Generalized Brownian motion from a logical point of view J¨org Kampen 2 January 9, 2012 1 0 2 Abstract n a We describe generalized Brownian motion related to parabolic equa- J tion systems from a logical point of view, i.e., as a generalization of An- 6 derson’srandom walk. Theconnection toclassical spaces isbased on the Loeb measure. It seems that the construction of Roux in [11] is the only ] attemptintheliteraturetodefinegeneralizedBrownianmotionrelatedto R parabolic systems with coupled second order terms, where Lam´e’s equa- P tion of elastic mechanics is considered as an example. In this paper we . providean exact construction from alogical point of viewin amore gen- h eral situation. A Feynman-Kacformula for generalized Brownian motion t a isderivedwhichisausefultoolinordertodesignprobabilisticalgorithms m for Cauchy problems and initial-boundary value (of a class of) parabolic [ systemsaswellasforstationaryboundaryproblemsof(aclassof)elliptic equation systems. The article includes a selfcontained introduction into 3 all tools of nonstandard analysis needed, and which can be read with a v minimum knowledge of logic in order to make the results available to a 0 6 wider audience. 6 Keywords:[class=AMS] [Primary ]60G05[; secondary ]03H05 4 fundamental solutions of parabolic systems, nonstandard analysis . 2 tensorial processes, Loeb measure 1 1 1 1 Introduction : v i Parabolic systems of equations are ubiquitous in sciences: they describe fields X in physics and mechanics such as displacement fields of solids in elasticity, are r a used for modelling reaction-diffusion in chemical and biological processes, and eveninmathematicalfinancecertainsystemswithinteractingItoˆandpointpro- cessesarerelatedto parabolicsystems andthereforebeyondthe realmofscalar equations (cf. [3]). On the other hand, although the powerful concept of Brow- nian motion entered many branches of physics, finance, and engineering such as heat transfer, dispersion (cf. Einstein,[4]), electrostatics and equity markets (Bachelier,[1]),informationtheoryandnoise(Shannon,[13],Brillouin,[2]),and quantum mechanics (Nelson). However, except for some work related to fluid mechanics (probabilistic approaches to the Navier-Stokes equations based on Malliavin calculus such as in Mattingly) and more abstract articles related to parabolic systems with coupling of lower order terms it seems that the work of Roux is the only attempt to construct a generalized Brownian motion related 1 to parabolic systems with coupled terms (such as Lam´e’s equation) of second orderinthesensethatcertainexpectationsoffunctionsofgeneralizedprocesses solves Cauchy problems of a certain class of parabolic systems (to start with). It seems that the construction of Roux in [11] is the only published attempt in this direction. However, it is clear that this construction is not in an exact measure theoretical sense. This may be a parallel to the situation of the clas- sical Brownian motion where Winer’s construction of Wiener in 1923 was the first exact functional analytic description of Brownianmotion after it has been introducedas a mathematicalobjectona moreheuristic levelby Bachelier and Einstein a long time before. However, it is not even clear on a heuristic level whether the constructionofRouxreallyworks(inthis contextwemaysaythat aconstructionis’correctonaheuristiclevel’ifitleadstoanalgorithmicscheme (a Monte-Carlomethod)where thereis (atleast)experimentalevidence ofcon- vergence in a probabilistic sense. Well it seems by no means clear whether the Landauremainderterms inequations (29)and (30)in [11]become smallas the timestepsizeoftheschemebecomessmall. Inthispaperwedescribeadifferent approach to the problem stated. It is based on a generalized θ function corre- sponding to stochastic processesonthe n-dimensionaltorusTn =S1 S1 ×···× (note that the θ-function on the circle S1 is the analogue of the fundamental solutiononR). ItseemseasiertoconstructgeneralizedBrownianmotionsinthe context of non-standard analysis. Since this theory is not that well-known this article provides a self-contained introduction to the subject including elemen- tary stochastic analysissuchas non-standarddefinitions of stochastic integrals, non-standard densities and Itoˆ formulas. The theory is quite appealing in this respect since proofs are easier once the framework is established. Moreover, a very modest knowledge of logic is required to follow our introduction into the subject. Note that the non-standard theory used here is a consequence of the Zermelo-Fr¨ankel theory together with the axiom of choice (so-called ZFC). We do not need additional assumptions. Other theories with explicit infinitesimals maybeused,forexampletopoiusedintheconextofsyntheticdifferentialgeom- etry or Connes functional analytic theory (called non-commutative geometry), where compact operators play the role of infinitesimals. In any case we may interpret the part of nonstandard analysis as a part of ZFC-theory, although heavy use of the axiom of choice maybe against the taste of some advocates of classical descriptions. From a logical point of view this taste does not matter as does Connes remark that dart play may be better described by his theory (without proof). Similar as intuitionist mathematics nonstandard theory has been criticised to be not productive. In the latter case this criticism is stated in the sense that nonstandard analysis does not lead to results which do not have their classical counterparts. Adherents to this view are asked to give a classical counterpart to the construction of generalized Brownian motion and Feynman-Kacformuladescribedinthispaper. Weclosethisfirstsectionwitha description of the problem of generalizedBrownianmotion related to parabolic systems in a classical context. The parabolic equation systems considered in this paper the quadratic form ∂u n ∂2u n = Aij + Bj u+Cu. (1) ∂t ∂x ∂x ∇ i j i,j=1 j=1 X X(cid:0) (cid:1) 2 (We use the quadratic form for simplicity). Here, u=(u , ,u )T (2) 1 n ··· is a vector-valued function where for each x Rn and each 1 i,j n ∈ ≤ ≤ aij(x) aij (x) 11 ··· 1n x Aij(x):= .. .. (3)  . .  → aij (x) aij (x)  n1 ··· nn    may be assumed to be bounded C . Similarly x Bj(x),j = 1, ,n and ∞ → ··· x C(x) may be assumed to be a n n-matrix-valued function respectively → × with bounded C -entries. If certain ellipticity conditions are satisfied then ∞ equations of the form (1) are called parabolic. Such conditions can be found in manyclassicaltextbookssuchas([6]). Inanalogytotheconstructionofclassical BrownianmotionourconstructionofgeneralizedBrownianmotionisinrelation to equation with constant coefficients. The generalization to processes related to parabolic systems is quite straightforward. So let us assume for a moment that aij(x) aij. What we need for our construction is global existence and kl ≡ kl the requirement that for all 1 i,j n the matrices ≤ ≤ A =(Aij):= aijα α (4) α α kl k l ! kl X are strictly elliptic for all α where all α =0. i 6 Nowinthecaseofscalarequationsthereisanaturalcorrespondencebetween solutions of certain parabolic Cauchy problems and stochastic diffusion pro- cesses. We may put it this way: the fundamental solution of a scalar parabolic equation of type ∂u = n a ∂2u + n b ∂u (5) ∂t i,j=1 ij∂xi∂xj i=1 i∂xi equals the transition densitPy of a diffusion procPess of the form n n dX = b dt+ σ dW , (6) t i ij j i=1 i,j=1 X X where W =(W , ,W ) is a standard Brownian motion, and 1 n ··· (a )=σσT. (7) ij A decomposition as in (7) exists if a certain regularity requirement is satisfied. More precisely the fundamental solution of (5) is the solution of the family of Cauchy problems ∂p = n a ∂2p + n b ∂p ∂t i,j=1 ij∂xi∂xj i=1 i∂xi (8)  P P  p(0,x,y)=δy(x), where for each y Rn δ (x)=δ(y x) along with the Dirac delta distribution y δ. On the other∈hand, if for each−x Rn the stochastic differential equation ∈ 3 (6) with initial data Xx = x has a strong solution X = (Xx) which is 0 t 0 t< adsesnoscitiaytepdshwoiuthldtshaetisMfyarkov family (X,Ω,F = (F)0≤t<∞,Px)x≤∈Rn∞, then the Px(Xx dy)=p(t,x,y)dy. (9) t ∈ For example, the transition density related to the the family of Brownian mo- tions (Wx)x Rn is the fundamental solution of the heat equation. This rela- ∈ tionship between stochastic processes and partial differential equations is quite useful as it provides us with probabilistic algorithm for the solution of Cauchy problems and boundary value problems. Moreover, analytic approximations of densities can be used in order to improve such probabilistic schemes (cf. ([7])). Our question is: Is there a analogous relationship for parabolic systems? Note that there are constructionof fundamental solutionsfor parabolic systems such as(1)bytheparametrixmethod. Howdoesastochasticprocesslooklikewhich satisfies a relation analogous to (9)? It seems that the paper of Roux (cf. [11]) is the only paper which has posed this question. However, as we have said, it seems that the definition given there needs some mathematical clarification since the Landau terms in equations (29) and equation (30) are not estimated. Maybe the situation is similar as with Bachelier’s and Einstein’s early work on Brownian motion. Here the work of Wiener in [14] provided the first exact de- scriptionoftheWienermeasure. TheheatequationrelatedtoBrownianmotion has constant second order coefficients. Hence, similar as in the scalar case we may consider parabolic systems with constant coefficients. Well the probabilis- tic interpretation of the fundamental solution is far less obvious. However,it is astart. Asanexample,letusconsiderLam´e’sequationdescribingdisplacement fields in elasticity. In its time- dependent version it is ∂v =[1 ν(n 1)] 2v+ [ v], (10) ∂t − − ∇ ∇ ∇· and a typical problem is to solve it in some bounded domain D = (0,T) Ω with Ω Rn, and where × ⊂ v(t,x)=B(x) for all x ∂ D p ∈ (11) v(0,x)=A(x) for all x Ω. ∈ Here,∂ DistheparabolicboundaryofDandtheinitialandboundarycondition p fields A and B will satisfy some compatibility conditions. In coordinates (10) reads (we use Einstein summation) ∂vi = [1 ν(n 1)]v +v (12) ∂t j − − i,jj j,ji and in case of dimension nP= 2 we see, with a = [1 ν(n 1)] we have the − − representation v a+1 0 v 0 1 v 1,t = 1,11 + 1,12 v 0 a v 1 0 v 2,t 2,11 2,12 (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (13) a 0 v + 1,22 0 a+1 v . 2,22 (cid:18) (cid:19)(cid:18) (cid:19) 4 In this case our ellipticity assumption (4) amounts to the assumption that for all multiindices α Zn the matrix ∈ α2(a+1)+aα2 α α 1 2 1 2 (14) α α aα2+(a+1)α2 (cid:18) 1 2 1 2(cid:19) is strictly elliptic (which is certainly true for a>0). Note that this example is of the form (1) along with B = 0 and C = 0 (where 0 denotes a matrix with zero entries in the former case and a vector with zero entries in the latter case. It is natural to define generalized Brownian motions to be processes which are relatedtothelatterclassofparabolicsystems. Notethatinelasticitystationary problems of the form [1 ν(n 1)] 2v+ [ v]=0 in Ω − − ∇ ∇ ∇· (15)  v=g for all x ∂Ω,  ∈ are of special interest. Here Ω Rn is some open domain. The probabilistic ⊂ representation in terms of expectations with stopping times (first exit time) is quiteappealing(alsofromaalgorithmicpointofview). Inanycase,thequestion is whether we find a a family of processes, say B ,A (where A encodes the ⊗ information of constant coefficients aij of the diffusion), such that for D = Rn kl the function (t,x)→Ex F f,Bt⊗,A , (16) (cid:16) (cid:16) (cid:17)(cid:17) where F is a rather simple functional with values in Rn. We shall define Bt⊗,A as an infinite vector of n by n matrices. Then F will be just the infinite sum over all multiindices α of products of each n by n matrix entry of Bt⊗,A with the vector entries of the infinite vector of vectors of the form n fˆ exp(i2παx)e (17) iα i ! Xi=1 α∈Zn encoding the information of the initial data f. Here, fˆ is the αth Fourier iα coefficient vector of the ith component f of the vector-valued function f and i e denotes the ith unit basis vector of Rn (cf. next section for more details of i this definition). Although it seems possible do to the constructionin a classical frameworkofWienermeasuresthenonstandardconstructionseemstobe easier in this context. The connection to standard spaces is via the Loeb measure. Therefore, in the next section we shall introduce Anderson’s random walk and the Loeb measure and make a precise definition of the generalized Brownian motion. 2 Anderson random walk and generalized An- derson random walk Thefollowingconstructionmaybeputintoamoreclassicalframework,butthe formulation seems more simple to me in the nonstandard framework. This is a matter of taste to some extent. In any case, from a logical point of view, we are working in ZFC. Moreover, we shall project to classical space finally. The 5 transitionfromnonstandardprobabilityspacestostandardprobabilityisviathe Loeb measure. Let us recall the idea of the Loeb measure first (Readers with no background in nonstandard analysis are advised to read our selfcontained introduction into the subject starting with the next section first). Let H be an hyperfinite, and let P (H) be the set of all internal subset of H. Define I µ:P (H) [0,1] I ∗ → (18) µ(S)= |S|, H | | where . denotes the cardinality of a hyperfinite set. The values of µ are in || [0,1] because subsets of a hyperfinite set H havea smaller internal cardinality ∗ than H. Loeb observed that the map µ :P (H) [0,1] L I → (19) µL(S)=sh |HS| , | | (cid:16) (cid:17) is a measure, i.e., the map µ satisfies the countable additivity axiom. This is L due to the fact that a family (Si)i N of mutually disjoint elements Si PI(H) ∈ ∈ with S := i NSi PI(H) (20) ∪∈ ∈ is a finite actually, i.e. S =S S (21) 1 k ∪···∪ for some k N (otherwise S would be external). Next let us recall the main ∈ theorem concerning the Loeb measure. Let (Ω, ,P) be an internal, finitely A additive probability space, i.e. i) Ω internal ii) is internal subalgebra of (Ω) A P iii) P : R is an internal function such that ∗ A→ iv) P( )=0, P (Ω)=1, A,B : P(A B)=P(A)+P(B) P(A B). ⊘ ∀ ∪ − ∩ The following theorem is the main theorem of non standard probability theory. Theorem1. Thereisastandard(σ-additive) probabilityspace(Ω, ,P )such L L A that i) is a σ-algebra with (Ω) L L A A⊆A ⊆P ii) P = P on A L ◦ iii) For every A and standard ǫ>0 there are and in such that L i o ∈A A A A A A A and P (A A )<ǫ i o o i ⊆ ⊆ \ iv) For every A there is B such that P (A∆B)=0 L L ∈A ∈A The space (Ω, ,P ) is called a Loeb probability space L L A 6 In order to introduce stochastic processes we consider hyperfinite timelines T = 0,t ,t , ,t , (22) 1 2 N { ··· } whereN isaninfiniteintegerandt t areinfinitesimalforalli 1,2, ,N . i+1 i − ∈{ ··· } In the following we assume that we have ∆t = t t for all i 1, ,N , i+1 i − ∈ { ··· } i.e., the discretization is uniform. An internal stochastic process is an internal map X :T Ω R. (23) ∗ × → We may assume that it is adapted to a certain filtration of the Loeb algebra. We may then model Anderson’s random walk (the nonstandard counterpart of Brownian motion) as an internal map B :T Ω R ∗ × → (24) B(t,ω)= ω(s)√∆t, s<t where we may model P Ω= ω :T 1,1 ω is internal (25) { →{− }}| } It is clear how the associated probability measure and the Loeb filtration looks like in this case. It is well known that standard Brownian motion is just the shadow of this process, i.e., we may define W(t,ω)= B(t ,ω), (26) ◦ − wheret isjustthelargestelementinT smallerorequaltot. Notethatin(26) − wedefinedtheBrownianmotionjustincaseofdimensionn=1. Generalization to higher dimension is straightforward, and we do not indicate the dimension whenweusethesymbolW inthefollowing. Asusualprocessesareoftenwritten asfamiliesofrandomvariablesintheformW orX withthesubscriptt,andwe t t adopt this convention. It will be clear from the context or irrelevant. This may be the most simple exact definition of a Brownian motion available. Note that Levy’stheoremcanbeprovedquiteeasilyinthisframework. Wemayconstruct morecomplicatedprocessesfromthiseasily,andweshallconsidermoregeneral internal martingales later, but for the moment let us look at the Feynman-Kac formula ina very simple form. Consider the solutionofthe Cauchy problemon [0, ) Rn ∞ × ∂u = n ∂2u ∂t j=1 ∂x2 j (27)  P  u(0,.)=f, where f ∈Hs(Rn) for arbitrary s ∈R (where Hs is the usual Sobolev Hilbert space)- especially this means that f is smooth and decays rather rapidly at infinity. We know that u has the representation u(t,x)=Ex(f(W )). (28) t whereEx istheexpectationandthesuperscriptxindicatesthatweletstartthe Brownian motion at x Rn. Similarly, if v :Ω Rn R solves the equation ∈ ⊂ → n ∂2u =f on Ω, (29) j=1 ∂x2 j P 7 then v has the representation u(t,x)=Ex(f(W )), (30) τΩ where τ denotes the first exit time from the domain Ω. Similarly, in order to Ω define a Wiener measure on the n-torus Tn we may consider the fundamental solution θ of the problem with periodic boundary conditions ∂u = n ∂2u, ∂t j=1 ∂x2 j (31)  P  u(0,.)=f, u(t,x+ei)=u(t,x) for all 1 i n, ≤ ≤ where e denotes the ith vector of the standard basis of Rn. This θ-function is i given for t>0 by θ(t,x)= exp 2πiαx 4πα2t (32) − i α Zn X∈ (cid:0) (cid:1) where the sum is over all multiindices α = (α , ,α ) with entries α in 1 n i the integers Z. Note that for θ(t,.) converges in ·d·is·tributive sense to the δ- distribution as t 0. Accordingly, ↓ u(t,x)= f(y)θ(t,x y)dy. (33) ZTn − Similar formulas hold for n-tori of any radius R of course, and we may define associated Wiener measure WTnR in the usual manner. As R we get ↑∞ the standard Wiener measure. For elliptic problems such as (29) on bounded domains Ω we find equivalent representation of the form (34) or of the form v(t,x)=Ex f(WTnR) , (34) τΩ (cid:16) (cid:17) forRlargeenough. Weshallconsidermeasurewhicharedefinedonthen-torus. They areeasilydefinedinthe frameworkofnonstandardanalysisandthey lead to a description of generalized Brownian motion as we point out next. From the pointofviewofnonstandardanalysiswemayconsiderthe functions u,f to be standardpartsof internalfunctions whichwe denote with the same symbols u and f for the sake of simplicity of notation. Let the time t be the standard partofa time t T. Ifwe observethe processBx,i.e., the Andersonrandom M ∈ walk starting at x, up to time t , then we know the value of ω(s) up to time t M and we nothing about ω for s>t. We may consider the equivalence classes ω ω˜ iff s<t:ω(s)=ω˜(s), (35) ∼ ∀ and denote the corresponding equivalence classes by [ω] . We define t Ω = [ω] ω Ω (36) M { tM| ∈ } and P :Ω [0,1] n M → (37) P ([ω] )= 1 for all [ω] Ω . n tM 2M tM ∈ M 8 Furthermore let us define the random variable Bx :Ω R tM →∗ (38) Bx ([ω] ):=x+ ω(s)√∆t, tM tM s<t whereωissomeinternalfunctionwithω P[ω] (thisiswelldefinedsinceweget ∈ tM the same result for all ω [ω] by definition of the equivalence relation [] ). Next we consider a hyper∈finitetMdiscretization of R. Let ∆x be an infinitesitmMal ∗ hyperreal number and define R := k∆xk Z , (39) ∗ ∆x ∗ { | ∈ } where Z denotes the set of hyperintegers. ∗ This discretization has the advantage that values of integrals in classical calculusarethestandardpartsofhyperfinitesums. Thenwemayintroducethe density function p defined on T R R by ∗ ∆x ∗ ∆x × × p(t ,x,y):= δ Bx ([ω] ) P([ω] ), (40) M y tM tM tM [ω]tXM∈ΩM (cid:0) (cid:1) where for each y R ∗ ∆x ∈ 1 iff z =y δ (z):= (41) y  0 iff z =y  6 denotes a hyperfinite Kronecker delta translated by y. Note the difference to classicalcalculuswherethedensityisdefinedbyaCauchyproblemwithadelta distributionasinitialdata. Thestandardpartofthefunctionpcanbecomputed asalimitsimilarasinthenonstandardproofofthecentrallimittheorembelow. Furthermore, the internal function u has a representation u(t ,x)= f(y)δ Bx ([ω] ) P([ω] ), (42) M y tM tM tM yX∈∗R[ω]tXM∈ΩM (cid:0) (cid:1) and this representation may be used to get another proof of the Feynman- Kac formula. These representations of the density p and the value function u motivate analogous definitions in the context of the linear parabolic systems consideredintheintroduction. InordertodothiswefirstconsideranAnderson random walk in dimension n. We define Ω = ω :T 1,1 n ω internal , (43) n { →{− } | } and define B :T Ω Rn n ∗ × → (44) B(t,ω)= n ω (s)√∆t e i=1 s<t i i (cid:16) (cid:17) where ω is the ith componentPof thePfunction ω, and e is the ith element i i of the standard basis in Rn. Next for any positive matrix A we consider its ∗ 9 representationA=QΛQT withΛ=diag(λ )thediagonalmatrixwithdiagonal i entries λ >0 and define i BΛ :T Ω Rn Rn n ∗ ∗ × → ⊗ (45) BΛ(t,ω) = n ω (s)√∆t λ δ . ij i=1 s<t i i ij (cid:16) (cid:17) P P with the Kronecker δ-function δ (the subscript ij indicates that the entry of ij the ith row and the jth clumn is defined). Furthermore, we define BA(t,ω)=QBΛ(t,ω)QT (46) Generalized Brownian motions related to linear parabolic systems will be de- finedbyaninfinitevectorofsuchmatrix-valuedAndersonrandomwalks,where eachentryencodesinformationofthe diffusioncoefficientsaij. The generalized kl Brownian motion we are going to construct can be represented by an infinite vector of n by n matrices. This may be described analogously to the so-called Kronecker description of tensor products but we do not go into this. For some total ordering of multiindices α Zn we may interpret BZn = (Bα)α Zn to be ∈ ∈ some infinite vector of n by n matrices matrices with entries in the complex numbers C and VZn = (vα)α Zn some infinite vector of n-dimensional vectors ∈ with entries in the complex numbers. Then we define BZn,VZn := BαCα. (47) h i α X Moreover,the components of the product are denoted by hBZn,VZnii := (BαCα)i, (48) α X andwhere(B C ) denotestheithentryofthevectorB C . LetA =(Aij):= α α i α α α α aij4π2α α . Then for A = Q Λ QT for diagonal Λ with positive kl kl k l α α α α α e(cid:16)nPtries λi >0 and(cid:17)orthogonal Q define (with ∆x=√∆t B√Λ :T Ω Rn Rn × n →∗ ∆x⊗∗ ∆x (49) B√Λ(t,ω) = n ω (s)√∆t √λ δ , i=1 s<t i i ij ij (cid:16) (cid:17) (cid:16) (cid:17) P P whereδ denotestheclassicalKroneckerdelta,andforapositivedefinitematrix ij A with decomposition QΛQT and Λ=diag(λ ) define i B√A :T Ω Rn Rn × n →∗ ∆x⊗∗ ∆x (50) B√A(t,ω):=QBΛ1/2(t,ω)QT. A crucial observation is that ΘA(t,x−y)≈E α Znexp iαB√Aα(t,.) e→xp(i2πα(x−y)) (51) ∈ h (cid:16) (cid:17) i P 10