Mean-field dual of cooperative reproduction

Size: px

Start display at page:

Download "Mean-field dual of cooperative reproduction"

Alvin Pitts
5 years ago
Views:

1 The mean-field dual of systems with cooperative reproduction joint with Tibor Mach (Prague) A. Sturm (Göttingen) Friday, July 6th, 2018

2 Poisson construction of Markov processes Let (X t ) t 0 be a continuous-time Markov process with finite state space S, semigroup (P t ) t 0 of transition probabilities P t (x, y), and generator G, i.e., P t = e tg = n=0 1 n! tn G n. We view P t and G as matrices and use the notation µp t (y) := x S µ(x)p t (x, y), P t f (x) := y S P t (x, y)f (y). A Markov generator satisfies G(x, y) 0 (x y) and G(x, y) = 0. y We call G(x, y) (x y) the rate of jumps from x to y.

3 Poisson construction of Markov processes Let M be a collection of maps m : S S and let (r m ) m M be nonnegative constants. Then Gf (x) := { ( ) ( )} r m f m(x) f x m M defines a Markov generator. Conversely, every Markov generator can (not uniquely) be written in the form (1). We call (1) a random mapping representation. Let ω be a Poisson point set on M R with intensity (1) µ ( {m} A ) = r m l(a) ( A B(R) ), where B(R) is the Borel-σ-field on R and l denotes Lebesgue measure. If m M r m <, then we may order the elements of ω s,t := ω M (s, t] = { (m 1, t 1 ),..., (m n, t n ) } such that t 1 < < t n.

4 Poisson construction of Markov processes Define random maps X s,t : S S (s t) by X s,t := m n m 1. (Poisson construction of Markov processes) Define maps (X s,t ) s t as above in terms of a Poisson point set ω. Let X 0 be an S-valued random variable, independent of ω. Then X t := X 0,t (X 0 ) (t 0) is a Markov process with generator G. Remark The sample paths of X are right-continuous. We get left-continuous paths by defining X s,t in terms of ω s,t := ω M (s, t).

5 Poisson construction of Markov processes The random maps (X s,t ) s t form a stochastic flow X s,s = 1 and X t,u X s,t = X s,u with independent increments, in the sense that X t0,t 1,..., X tn 1,t n are independent for each t 0 < < t n. Different stochastic flows can define the same Markov process, as there may be many different ways of writing down a random mapping representation Gf (x) = { ( ) ( )} r m f m(x) f x, for a given generator G. m M Compare: Different SDE s can define the same diffusion process.

6 Cooperative branching Let Λ be a finite set and let S := {0, 1} Λ be the space of all functions Λ i x(i) {0, 1}. For each i, j, k Λ, we define a cooperative branching map coop ijk : S S by { ( ) x(i) x(j) x(k) if l = k, coop ijk x(l) := x(l) otherwise. For each i Λ, we define a death map death i : S S by { 0 if k = i, death i x(k) := x(k) otherwise. On {0,..., n 1} (with periodic boundary conditions) we define Gf (x) := { ( f deathi (x) ) f ( x )} { ( i ) ( )} { ( ) ( )} f coopi 1,i,i+1 f x + α f coopi+1,i,i 1 f x. +α i i

7 A graphical representation time space We denote coop ijk and death i by suitable symbols.

8 A graphical representation X t = X 0,t (X 0 ) time X 0 The map x X 0,t (x). space

9 Pathwise duality Def Two maps m : S S and ˆm : T T are dual with duality function ψ iff ψ ( m(x), y ) = ψ ( x, ˆm(y) ) (x S, y T ). Let (X s,t ) s t be a stochastic flow defined in terms of a Poisson point set ω whose elements are pairs (m, t) with m M. Fix some duality function ψ and assume that each m M has some dual ˆm w.r.t. ψ. Then we can define a dual flow (X t,s ) t s by X s,t := m n m 1 and X t,s := ˆm 1 ˆm n, with ω s,t := ω M (s, t] = { (m 1, t 1 ),..., (m n, t n ) } and t 1 < < t n.

10 Pathwise duality Let X 0 and Y 0 be independent of ω. For fixed s R, setting X t := X s,s+t (X 0 ) and Y t := X s,s t (Y 0 ) (t 0) defines Markov processes with generators Gf (x) = { ( ) ( )} r m f m(x) f x, m M Gf (y) = m M r m { f ( ˆm(y) ) f ( y )}. Then X and Y are pathwise dual in the sense that ψ ( X s,t (x), y ) = ψ ( x, X t,s (y) ). Slight complication: (Y t ) t 0 has left-continuous sample paths.

11 Pathwise duality Trivial example Set T := P(S) and ψ(x, A) := 1 {x A}. Then each map m : S S is dual w.r.t. ψ to the inverse image map m 1 : P(S) P(S): ψ ( m(x), y ) = 1 {m(x) A} = 1 {x m 1 (A)} = ψ( x, m 1 (y) ). Nontrivial examples are associated with invariant subspaces of P(S).

12 Monotone systems duality Let S be equipped with a partial order. A set A S is increasing if A x y implies y A. A map m : S S is monotone if x y implies m(x) m(y) or equivalently A increasing implies m 1 (A) increasing. We can encode increasing subsets by their minimal elements. This leads to the duality function ψ ( x, Y ) := 1 {x y for some y Y }. Note: The maps coop ijk and death i are monotone.

13 Monotone systems duality time 1 1 1

14 Monotone systems duality time

15 Monotone systems duality time

16 Monotone systems duality time

17 Monotone systems duality time dual process Y t P [ X t y for some y Y 0 ] = P [ X0 y for some y Y t ].

18 The mean-field limit Let S be a Polish space, let G be a collection of measurable maps g : S k S, where k = k g 0 can depend on g, and let (r g ) g G be nonnegative constants. We will be interested in Markov processes with state space S N that are constructed in the following way. With rate r g, we sample a collection (i 1,..., i kg ) of elements of {1,..., N}, all different. We calculate g ( x(i 1 ),..., x(i kg ) ). At some site j, we replace the value x(j) by g ( x(i 1 ),..., x(i kg ) ). The site j may or may not be an element of {i 1,..., i kg }, depending on the process we wish to construct.

19 The mean-field limit In particular, we are interested in S = {0, 1} and the maps cob : S 3 S and dth : S 0 S defined as cob ( x(1), x(2), x(3) ) := ( x(1) x(2) ) x(3) dth ( ) := 0. Here we view S 0 as a set with a single element, the empty sequence ( ).

20 The mean-field limit Let [N] := {1,..., N} and [N] k := { i = (i 1,..., i k ) [N] k : i m i n n m }, which has N k := N(N 1) (N k + 1) elements. For g : S k S, N k, i [N] k, and j [N], define g i,j : S N S N by { ( g x(i1 g i,j x(j ),..., x(i k ) ) if j = j, ) := x(j ) otherwise.

21 The mean-field limit We are interested in the Markov process with state space {0, 1} N and generator Note that Gf (x) := αn { ( f cob i,i 3 N 3 (x) ) f ( x )} i [N] 3 + { ( f dth (),i (x) ) f ( x )}. i [N] cob i,i 3 = coop i1,i 2,i 3 since we replace the value x(i 3 ) by ( x(i 1 ) x(i 2 ) ) x(i 3 ).

22 The mean-field limit In general, we are interested in generators of the form Gf (x) = N g G In words: r g N kg + ( 1 i [N] kg k g m=1 [ kg { ( p m f g i,i m (x) ) f ( x )} m=1 p m ) 1 N { ( f g i,j (x) ) f ( x )}]. j [N] With rate Nr g, we select a random k g -tuple i = (i 1,..., i kg ) and calculate g ( x(i 1 ),..., x(i kg ) ). With probability p m, we write the outcome in place of x(i m ). With the remaining probability 1 k g m=1 p m, we write the outcome at a randomly chosen site j [N].

23 The mean-field limit Let (X N t ) t 0 be the resulting Markov process and let µ N t := µ [ X N ] t (t 0) with µ[x] := 1 N i [N] δ x(i) denote the associated empirical process. In the mean-field limit N, we expect (µ N t ) t 0 to be close to the solution of an ODE.

24 The mean-field ODE Let M 1 (S) denote the space of all probability measures on S, equipped with the topology of weak convergence and the associated Borel-σ-algebra. For each measurable map g : S k S, we define a measurable map ǧ : M 1 (S) k M 1 (S) by ǧ(µ 1,..., µ k ) := P [ g(x 1,..., X k ) ] where X 1,..., X k are indep. with P[X i ] = µ i. We also define T g : M 1 (S) M 1 (S) by T g (µ) := ǧ(µ,..., µ). Note that T g is in general nonlinear, unless k = 1.

25 The mean-field ODE Theorem [Mach, Sturm & S. 18] Assume that r g k g <. g G Then, for each initial state µ 0 M 1 (S), the mean-field ODE t µ t = { } r g Tg (µ t ) µ t g G (t 0) (2) has a unique solution. Here, writing µ, φ := φ dµ, we interpret (2) in a weak sense: for each bounded measurable φ : S R, the function t µ t, φ is continuously differentiable and t µ t, φ = { r g Tg (µ t ), φ µ t, φ } (t 0). g G

26 The mean-field ODE Theorem [Mach, Sturm & S. 18] Assume that g G r g k g < and let S be finite. Let (µ N t ) t 0 be empirical processes such that µ N 0 µ 0 for some µ 0 M 1 (S). Then P [ sup µ N (t) µ t ε ] 0 ε > 0, T <, 0 t T N where (µ t ) t 0 solves the mean-field ODE (2). Note Something similar should hold for infinite (even uncountable) S. Note The probabilities p m drop out of the limiting ODE.

27 The mean-field ODE Recall that in our example S = {0, 1}, G = {cob, dth}, r cob = α and r dth = 1 A probability measure µ on {0, 1} is uniquely determined by µ({1}). Setting X t := µ t ({1}), we find the mean-field ODE t X t = αx 2 t (1 X t ) X t =: F α (X t ).

28 Cooperative branching F α (x) α = 3 x For α < 4, the equation t X t = F α (X t ) has a single, stable fixed point x = 0.

29 Cooperative branching 0.3 F α (x) α = x For α = 4, a second fixed point appears at x = 0.5.

30 Cooperative branching F α (x) α = x For α > 4, there are two stable fixed points and one unstable fixed point, which separates the domains of attraction of the other two.

31 Cooperative branching x z upp 0.4 z mid z low Fixed points of t X t = F α (X t ) for different values of α. α

32 A recursive tree representation We can construct the process (X N t ) t 0 from a stochastic flow (X N s,t) s t which is defined in terms of a Poisson point set ω. We aim to find a similar stochastic representation for solutions (µ t ) t 0 to the mean-field ODE (2). In the limit N, the state X N t (j) of a randomly chosen site j [N] is approximatey distributed according to µ t. We trace this site back in time till the first time when its state is changed by the application of a map g i,j. From this moment on, we trace back the sites i 1,..., i kg, and so on...

33 A recursive tree representation X t (j) has law µ t coop coop coop coop death coop death death death death This leads to a representation of (µ t ) t 0 in terms of a marked branching process.

34 A recursive tree representation Let T be the set of all finite words i = i 1 i n (n 0) made up from the alphabet N + = {1, 2,...}. We view T as a tree with root, the word of length zero, in which each individual i has infinitely many offspring i1, i2,... Let r := g G r g and let (γ i ) i T be i.i.d. with law P[γ i = g] = r 1 r g (g G). We inductively define a random subtree T T which contains the root and satisfies ij T iff i T and j k i, where k i := k γi is the integer such that γ i : S k S. Then T is the family tree of a branching process with maps (γ i ) i T attached to its vertices, such that the individual i has k i offspring.

35 A recursive tree representation For any subtree U T that contains the root, we write U := {ij T\U : i U}. Let (σ i ) i T be i.i.d. exponentially distributed random variables with mean r 1, independent of (γ i ) i T. We interpret σ i as the lifetime of i and let τi 1 i n := σ + σ i1 + + σ i1 i n 1 and τ i := τi + σ i denote its birth and death time. Then T t := {i T : τ i t} and T t denote the set of individuals that have died before time t resp. are alive at time t. In particular, ( Tt ) is a branching process where each individual i gives with rate r g birth to offspring i1,..., ik g, for each g G. t 0

36 A recursive tree representation X law µ t coop coop coop coop death coop death death death death X 32 X 21 X 23 i.i.d. µ 0 X 131 X 132 X 133

37 A recursive tree representation We let F t := σ ( ( T s ) 0 s t, (γ i ) i Tt ) denote the filtration generated by the branching process ( T t ) t 0 as well as the maps attached to the particles that have died by time t.

38 A recursive tree representation Given a finite subtree U T that contains the root, we define a map G U : S U S by G U := x where (x i ) i U satisfy x i = γ i (x i1,..., x iki ) (i U). In particular, we set G t := G Tt. Theorem [Mach, Sturm & S. 18] Assume that g G r g k g <. Then the solution to the mean-field equation (2) is given by µ t = E [ T Gt (µ 0 ) ] (t 0), i.e., µ t = P[X ] where (X i ) i Tt Tt satisfy (i) Conditional on F t, the (X i ) i Tt are i.i.d. with law µ 0. (ii) X i = γ i (X i1,..., X iki ) (i T t ).

39 A recursive tree representation In the special case that k g = 1 for each g G, the mean-field ODE (2) is just the backward equation of a continuous-time Markov chain where each map g G is applied with Poisson rate r g. We can think of the collection of random variables (γ i, σ i ) i T as a generalization of the Poisson construction of a continuous-time Markov chain, where time now has a tree-like structure.

40 Unique ergodicity Lemma Assume that P [ t < s.t. G t is constant ] = 1. Then the mean-field ODE (2) has a unique fixed point ν and solutions started in an arbitrary initial law µ 0 satisfy µ t ν t 0, where denotes the total variation norm.

41 Unique ergodicity For our process with cooperative branching and deaths, say that S T is a good subtree if S and (i) γ i death for all i S (ii) i S, {i1, i2, i3} S is either {i1} or {i2, i3}. Lemma The following events are a.s. equal: (i) T contains no good subtrees. (ii) G t is constant for some t <. Moreover, P[T contains a good subtree] > 0 iff α 4.

42 Good subtrees X law µ t coop coop coop coop death coop death death death death X 32 X 21 X 23 X 131 X 132 X 133 i.i.d. µ 0

43 Good subtrees X law µ t X 32 X 21 X 23 X 131 X 132 X 133 i.i.d. µ 0

44 Good subtrees a good subtree

45 Good subtrees dual process Y t

46 A Recursive Distributional Equation Fixed points ν of the mean-field ODE (2) solve the Recursive Distributional Equation (RDE) µ = r 1 g G T g (µ). (3) For each solution ν to the RDE, it is possible to define a collection of random variables (γ i, X i ) i T such that: (i) (γ i ) i T is an i.i.d. collection of G-valued random variables with law P[γ i = g] = r 1 r g (g G). (ii) For each finite subtree U T that contains the root, (X i ) i U are i.i.d. with common law ν and independent of (γ i ) i U. (iii) X i = γ i (X i1,..., X iki ) (i T). Following Aldous and Bandyopadhyay (2005), we call such a collection of r.v. s a Recursive Tree Process (RTP).

47 Recursive Tree Processes Let X have law µ. Then the RDE (3) says X = d γ ( ) X 1,..., X kγ with X 1, X 2,... i.i.d. µ, indep. of γ with P[γ = g] = r 1 r g. We can think of fixed points ν of the mean-field ODE (2) as a generalization of the invariant law of a (continuous-time) Markov chain. Then a Recursive Tree Process (RTP) is a generalization of a stationary (continuous-time) Markov chain. If we add independent exponentially distributed lifetimes (σ i ) i T as before, then for each t 0: Conditional on the σ-field F t of events measurable before time t, the (X i ) i Tt are i.i.d. with common law ν.

48 Endogeny Let (γ i, X i ) i T be the Recursive Tree Process (RTP) corresponding to a solution ν of the Recursive Distributional Equation (RDE) (3). Following Aldous and Bandyopadhyay (2005), we say that the RTP correspondig to ν is endogenous if X is measurabe w.r.t. the σ-field generated by the maps (γ i ) i T. In [AB05], it is proved that endogeny is equivalent to bivariate uniqueness.

49 The n-variate ODE Recall that the mean-field ODE (2) describes the Markov process (X N t ) t 0 on the complete graph in the mean-field limit N. The Markov process (X N t ) t 0 is defined in terms of a stochastic flow (X N s,t) s t. The stochastic flow (X N s,t) s t contains more information than the Markov process (X N t ) t 0 alone; in particular, the stochastic flow provides us with a natural way of coupling processes with different initial states. We would like to understand this coupling in the mean-field limit N.

50 The n-variate ODE For each measurable map g : S k S and n 1, we define an n-variate map g (n) : (S n ) k S n by g (n)( x 1,..., x n ) := ( g(x 1 ),..., g(x n ) ) (x 1,..., x n S k ). Let G and (r g ) g G be as before. We will be interested in the n-variate ODE t µ(n) t = g G r g { Tg (n)(µ (n) t ) µ (n) } t (t 0), that describes the mean-field limit of n coupled Markov processes, that are constructed from the same stochastic flow but have different initial states.

51 The n-variate ODE Let M 1 sym(s n ) be the space of all probability measures on S n that are symmetric under a permutation of the coordinates. For any µ M 1 (S), let M 1 sym(s n ) µ be the set of all symmetric µ (n) whose one-dimensional marginals are given by µ. Let S n diag := { (x 1,..., x n ) S n : x 1 = = x n}. Observations Let (µ (n) t ) t 0 solve the n-variate ODE. Then: Its m-dimensional marginals solve the m-variate ODE. µ (n) 0 M 1 sym(s n ) implies µ (n) t M 1 sym(s n ) (t 0). If ν solves the RDE (3), then µ (n) 0 M 1 sym(s n ) ν implies µ (n) t M 1 sym(s n ) ν (t 0). If µ (n) 0 is concentrated on Sdiag n then so is µ(n) t (t 0).

52 The n-variate ODE In particular, these observations show that if ν (n) solves the n-variate RDE, then its marginals must solve the RDE (3). Conversely, if ν solves the RDE (3) and X is a random variable with law ν, then ν (n) := P [ (X,..., X ) ] solves the n-variate RDE. Question Are all fixed points of the n-variate RDE of this form?

53 The bivariate ODE for cooperative branching x z upp 0.4 z mid z low Fixed points of t X t = F α (X t ) for different values of α. α

54 The bivariate ODE for cooperative branching For our system with cooperative branching and deaths, set z low := 0, z mid := α, and z upp := α, where z mid and z upp are only defined for α 4 and satisfy z mid = z upp for α = 4 and z mid < z upp for α > 4. Let ν low, ν mid, ν upp be the measures on {0, 1} with these intensities. Then ν low, ν mid, ν upp constitute all fixed points of the mean-field ODE (2).

55 The bivariate ODE for cooperative branching Proposition The measures ν (2) low, ν(2) mid, ν(2) upp defined as ν (2) low := P[ (X, X ) ] with P[X ] = ν low etc. are fixed points of the bivariate ODE. In addition, for α > 4, there exists one more fixed point ν (2) mid M1 sym({0, 1} 2 ) that has marginals ν mid but differs from ν (2) mid. Any solution (ν (2) t ) t 0 to the bivariate ODE with µ (2) 0 M 1 sym({0, 1}) νmid and µ (2) 0 ν (2) mid satisfies µ (2) t = ν (2) t mid. Unlike ν (2) mid, the fixed point ν(2) mid is not concentrated on S diag.

56 The bivariate ODE for cooperative branching Interpretation For large N, let x, x {0, 1} N be initial states such that 1 N x(i) = z mid = 1 N x (i), N N and i=1 1 N i=1 N 1 {x(i) x (i)} > 0, i=1 but arbitrarily small. Then 1 N N 1 {X N 0,t (x)(i) X N 0,t (x )(i)} i=1 converges in probability as N and then t to ν (2) mid ({(0, 1), (1, 0)}) > 0. In particular, the evolution under the stochastic flow is unstable in the sense that small differences in the initial states are multiplied, provided the initial density is z mid.

57 Endogeny Recall that a Recursive Tree Process (RTP) (γ i, X i ) i T is endogenous if X is measurable w.r.t. the σ-field generated by (γ i ) i T. Theorem [AB 05, MSS 18] For any solution ν to the RDE (3), the following statements are equivalent. (i) The RTP corresponding to ν is endogenous. (ii) The measure ν (2) is the only solution of the bivariate RDE in the space M 1 sym(s 2 ) ν. In our example of a system with cooperative branching and deaths, the RTPs corresponding to ν low and ν upp are endogenous, but for α > 4, the RTP corresponding to ν mid is not endogenous.

58 A higher-level ODE Recall that for each measurable map g : S k S, we have defined a measurable map ǧ : M 1 (S) k M 1 (S) by ǧ(µ 1,..., µ k ) := P [ g(x 1,..., X k ) ] where X 1,..., X k are indep. with P[X i ] = µ i. In particular, T g : M 1 (S) M 1 (S) is defined by T g (µ) := ǧ(µ,..., µ), and the mean-field ODE reads t µ t = { } r g Tg (µ t ) µ t g G (t 0). We will be interested in the higher-level ODE t ρ t = { } r g Tǧ (ρ t ) ρ t (t 0), g G with ρ t M 1 (M 1 (S)).

59 A higher-level ODE Let ξ be a M 1 (S)-valued random variable, i.e., a random probability measure on S, and let ρ M 1 (M 1 (S)) denote its law. Conditional on ξ, let X 1,..., X n be independent with law ξ. Then ρ (n) := P [ (X 1,..., X n ) ] = E [ ξ ξ] }{{} n times is called the n-th moment measure of ξ. Lemma If (ρ t ) t 0 solves the higher level ODE, then its n-th moment measures solve the n-variate ODE.

60 Higher-level RTPs Theorem [MSS 18] Let ν be a fixed point of the mean-field ODE (2). Let (γ i, X i ) i T be the RTP corresponding to ν. Set and ν := P[ξ ] ξ i := P [ X i (γ ij ) j T ]. and ν := P[δ X ]. Then ν, ν are fixed points of the higher-level ODE and (ˇγ i, ξ i ) i T and (ˇγ i, δ Xi ) i T are higher level RTPs corresponding to ν, ν. The original RTP is endogenous if and only if ν = ν. Remark 1 ν and ν correspond to minimal and maximal knowledge about X. The former describes the knowledge contained in (γ i ) i T, the latter represents perfect knowledge. Remark 2 ν (n) := P [ (X,..., X ) ] where X has law ν, in line with earlier notation.

61 The convex order [Strassen 65, MSS 18] Let S be a compact metrizable space and let C cv (M 1 (S)) denote the space of all convex continuous φ : M 1 (S) R. Then for ρ 1, ρ 2 M 1 (M 1 (S)), the following are equivalent: (i) φ dρ 1 φ dρ 2 for all φ C cv (M 1 (S)). (ii) There exists a random variable X and σ-fields F 1 F 2 such that ρ i = P [ P[X F i ] ] (i = 1, 2). [MSS 18] If ρ M 1 (M 1 (S)) ν solves the higher-level RDE, then ν cv ρ cv ν, where cv denotes the convex order.

62 The higher-level ODE In our example of a system with cooperative branching and deaths, we identify M 1 ({0, 1}) µ µ({1}) [0, 1] and correspondingly M 1 (M 1 ({0, 1})) = M 1 [0, 1]. If g = cob, then ǧ : [0, 1] 3 [0, 1] is given by ǧ(ω 1, ω 2, ω 3 ) := P[X 1 (X 2 X 3 ) = 1] with P[X i = 1] = ω i, which gives ǧ(ω 1, ω 2, ω 3 ) = ω 1 + (1 ω 1 )ω 2 ω 3 and, for any ρ M 1 [0, 1], Tǧ (ρ) = P [ ω 1 +(1 ω 1 )ω 2 ω 3 ] with ω 1, ω 2, ω 3 i.i.d. with law ρ.

63 The higher-level ODE Proposition For our system with cooperative branching and deaths, for α > 4, the higher-level ODE t ρ t = α { Tǧ (ρ) ρ } + { δ 0 ρ } has precisely four fixed points. Three trivial fixed points of the form ρ = ν... = (1 z... )δ 0 + z... δ 1 with z... = z low, z mid, z upp, and a nontrivial fixed point ν mid which is the law of the [0, 1]-valued random variable P [ X = 1 (γ i ) i T ] where (γ i, X i ) i T is the RTP with P[X = 1] = z mid.

64 Numerical results Inductively define measures µ n as µ 0 = δ zmid and Then µ n = α α + 1 T ǧ (µ n 1 ) + 1 α + 1 δ 0. P [ X = 1 {γi1 i m : m < n} ] = µ n n ν mid. We plot the distribution functions F n (s) := µ n ( [0, s] ) ( s [0, 1] ) for the parameters α = 9/2, z mid = 1/3, z upp = 2/3 and various values of n.

65 Numerical results F

66 Numerical results F

67 Numerical results F

68 Numerical results F

69 Numerical results F

70 Numerical results F

71 Numerical results F

72 Numerical results F

73 Numerical results F

74 Numerical results F

75 Numerical results F

76 Numerical results F

77 Numerical results α =

78 Numerical results α =

79 Numerical results α =

80 Numerical results α =

81 Numerical results α =

82 Numerical results α =

83 Numerical results α =

84 Numerical results α =

85 Numerical results α =

86 Numerical results α =

87 Numerical results α =

88 Numerical results α =

89 Numerical results α =

90 Numerical results α =

Recursive tree processes and the mean-field limit of stochastic flows

Recursive tree processes and the mean-field limit of stochastic flows Tibor Mach Anja Sturm Jan M. Swart December 26, 2018 Abstract Interacting particle systems can often be constructed from a graphical