Stochastic solutions of nonlinear pde s: McKean versus superprocesses

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1 Stochastic solutions of nonlinear pde s: McKean versus superprocesses R. Vilela Mendes CMAF - Complexo Interdisciplinar, Universidade de Lisboa (Av. Gama Pinto 2, , Lisbon) Instituto de Plasmas e Fusão Nuclear, IST (Av. Rovisco Pais, Lisbon) Abstract Stochastic solutions not only provide new rigorous results for nonlinear pde s but also, through its local non-grid nature, are a natural tool for parallel computation. Here a comparison is made of two di erent approaches for the construction of stochastic solutions. An extension of superprocesses, as stochastic processes on signed measures and on distributions, is proposed to extend the stochastic solution approach to a wider class of pde s. 1 Introduction: Stochastic solutions and their uses A stochastic solution of a linear or nonlinear partial di erential equation is a stochastic process which, when started from a particular point in the domain generates after a time t a boundary measure which, when integrated over the initial condition at t =, provides the solution at the point x and time t. For example for the heat t u(t; x) = 1 u(t; x) with u(; x) = f(x) (1) the stochastic process is Brownian motion and the solution is u(t; x) = E x f(x t ) (2) E x meaning the expectation value, starting from x, of the process vilela@cii.fc.ul.pt, vilela.mendes@ist.utl.pt dx t = db t (3) 1

2 The domain here is R[; t) and the expectation value in (2) is indeed the inner product h t ; fi of the initial condition f with the measure t generated by the Brownian motion at the t boundary. The usual integral solution, u (t; x) = 1 Z! 1pt 2 p exp (x y) 2 f (y) dy (4) 4t with the heat kernel, has exactly the same interpretation. Of course, an important condition for the stochastic process (Brownian motion in this case) to be considered the solution of the equation is the fact that the same process works for any initial condition. This should be contrasted with stochastic processes constructed from particular solutions. That the solutions of linear elliptic and parabolic equations, both with Cauchy and Dirichlet boundary conditions, have a probabilistic interpretation is a classical result and a standard tool in potential theory [1] [2] [3]. In contrast with the linear problems, explicit solutions in terms of elementary functions or integrals for nonlinear partial di erential equations are only known in very particular cases. Therefore the construction of solutions through stochastic processes, for nonlinear equations, has become an active eld in recent years. The rst stochastic solution for a nonlinear pde was constructed by McKean [4] for the KPP equation. Later on, the exit measures provided by di usion plus branching processes [5] [6] as well as the stochastic representations recently constructed for the Navier-Stokes [7] [8] [9] [1] [11], the Vlasov-Poisson [12] [13] [15], the Euler [14] and a fractional version of the KPP equation [16] de ne solution-independent processes for which the mean values of some functionals are solutions to these equations. Therefore, they are exact stochastic solutions. In the stochastic solutions one deals with a process that starts from the point where the solution is to be found, a functional being then computed on the boundary or in some cases along the whole sample path. In addition to providing new exact results, the stochastic solutions are also a promising tool for numerical implementation. This is because: (i) Deterministic algorithms grow exponentially with the dimension d of the space, roughly N d ( L N being the linear size of the grid). This implies that to have reasonable computing times, the number of grid points may not be su cient to obtain a good local resolution for the solution. In contrast a stochastic simulation only grows with the dimension of the process, typically of order d. (ii) In general, deterministic algorithms aim at obtaining the global behavior of the solution in the whole domain. That means that even if an e cient deterministic algorithm exists for the problem, a stochastic algorithm might still be competitive if only localized values of the solution are desired. This comes from the very nature of the stochastic representation processes that always start from a de nite point of the domain. According to what is desired, real or Fourier space representations should be used. For example by studying only a few high Fourier modes one may obtain information on the small scale uctuations that only a very ne grid would provide in a deterministic algorithm. 2

3 (iii) Each time a sample path of the process is implemented, it is independent from any other sample paths that are used to obtain the expectation value. Likewise, paths starting from di erent points are independent from each other. Therefore the stochastic algorithms are a natural choice for parallel and distributed implementation. Provided some di erentiability conditions are satis ed, the process also handles equally well simple or complex boundary conditions. (iv) Stochastic algorithms may also be used for domain decomposition purposes [17] [18] [19]. One may, for example, decompose the space in subdomains and then use in each one a deterministic algorithm with Dirichlet boundary conditions, the values on the boundaries being determined by a stochastic algorithm, thus minimizing the time-consuming communication problem between domains. There are basically two methods to construct stochastic solutions. The rst method, which will be called the McKean method, is essentially a probabilistic interpretation of the Picard series. The di erential equation is written as an integral equation which is rearranged in a such a way that the coe cients of the successive terms in the Picard iteration obey a normalization condition. The Picard iteration is then interpreted as an evolution and branching process, the stochastic solution being equivalent to importance sampling of the normalized Picard series. The second method constructs the boundary measures of a measure-valued stochastic process (a superprocess) and obtain the solutions of the di erential equation by a scaling procedure. In this paper the two methods are compared and in the nal sections one shows how the superprocess construction may be extended to larger classes of partial di erential equations by going from process on measures to processes on signed measures and processes on distributions. 2 McKean and superprocesses: The KPP equation 2.1 The KPP equation: McKean s formulation To illustrate the two methods for the construction of stochastic solutions a classical example will be used, namely the KPP = 2 v 2 + v2 v (5) with initial data v (; x) = g (x) Let G (t; x) be the Green s operator for the heat t v(t; x) = 2 v(t; x) 2 Then the equation in integral form is v (t; x) = e t G (t; x) g (x) + G (t; x) = e 1 2 (6) 3 e (t s) G (t s; x) v 2 (s; x) ds (7)

4 Figure 1: The McKean process for the KPP equation Denoting by ( t ; x ) a Brownian motion started from time zero and coordinate x, Eq.(7) may be rewritten v (t; x) = x e t g ( t ) + e (t s) v 2 s; t s ds = x e t g ( t ) + e s v 2 (t s; s ) ds Therefore the solution is obtained by the following process: At the initial time, a single particle begins a Brownian motion, starting from x and continuing for an exponential holding time T with P (T > t) = e t. Then, at T, the particle splits into two, the new particles continuing along independent Brownian paths starting from x (T ). These particles, in turn, are subjected to the same splitting rule, meaning that after an elapsed time t > one has n particles located at x 1 (t) ; x 2 (t) ; x n (t) with P (n = k) = e t (1 e t ) k 1. The solution of (7) is obtained by v (t; x) = E fg (x 1 (t)) g (x 2 (t)) g (x n (t))g (9) An equivalent interpretation consists in considering the process propagating backwards in time from time t at the point x and, when it reaches time zero, it samples the initial condition. That is, the process generates a measure at the t = boundary which is then applied to the function g (x) = v (; x). This construction, which expresses the solution as a stochastic multiplicative functional of the initial condition, is also qualitatively equivalent to importance sampling of the Picard iteration of Eq.(7). A su cient condition for the existence of (9) is jg (x)j 1 or, almost surely, jg (x)j (1 e t ) 1. Another probabilistic approach to this type of equations is through the construction of superprocesses. In many cases a superprocess may be looked at as the scaling limit of a branching particle system. The point of view used in the derivation of superprocesses is di erent from the derivation above. In the next subsection a short introduction to superprocesses is sketched and then the KPP solution is constructed via superprocesses. (8) 4

5 2.2 Branching exit measures and superprocesses Let (E; B) be a measurable space and M + (E) the space of nite measures in E. Denote by (X t ; P ; ) a branching stochastic process with values in M + (E) and transition probability P ; starting from time and measure. The process is said to satisfy the branching property if given = P ; = P ;1 P ;2 (1) that is, after the branching X 1 t ; P ;1 and X 2 t ; P ;2 are independent and X 1 t + X 2 t has the same law as (X t ; P r; ). In terms of the transition operator V t operating on functions on E this is where e hvtf;i $ P r; e hf;xti or V t f ( ) = V t f ( 1 ) + V t f ( 2 ) (11) V t f () = log P ; e hf;xti (12) V t is called the log-laplace semigroup associated to X t. In (12) if the initial measure is x one writes V t f (x) = log P ;x e hf;xti (13) By (1) the probability law of X t is in nitely divisible. Now in S = [; 1) E consider a set Q S and the associated branching exit process (X Q ; P ) composed of a propagating Markov process in E, = ( t ; ;x ), a set of probabilities p n (t; x) describing the branching and a parameter k de ning the lifetime. u (x) = V Q f (x) = log P ;x e hf;x Qi (14) hf; X Q i is the integral of the function f on the (space-time) boundary with the boundary exit measure generated by the process. One says that this branching exit process is a (; ) superprocess if u (x) satis es the equation where G Q is the Green operator, K Q the Poisson operator u + G Q (u) = K Q f (15) Z G Q f (r; x) = ;x f (s; s ) ds (16) K Q f (x) = ;x 1 <1 f ( ) (17) (u) means (; x; u (; x)) and is the exit time from Q. 5

6 The superprocess is constructed as follows: Let ' (s; x; z) be the o spring generating function at time s and point x 1X ' (s; x; z) = c p n (s; x)z n (18) where P n p n = 1 and c denotes the branching intensity. Then for e w(;x) $ P ;x e hf;xqi one has Z P ;x e hf;xqi $ e w(;x) = ;x e k e f(; ) + dske ks ' s; s ; e w( s; ) s (19) The measure-valued process starts from x at time, is the rst exit time from Q and f (; ) the value of a function in the Using R ke ks ds = 1 e k and the Markov property ;x 1 s< s;s = ;x 1 s<, Eq.(19) for e w(;x) is converted into Z h e w(;x) = ;x e f(; ) + k ds ' s; s ; e w( s; ) s e w( s; )i s (2) This is lemma 1.2 in ch.4 of Ref.[5]. Because of the central role of this result for the construction of superprocesses, a proof is included in the Appendix with the notations used in this paper. Eq.(2.11) is now obtained by a limiting process. Let in (2) replace w (; x) by w (; x) and f by f. is interpreted as the mass of the particles and when the measure-valued process X Q! X Q then P! P. e w(;x) = ;x e f(; ) + k Z De ning and ds h' s; s ; e w( s; ) s e w( s; )i s (21) u = 1 e w = ; f = 1 e f = (22) (r; x; u ) = k (' (r; x; 1 u ) 1 + u ) (23) one obtains from (21) that is Z u (; x) + ;x ds (s; s ; u ) = ;x f (; ) (24) u + G Q (u ) = K Q f (25) When!, f! f and if goes to a well de ned limit then u tends to a limit u solution of (15) associated to a superprocess. Also one sees from (22) that in the! limit u! w = log P ;x e hf;x Qi (26) as in Eq.(14). The superprocess corresponds to a cloud of particles for which both the mass and the lifetime tend to zero. 6

7 2.3 The KPP equation as a superprocess When the integral Eq.(7) is interpreted probabilistically, it may be identi ed with Eq.(19) with k = 1, e w(;x) = v (; x), e f(; ) = g ( ), ' s; s ; e w( s; ) s = v 2 ( s; s ). Therefore the McKean probabilistic construction corresponds to an intermediate step in the superprocess construction. At this level the process that is considered in Eq.(19) is the same as in McKean s construction. Summing over the exit measure, the solution is v (t; x) = e hf;x Qi = e P i f( i ) = e P i log g( i ) = i g i (27) essentially the same as in (9). However, there are two di erences. First, the initial condition g must be positive to have a well-de ned logarithm. This is a restriction as compared to McKean s construction. But, on the other hand, the interpretation as an exit measure, allows to deal with Cauchy problems with boundary conditions. The exit measure is from the set Q = [; t], being the time at which the (t; t ) process or = t inside. For the superprocess, let u (t; x) = 1 v (t; x), which satis es the equation and the = 2 u 2 u 2 + u (28) or u (t; x) = G (t; x) (1 g (x)) + u (t; x) + x that is for KPP Equating with (23) one obtains G (s; x) u (t s; x) u 2 (t s; x) ds (29) u 2 (t s; s ) u (t s; s ) ds = x (1 g ( t )) (3) (; x; u) = u 2 u (31) (; x; u ) = k (' (; x; 1 u ) 1 + u ) = k c X p n (1 u ) n 1 + u = k c 2 u 2 u = u 2 u (32) with p n = n;2. Therefore c = = 1 and k = 1. That is, for KPP the superprocess is not a scaling limit. It coincides with the McKean process. However in this case, because = 1 instead of!, the solution is given by (1 e w ) instead of (14). 7

8 However, the power of the superprocesses is that, with other limiting choices of, stochastic solutions may be constructed for other equations, in particular for solutions without the natural Poisson clock provided by the term v which is present in the KPP equation. For example for with u = one has v which equated with (23) (; x; u ) = 2 v 2 + = 2 u 2 u 2 (34) u 1 + (; x; u) = u 2 (35)! 2X p + p 1 (1 u ) + p 2 (1 u ) 2 n= = u 2 (36) leads to p 1 = ; p = p 2 = 1 2 ; k = 2 (37) In this case one may let!, the solution is given by (14) and the superprocess corresponds to the scaling limit (n! 1 in Fig.2) of a process where both the mass and the lifetime of the particles tends to zero and at each bifurcation point one has equal probability of either dying without o spring or having two children (Fig.2)This construction may be generalized for interactions u with 1 < 2. With z = 1 u one has ' (; x; z) = X n = z + p n z n = z + u = z + k (1 1 k 1 1 z + ( 1) 2 z) z 2 ( 1) ( 2) 3! z 3 + (38) Choosing k = the terms in z cancel and for 1 < 2 the coe cients 1 of all the remaining z powers are positive and may be interpreted as branching probabilities. It would not be so for > 2. Then p = 1 ; p 1 = ; p n = ( 1)n n 2 (39) n with P n p n = 1. With this choice of branching probabilities, k = and 1! one obtains a superprocess which, through (14), provides a solution to = 2 u 2 u (4) for 1 < 2. 8

9 Figure 2: The branching process which in the scaling limit n! 1 leads to the superprocess solution of Eq.(34) 3 Signed measures and superprocesses As seen in the previous section, the superprocess cannot be constructed for > 2 because some of the z n coe cients in the o spring generating function ' (; x; z) would be negative. This suggests an extension of the superprocess construction from processes in the space of measures to processes in signed measures. Consider in the measurable space (E; B) the space of nite signed measures M (E) in E. As before denote by (X t ; P ; ) a stochastic process with values in M (E) and transition probability P ; starting from time and measure. Here it is also assumed that the process satis es a branching property and V t is the transition operator operating on functions on E, that is or e hvtf;i = P ; e hf;xti (41) V t f (x) = log P ;x e hf;xti (42) if the initial measure is x. Now X t is a signed measure but otherwise the framework is similar to the previous one. Now in S = [r; 1) E consider a set Q S and the associated branching exit process (X Q ; P ) composed of a propagating Markov process in E, = ( t ; ;x ), a set of probabilities p n (t; x), as well as signs and intensities describing the branching and a parameter k de ning the life time. u (x) = V Q f (x) = log P r;x e hf;x Qi (43) 9

10 Then for e w(;x) $ P ;x e hf;x Qi one has Z P ;x e hf;xqi $ e w(;x) = ;x e k e f(; ) + dske ks s; s ; e w( s; ) s (44) where now (; s ; z) = c X " n p n z n n (45) p n denotes the branching probability into n particles, c denotes the intensity and " n the signs of the branched measures. The superprocess construction proceeds now as in measure-valued case. As an example consider the search for a superprocess = 2 u 2 u 3 (46) As in (38) (; x; z) = c X n " n p n z n = z + u 3 = z + k (1 z) 3 = z + 1 k 2 1 3z + 3z2 z 3 (47) With k = 3 2 one obtains c = 5 3 ; p = 1 5 ; p 2 = 3 5 ; p 3 = 1 5 ; " 2 = 1; " 3 = 1 and in the! limit the signed measure superprocess yields, by (43), a solution of (3.6). In (43) hf; X Q i is the integral of f over the signed measure. This construction extends the superprocess construction to a much wider class of nonlinear terms. 4 Distribution-valued superprocesses The construction of superprocesses using signed measure-valued processes already extends the method to a wider class of partial di erential equations. Nevertheless it is not yet su ciently general to handle nonlinear terms with derivatives as they commonly occur, for example, in kinetic equations. In the construction of Section 2, the exit boundary measure generated by the measurevalued process (X t ; P ; ) is typically generated by starting at the point x with a x measure which then propagates along the trajectories of the ( t ; ;x ) process and, after an exponentially controlled time, bifurcates into n s measures with probabilities given by the generating function ' (; x; z). For signed measures the only modi cation is that in the branching some of the o spring local measures acquire a charge " n. As long as the nonlinear terms have a local nature the construction may be simply extended to processes in more general distributions, by allowing the x measures to bifurcate into derivatives (n) s. As long as the distribution-valued process is restricted to distribution with support at a single point, the whole 3 1

11 construction leading to (21) and (24) is the same as before. As an example consider = 2 u 2 Then (; x; u ) = k ( (; x; 1 u ) 1 + u ) = x u with z = 1 u leads to (; x; z) = 1 k (z x z + z and with k = 1 1 (; x; z) = 3 3 z 1 xz z@ xz That is, the solution of (48) is obtained by the scaling limit! ; k = 1! 1 of a process where, starting from x, each point measure s (n) propagates for an holding time T with P (T > t) = e kt and then, at the branching point, with equal probabilities, it either continuesunchanged or becomes (n+1) s changing its charge sign or gives raise to a pair s (n) ; (n+1) s. One sees that, at least for local nonlinearities, the superprocess construction implies a simple extension from processes on measures to processes on distributions with point support. It is probable that processes on more general distributions would be relevant to deal with nonlocal nonlinearities. The need to extend the processes from measures to processes on distributions for interactions involving derivatives has in fact been anticipated in some of the constructions using the McKean approach [14] [15]. There, whenever a derivative interaction is found in the evolution of the process, the o spring acquires a cumulative label which, when the particle nally reaches time zero, it samples the derivatives of the initial condition. This is in fact equivalent to the construction above. 5 Appendix: Proof of the lemma Let Then u (x; t) = ;x e kt u ( t ; ) + ke ks ( s ; t s) ds ;x ku ( s ; t s) ds = ;x ke k(t s) u s+t s ; ds + kds s (49) kds e ks s+s ; t s s (5) 11

12 Summing (49) and (5) u (x; t) + ;x ku ( s ; t = ;x e kt + +k e ks ( s ; t ke k(t s) ds + k s) ds s) ds u ( t ; ) ds s kds e ks s+s ; t s s ds Changing variables in the last integral in (51) from (s; s ) to (s; = s + s ) one obtains for the last term and nally k d Z kdse k( s) ( ; t ) ds (51) u (x; t) + ;x k u ( s ; t s) ds = ;x u ( t ; ) + k ( s ; t s) ds (52) References [1] R. M. Blumenthal and R. K. Getoor; Markov processes and potential theory, Academic Press, New York [2] R. F. Bass; Probabilistic techniques in analysis, Springer, New York [3] R. F. Bass; Di usions and elliptic operators, Springer, New York [4] H. P. McKean; Comm. on Pure and Appl. Math. 28 (1975) , 29 (1976) [5] E. B. Dynkin; Di usions, Superdi usions and Partial Di erential Equations, AMS Colloquium Pubs., Providence 22. [6] E. B.Dynkin; Superdi usions and positive solutions of nonlinear partial di erential equations, AMS, Providence.24. [7] Y. LeJan and A. S. Sznitman ; Prob. Theory and Relat. Fields 19 (1997) [8] E. C. Waymire; Prob. Surveys 2 (25) [9] R. N. Bhattacharya et al. ; Trans. Amer. Math. Soc. 355 (23) [1] M. Ossiander ; Prob. Theory and Relat. Fields 133 (25)

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