, it was shown in [5] that whenever one of the twointegrals dx5.1) f a~ll2(x)dx, f_ a~ll2(x)converges, an appropriate b(x) can be found such that (3.12) admits of a solution which is non-negative, integrable, and such that w(0, x) = 0. Thus in thiscase the equation (3.12) does not suffice to describe our process. However, assuming that both integrals (5.1) diverge, the construction of [5] shows that ourconjecture is valid under very mild regularity restrictions on a(x) and b(x).In other words, in case of divergence of (5.1) the whole classical theory carriesover to our diffusion equations.8 The problem was taken up by Yosida [26] andHille [11, 12] who obtained similar results unde r slightly different conditions.In particular, Hille showed that (3.13) has a unique, non-negative solutionu(i, x) satisfying the conditions

Z +oo --Hu(t} x) dx = / f(x) dxoo Jooprovided that(5.3) \a'(x) - b(x) \ _S K{ \ x | + 1}and tha t the integral(5.4) f - T N ^J a(x)diverges both at + oo and oo ; conversely, if (5.3) holds, the divergence of(5.4) is a necessary condition for the theorem.9These unexpected results were at first rather disturbing since they seemed toindicate that the diffusion equations are, after all, an inadequate description ofthe actual process. Fortunately a satisfactory explanation can be found whichmakes the theory more harmonious than it would be if our diffusion equa tionsalways behaved as the classical equation ut = uxx . At the same time we shallbe led to a new type of solution of the backward equation and to an interestingphenomenon concerning the forward equation.The singular diffusion equations discussed in the preceding section point in

8 In [5] the more general case is treated where the coefficients a and b depend on t. Thelater investigations depend on the theoiy of semigroups and apply therefore only to thetemporally homogeneous case treated in the text.9 The apparent discrepancy between Hille's conditions and Feller's condition (5.1) isreadily explained. If one of the integrals in (5.1) converges, then it is impossible that| a'{x) | < K ( | x | -f 1), and hence (5.3) can hold only if the growth of b(x) is adjusted ina special way to that of a(x). On the other hand, if | a'{x) \ < K (| x | + 1), the n a(x) 0(x 2) and the divergence of (5.1) implies that of (5.4). [Added in proof. Unfor tunatelythe text takes account only of Hille's paper [12] in which (3.13) is treated. The authoris indebted to Hille for the manuscript and for many inspiring discussions. SubsequentlyHille treated also (3.12) and obtained conclusive results concerning the uniqueness problems for the two equations; they are announced in [11].]


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330 WILLIAM FELLERthe proper direction. T?he first point to be noted,is that there is.no essentialdiference between diffusion in a finite or infinite interval. In fact, let 7(5) be anarbitrary positive function and consider the transformation of variables

y(s) ds, n = / y(s) ds., . . , . , . , ' ; ; , > , Jo . . ' r 1 . , \ 'MiJ'X$satisfies the diffusion equations 5(i 12) and (3.13), then u*(i;t, )/y(i)(as a'functinf H, rj and t\ , respectively) satisfies similar*1 equations with(y) and 8"(#)" replcedby a(y)-yy2(y) and1Xid(!y)'y'(y) + b(y)-y(y). Moreover1, ifu(y\ V,'x) is,'s a'funotii f xx a probability density; then the same'is tnieofti* as a function of'. Thus w hd,v a whole group of trarisformaton whichchange'th pair of diffusion equatiohs'into^equivalent equation^. We can makeuse of this fact either to simplify the equations or to change the interval. Inparticular, it is always possible to transform a finite interval into the1 entire rlaxis and vice versa. However, even if the coefficients are regular in" the interior,the transformed equations w ll in general be of a singular type, jvith the coefficients unbounded at ,the endpints, or. d(x) vanishing at an endpoint.In particular, we may take the ordinary diffusion equation ut = .u xx fpr thefinite interva l 1 < x


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MATHEMATICAL THEO RY OF DIFFU SION 331Following a remark of Doob10 we can modify our diffusion process by stipu

lating that whenever the particle reaches a boundary, it is instantaneouslytransferred back to the interior. More precisely, let i(x) (i = 1, 2) be two density functions and let us assume that if the particle reaches the right or leftboundary, it jumps to a position Yi or Y2 , where the Yi are random variableswith probability densities i(x). It is easy to calculate the transition probabilityu*(y; t, x) of the so modified process, but for our purposes the explicit form is ofno particular interest. It is rather obvious that u*(y; t, x) behaves asymptotically for t 0 as the transition probability u(y; t, x) of the original process, sothat according to the theory of 3 also u*(y; t, x) must satisfy the backwardequation (3.12). It follows that w* u is a non-negative solution of (3.12) withidentically vanishing initial values, so that in this case the initial valus problemfor (3.12) cannot be determined. The interesting fact is that our phenomenon canoccur even when the initial value problem for (3.13) is uniquely determined.In this case u*(y; t, x) cannot satisfy (3.13) and we have thus a diffusion processwhere the transition probability satisfies the backward equation but not the forwardequation. Instead, the inequality sign _ holds in (3.13) and the right side is tobe modified by adding a positive operator.11 We are thus led to the conclusionthat the adjoint of the backward equation (3.12) is not always given by (3.13),and is in general not even a differential equation. This is a new phenomenonwhich seems of interest for the general theory of semigroups and of differentialoperators.

6. Ito's approach. Before passing to more general processes a word shouldbe said on a new way of describing our diffusion processes which is due to Ito[14,15]. He does not make direct use of the differential equations or the transitionprobabilities, but expresses (at least in certain cases) the path functions of ageneral process by means of those of the Wiener process.Let X(t) represent the path functions of the latter. We have seen that withprobability one X(t) is not of bounded variation so that the classical definitionof the integral(6.1) ff(r)dX(r)

J o10 Doob discussed in detail a similar phenomenon for discrete Markov chains, cf. [2J.11 For details cf. [8], In the case of discrete Markov chains one is led to two infinitesystems of ordinary differential equations playing the role of our diffusion equations. AsDoob has shown, the analogue of (3.12) holds always, but the analogue of (3,13) holds onlywith the equality replaced by the sign _t. We now see why a dire ct de riva tion of (3.13) isimpossible. In attempting a derivation along the lines which led to (3.12) one would introduce the function F{t, y) = J_M u(y; t, x)f(x) dx where f(x) vanishes for | x | > A and

has, say, three continuous derivatives. Then (*) F(t-\- At, y) =/i u{y; At, z)F(t, z) dz g/_> w(j/; t, z) dz'f\a-z\


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332 WILLIAM FELLERbreaks down, even if f(t) is continuous: in fact, for almost every X(t) we canfind a sequence of subdivisions ti

n)such that the corresponding Riemann sumswill diverge. Nevertheless, it is possible to give a meaning to the integral (6.1).In fact, consider a fixed sequence of subdivisions, say tin) = k/2 n. The corresponding Riemann sum

(6.2) HMn))AX(4n))k

is a random variable, and i t has been shown by var ious authors that i t converges in probability to a random variable which has all the desired propertiesof an integral.Using this notion I to shows that under cer tain restr ict ions on a an d b the sto

chast ic in tegral equation(6.3) Y(t) = c + [ b(Y(r)) dr+ [ a(Y(r)) dX(r)

J o Joadmits of an essentially unique solution Y(t). The la t te r is defined as a randomvariable on the Wiener space {X(t)}. It is fairly obvious from (6.3) that theinfinitesimal transition probabilities of Y(t) have the desired properties (3.4)and (3.8), and thus we have arrived at a representation of a stochastic process ofthe requ i red type .This approach has the advantage that i t permits a direct study of the properties of the path functions Y(t), such as their continuity, etc. In principle, wehave here a possibility of proving existence theorems for the partial differentialequations (3.12)-(3.13) directly from the properties of the path functions. However , the method is for the time being restricted to the infinite interval and theconditions on a an d b are such as to guarantee the uniqueness of the solution.So far, therefore, we cannot obtain any new information concerning the "pathological" cases.

7. Diffusion in phase space. Th e m etho d of 3 wo rks also in two an d m oredimensions. In two dimensions the transition probability is of the formu(Vi J 2/2 ; ty #1 j #2), and the equations corresponding to (3.12) and (3.13) are(7.1) Ut = ]C UiiUyiVj - Il >iUui , i,j = 1, 2a n d(7.2) Ut = 2 (aurxixj + _C Q>iu)x., i,j = 1, 2where the coefficients may depend on the two space variables and the matrix(flu) is symmetric and positive definite. The meaning of the coefficients an andbi is the same as before, and the mixed coefficient ai, 2 gives th e infinitesimalcovar iance :

12(2/1,2/2)(7 3) 1 ff= lim - / / (xi 2/1) (x 2 y2)u(yi, y2 ; At, xi, x2) dxi dx2.A*-Q /It JJ\Xiui]


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MATHEMATICAL THEOR Y OF DIFFU SION 333These eq uatio ns were first derived b y Kolmogorov [17]. Feller 's cons truction s

of the transition probability was generalized to two dimensions by Dressel [4],In the time homogeneous case the equations were integrated by Yosida [26]using the theory of semigroups. However, nothing is known for the case ofsingular equations or the case where no uniqueness exists, and practically nothing is known about the appropriate boundary condit ions in var ious problemsinvolving bounded domains.We shall not pursue this line of investigation but shall be content to indicatehow o ur equ atio ns lead to a refined mo del of diffusion in wh ich the p a th function shave der ivat ives.The fact that the original model of diffusion leads to the conclusion that the

particles have no velocities has been pointed out by many authors. Actuallythis fact is neither perturbing nor surprising: the whole theory is based on theassumption that the process is Markovian, that is , that the par t ic le has nomemory. Now if the particle had a finite velocity, shocks in the direction ofmotion would be less probable, so that any change of velocity would affect thechances of further changes. In other words, finite velocities would imply thatthe position of the particle is a random variable of a stochastic process withaftereffects.In a Section m eeting at th is C ongress two propos als Avere m ad e t o modifythe basic assumptions so as to endow the particles with finite velocities. In

both cases the diffusion equations would be replaced by equations of the hyperbolic type. Now the general solution of the initial value problem for such equations does not preserve positivity or the integral mean. Moreover, it dependsno t only on th e initial value but also on their derivative s, which ha ve no ap pa ren tprobabilistic meaning. In short, the hyperbolic equations in question cannotserve as appropriate descriptions of a stochastic process, and if a particulartransition probability is a solution, this is a lucky coincidence.However, a refined model of diffusion in which the particles have finite velocities (but no accelerations) is due to Ornstein and Uhlenbeck.12 This theoryassumes that the par t ic le has a velocity X2(t) which is the subject of an ordinarydiffusion process regulated by (2.1). The position Xi(i) of the particle is thenobtained from

(7.4) Xi(0 = f X2(r) dr ,J o

the integral having a meaning since X2(t) is continuous with probabil i ty one.As shown by Doob, Xi(t) is the variable of a Gaussian (non-Markovian) processwhose probability relations approach for large t those of the ordinary Brownianmotion process.An alternative way of reaching the same conclusion consists in studying thevector (Xi(t), X2(t)) as the varia ble of a two -dim ension al diffusion process (so12 Cf. [23]. A thoroug h discussion of this process which is more ap pr op ria te for our p urposes is contained in Doob [3].


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334 WILLIAM FELLERthat the plane is really the phase space of a particle in one-dimensional motion).Under the above assumptions this two-dimensional process is Markovian, andthe corresponding transition probability u(yi ,y 2;t,xi, x2) should satisfy a pairof equations of the form (7.1), (7.2), Now according to Ornstein and Uhlenbeckthe variable X2(t) is subject to a symmetric homogeneous diffusion'process, sotha t b2 = 0 and 022 is a constant which we can assume as unity. From (7.4) itfollows that if at any time Xi(t) = yi, X2(t) = y2, then

E(Xi{tx + At) - Xi(0) = y*At + o(At)E((Xi(t + At) - Xi(t))2) = o(At),

#,nd hence an = 0, i = y2 . From the second equation (7.5) and Schwarz' ine;quality it follows that also au = 0. Hence (7.1) reduces to(7.6) ' ut = uV2V2 - y2uyi .

This equation has first been given by Kolmogorov [19]. Ijf the condition ofhomogeneity is dropped, the same consideration leads to the more general equation(7.7) ut = b(y 1, y2)uy2y 2 + a(y x, y2)u y2 - y2uvi .This equation is of a degenerate type and practically nothing is known aboutthe appropriate boundary conditions for finite domains. A fundamental solutionfor the infinite plane was constructed by M. Weber [24].To prove the equivalence of the two approaches one should prove that thesolution of (7.7) satisfies the condition (7.4). This is intuitively rather obvious,bu t a sa tisfactory proof has not been given. Otherwise the same method could beapplied to the calculation of various Wiener functionals, which were the objectof investigation by Cameron and Martin, and by Kac (cf. [16]). Let againX2(t) be the random variable of the simplest diffusion process, and, pu t(7.8) Xi(t) = f V(X2(r)) dr,J owhere V(x) is a given function, say x2. Then the pair (Xi, X2) is the variable ofa two-dimensional diffusion process, and should satisfy an equation of the form(7.6) with the coefficient y2 replaced by V(y 2). The difficulty of this approach(as well as in Ito's approach) seems to he in a direct verification that the transition probability of the pair (X i, X2) has all the properties assumed in the derivation of (7.6). Formally, however, one can start with our diffusion equation andderive Kac's results from it.In the Ornstein-Uhlenbeck model of diffusion the position Xi(t) is not thevariable of a Markovian process, but the process becomes Markovian when thestate of the particle is defined so as to include position and velocity. Thus thesame physical process can be described as a Markovian or non-Markovianprocess depending on the parameters used for its description. For example, in


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MATHEMATICAL THEORY OF DIFFUSION 335cosmic ray showers the most interesting variable is the total number of particles,but it is not the variable of a Markov process. However, we may use the conceptual simplicity of and the tools available for Markov processes at the expenseof describing the state of the system by the positions, masses, and velocities ofeach particle. In this way we are led to a Markov process in a rather complicatedphase space. If the process is continuous (which excludes instantaneous changeslike splitting of particles ), we shall again be led to pa rtial differential equationsof the diffusion type.The most radical step in this direction was taken bj7" Feynman [9] in his newapproach to quantum physics. The starting point of Feynman's theory is theremark that the phase-time space can always be split into two parts involvingthe "past" and "future" in such a way that the physical process becomes whatwe would call a stochastic process of the Markov type. As a physicist Feynmandoes not mind generalizations and abstractions which would mak e m ere m athematicians shudder, and thus his "past" is permitted even to include futureevents (thus introducing advance effects instead of aftereffects). Feynman thenintroduces an assumption equivalent to continuity and shows heuristically thatth e diffusion equation to which one is led is essentially the Schrdinger equation.It has complex coefficients since Feynman deals with complex probabilityamplitudes rather than with classical transition probabilities. It is not clear atpresent whether this is only an excellent analytical tool or whether we have herean essential generalization of the classical concept of a stochastic process.

8. Generalizations by means of M. Riesz' potentials. Usually partial differential equations appear to play a special role among functional equations, butfrom a probabilistic point of view the diffusion processes are only a particulartype of a Markov process. Starting from the Chapman-Kolmogorov equation(3.2) we have derived a diffusion equation by imposing certain conditions on theasymptotic behavior of the transition probability u(y; t, x) as t 0. Probabilityconsiderations suggest to us also more general conditions which then lead to ageneralization of the diffusion equation in which certain integrals involving uare added t o the righ t-hand member (cf. [5] and Yosida [27]). How ever, even theintegro-differential equations thus obtained describe only a special class ofMarkov processes and represent from a certain point of view an unnatural typeof functional equations. Our real problem is to find a linear functional equationequivalent to the Chapman-Kolmogorov equation (3.2). In the language of thetheory of semigroups this amounts to saying that we desire to find the infinitesimal generators to the most general semigroup corresponding to (3.2).Before proceeding it should be emphasized that the Chapman-Kolmogorovequation (3.2) is by no means restricted to probability problems, but that ananalogous relation holds for all linear differential equations. In fact, suppose th atthe solution u(t, x) of an equation (whose coefficients do not depend on t) isdetermined by its initial values. Then u(t, x) will be given by an integral of theform (2.3), and the Chapman-Kolmogorov equation merely expresses the fact


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336 WILLIAM FE LLE Rthat the solution u(s + t, x) can be calculated either directly in terms of itsinitial values at time 0 or, alternatively, in terms of its values at time t which,in tu rn , are expressible in terms of the in itial value s. Equating the tw o expressionswe obtain (3.2). It is clear that a similar reasoning holds, for example, for ahyperbolic differential equation- except that a vector notation must be usedsince now the problem is determined by1 the initial values of the solution and itstime derivative.In short, a kernel u(y; t, x) induces two associated linear transformations

Ttf(x) = J f(y)u(y;t,x)dy,(8.1)

Stg(y) = J g(x)u(y; t, x) dxand the Chapman-Kolmogorov equation (3.2) merely expresses the fact that ast varies each of these transformations forms a semigroup. In operator theoreticallanguage the equation uy = uxx can be interpreted as stating th at th infinitesimalgenerator (time derivative) of Tt is given by th e operator d2/dx2, or sym bolically> T>-T-A formal integration leads to the symbolic equation(8.3) Tt = etdVdx\If /() is an entire function, then (8.3) gives(8.4) Ttf(x) = eldVdx*f(x) F= j _ tk/klfm(x),which agrees with the classical solution (2.2) and (2.3). The more general diffusion equations merely replace the operator d2/dx2 on the right by other differential operators. In the case of the integro-differential equations mentionedabove we have on the righ t a combination of a differential and in tegral operator.However, for a general solution we shall have to consider much more generaloperators.The stable distributions are the best known class of solutions of the Chapman-Kolmogorov equation which are not connected with either differential or integro-differential equa tions. Bochner [1] was the first to notice that the sym metricstable distributions can be interpreted as solutions of an operator equation

ut = Au,where the operator A is a certain fractional power of d2/dx2. A better understanding of this important observation may be derived from an application of atechnique developed by M. Riesz [22] which also points to various generalizations.


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MATHEMATICAL THEORY OF DIFFUSION 337A fractional potential of order a is defined, according to Riesz, by

( 8 . 5 ) F f (x ) = * r^y) \y-x I""1 dy.2T(a) cos ira/2 J-,This integral diverges in general, but Riesz showed that a meaning can beattached to it for a large class of functions f(x). In particular, If(x) = f(x)iand I~ 2n f(x) = ( l)nf(2n)(x), whenever n is a positive integer. This relation isinteresting for two reasons. First, it is clear tha t in general Ia is a global operator,that is, I"f(x) depends on all values of f(x). For the particular values a = 0,2, 4, , however, the operator takes on a purely local character. Secondly,since I~ 2 is the same as d2/dx2, we can interpret J - 1 as a square root of d2/dx2,and similarly for other powers. This enables us to pursue in more de tail Bochner'sremark. We are thus led to consider the general functional equation(8.6) ut = - J T VIf a = 2, this reduces to the classical diffusion equation ut = uxx , but for othervalues of a the functional equation (8.6) has quite different a character, sincethe operator on the right is of a global character.From Bochner's remark one should expect that for 0 < a g 1 the equation(8.6) should lead to symmetric stable distributions. This is so, and we proceed toverify it in a purely formal fashion for a = 1.

The symmetric stable distribution of order a = 1 is characterized by thedensity(8.7) u(y; t9 x) *(x-y)*+P'which is known as Cauchy distribution. It plays for the Cauchy process the roleof the normal density (2.2), and the solution (2.3) reduces in the present case tothe function u(t, x) which is harmonic in the half plane t > 0, vanishes at infinity,and reduces for t = 0 to f(x).If we integrate (8.6) for a = 1 formally by analogy with (8.4), we are led toan expansion( 8 . 8 ) u(t,x) =t r"f(x).Substituting from the definition (8.5) it is easily verified that the right sidereduces to the anticipated solution( 8 . 9 ) u>*)=llj{y)(y-l)>+l?dy-

The formal steps leading to this formula require justification. However, bysimilar calculations all symmetric stable distributions can be obtained, and bya slight generalization of Riesz' definitions of potentials we can derive also theunsymmetric stable distribution. I t appears, in fact, that all probabilistic solu-


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338 WILLIAM FELL ERtions of the Chapman-Kolmogorov equation can be obtained in thi way, exceptth at the parameters of the potentials on the right side will depend on x.It is of particular interest that we have obtained the solution of a Dirichletproblem for the Laplace equation Au = 0 as the solution of a generalized diffusionproblem. The explanation fies in the semigroup property of the Green function(8.7), and a similar statement applies also to the Dirichlet problem for closedcurves. The curve itself corresponds to th e a?-axis, and certain closed curves in thinterior to the lines t = const. These contract to a single point which correspondsto t oo. From the values of u along the boundary curve the values along thecurves in the interior can be calculated by the methods just described. Thefundamental fact that as t oo all the values tend to the same limit followsthen from the ergodic principle for Markov processes on closed manifolds. Wehave thus a direct connection of the Dirichlet problem for elliptic equations w ithcertain generalized diffusion equations, which in turn are closely connected withmeasure in function spaces where almost n o function is continuous.

R E F E R E N C E S1. S. BOCHNER, Diffusion equation and stochastic processes, Proc . N at . Acad. Sci. U.S.A.

vol. 35 (1949) pp. 368-370.2. J . L. DOOB, Markoff chains^enumerable case, Trans. Amer. Math. Soc. vol . 68(1945) p p . 455-473.3. , The Brownian mov ement and stochastic equations, Ann. of Math. vol. 43 (1942)

pp . 351-369.4. F. G. DRESSEL, The fundamental solution of the parabolic differential equation, DukeMath. J. vol. 7 (1940) pp. 186-203.5. W . FELLER, Zur Theorie der Stochastischen Prozesse (Existenz und Eindeutigkeitsstze),Math. Ann. vol. 113 (1936) pp. 113-160.6. , Diffusion processes in the mathematical theory of Evolution, Proceedings of theSecond Symposium on Probabil i ty and Mathematical Stat ist ics, Berkeley [to appear],7. , Two singidar diffusion problems, Ann. of Math. vol. 52 (1951), to appear8. , The parabolic differential equations and the associated semi-groups of transfer*mations, to appear, with a companion paper, in Ann. of Math.9 . R. P . FEYNMAN, Space-time approach to non-relativistic quantum mechanics, Reviewsof M ode rn Phy sics vol . 20 (1948) pp . 367-387.10. R . FORTET, Les fonctions alatoires du type de Markoff associes certaines quationslinaires aux drives partielles du type parabolique, J . M a t h . Pures Appi. vol. 22 (1943)

pp . 177-243.11. E . HILLE, Les probabilits continues en chane, G. R. Ac ad. S ci. Paris vol. 230 (1950)

pp . 34-35.X2. , On the integration problem for Fokker-Planck's equation in the theory of stochastic processes, [To appear in the proceedings of the Tenth Congress of ScandinavianMathematicians in Trondheim, 1950].13 . , "Explosive" solutions of Fokker-Plahck's equation, Proceedings of the Interna tiona l Congress of Ma thematicians, Cam bridge, Mass. , 1950.14. KIYOSI ITO, On a stochastic integral e quation, Proceedings of the Japan Academy of

Sciences vo l. 22 (1946) pp . 32-35.115 . , On stochastic differential equations, Memoirs of the American MathematicalSociety, no. 4 (1951). ,16. M , KAO, On the distribution of certain Wiener functionals, Trans , Amer. Math. Soc.

vol. 65 (1949) pp. 1-13.


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MATHEMATICAL THEORY OF DIFFUSION 33917. A. KOLMOGOROFF, lieber die analytischen Methoden in der Wahrscheinlichkeitsrech~nung, Math. Ann. vol. 104 (1931) pp. 415-458.18f 1 Zur Theorie der stetigen zuflligen Prozesse, Math. Ann. vol. 108 (1933) pp.149-160.19, > Zufllige Bewegungen, Ann. of Math. vol. 35 (1934) pp. 116-117,20. P. L E V Y , Thorie de Vaddition des variables alatoires, Paris, Gauthier-Vili ars, 1937.21 . , Processus stochastiques et mouvement Brownien, Paris, Gauthier-Villars,

1948.22. M. RIESZ, L*intgrale de Riemann-Liouville et le problme de Cauchy, Acta Math.vol. 81 (1948) pp. 1-223,23. G. E. UHLENBEOK and L. S. ORNSTEIN, On the theory of the Brownian motion, PhysicalReviews vol. 36 (1930) pp. 823-841.24. M. W EBER, The fundamental solution of the diffusion equation in phase space [To

appear ] .25. N. WIENER, Generalized harmonic analysis, Acta Math. vol. 55 (1930) pp. 117-258.26. K. YOSIDA, An operator-theoretical treatment of temporally homogeneous Markoffprocess, Journal of the Mathematical Society of Japan vol. 1 (1949) pp. 244-253.27 . 9 An extension of Fokker-Planck's equation, Proceedings of the Japan Academyof Sciences vol. 25 (1949) pp. 1-3.28. , On the differentiability and the representation of one-parameter semi-group oflinear operators, Journal of the Mathematical Society of Japan vol. 1 (1948) pp. 15-21.P R I N C E T O N U N I V E R S I T Y ,

PRINCETON, N. J., U, S. A.


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INTERNATIONAL CONGRESSOF

MATHEMATICIANS

Cambridge, Massachusetts, U. S. A.1950

CONFERENCE IN TOPOLOGY

CommitteeHassler Whitney (Chairman)

Deane Montgomery N . E . Steenrod


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some recent trends in the mathematical theory of diffusion

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