distribution dependent sdes driven by additive fbm · 2021. 7. 6. · fbm can also be seen as a...

$: Distribution dependent SDEs driven by additive fBm · 2021. 7. 6. · FBm can also be seen as a case study of what should happen for more complicated choices (mixed fBm, fractional$
Distribution dependent SDEs driven by additive fBm

Lucio Galeati

Based on ongoing joint work with F. Harang and A. Mayorcas

8 March 2021

CIRM - Pathwise stochastic analysis and applications

Lucio Galeati Distribution dependent SDEs driven by additive fBm 1 / 22

Mean field limits

In classical stochastic analysis there is a well-developed theory connecting:

a) N-particle interacting system (IPS) driven by i.i.d. BMs:

dX i ,Nt = B(X i ,N

t , µNt )dt + dW i ,Nt , µNt :=

1

N

N∑i=1

δX i,Nt

;

b) McKean-Vlasov equation (MKV), also called distribution-dependentSDE (DDSDE) or nonlinear process (L(X ) = law of X )

dXt = B(Xt ,L(Xt))dt + dWt ;

c) Nonlinear Fokker-Planck equation (FP):

∂tµt +∇ · (B(·, µt)µt) =1

2∆µt .

Roughly speaking: in the mean field limit N →∞, the one particle X 1,N

of (IPS) converges in law to X solving (MKV), while µNt converges to µtsolving (FP); if X solves (MKV), then µt = L(Xt) solves (FP).


Mean field limits - II

Meaning: macroscopic description of the large system (IPS) can be givenby either (MKV) or (FP), reducing its complexity.

Of most interest is the case of pairwise interacting particles, possibly inthe presence of a confining potential:

dX i ,Nt =

1

N

N∑j=1

K (X i ,Nt − X j ,N

t )dt + V (X i ,Nt )dt + dW i ,N

t

which corresponds to B(x , µ) = (K ∗ µ)(x) + V (x).

Applications:

Plasma physics and astrophysics: Poisson kernel K (x) ∼ x/|x |d ;

Aggregation models: typically K = −∇F , V = −∇G ;

2D Euler/Navier-Stokes: Biot-Savart kernel K (x) ∼ x⊥/|x |2

Agent-based models: swarming, opinion dynamics, etc.


Some classical and modern results

Sznitman (Dobrushin for W = 0): full correspondence(IPS)-(MKV)-(FP) for globally Lipschitz B; directcomparison/coupling method, use of Wasserstein distances.

Gartner (1988): continuous coefficients with coercivity andmonotonicity properties.

Jabin, Wang (2018) (relative entropy), Serfaty (2020) (modulatedenergy): MFL µN → µ and propagation of chaos for irregular kernels(e.g. K ∈W−1,∞).

Rockner, Zhang (2018); Huang, Wang (2019); Mishura, Veretennikov(2020); Chaudru de Raynal (2020): study wellposedness of (MKV) forsingular coefficients B under several structural assumptions.

Hoesema, Holding, Maurelli, Tse (2020): large deviations for µN → µfor singular B (e.g. K ∈ Lqt L

px ).

All modern methods rely on either the connection to the PDE (FP) (whichis the easiest object to treat) or the power of Ito calculus (Zvonkin transf.).


What happens when W is not sampled as BM?

Coghi, Deuschel, Friz, Maurelli (2020) showed (revisiting ideas fromTanaka, Sznitman) that if B is globally Lipschitz, then the connectionbetween (IPS) and (MKV) holds for any given (cadlag) W .

Once written in integral form, (MKV) becomes

Xt = ξ +

∫ t

0B(Xs , µs)ds + Wt , µs = L(Xs)

where the imput data (ξ,W ) are independent, ξ with prescribed law µ0.

Remarks:

no assumption needed on W in order for (MKV) to be meaningful;

if W is not Brownian, the connection to (FP) is lost;

results for singular B exploit crucially this connection: onceuniqueness for L(Xt) holds, (MKV) can be treated as a standardSDE; Zvonkin transform also strongly depends on PDE theory.

there are now examples of non-Markovian, non-martingale choices ofW for which ordinary SDEs with singular coefficients can be solved,in a regularisation by noise fashion.


The problem

In view of the above, we want to study (MKV) for W sampled as afractional Brownian motion (fBm) of parameter H ∈ (0, 1). Reasons:

similar features to standard BM: Gaussian, self-similar, with almostH-Holder continuous trajectories.

for H 6= 1/2 it’s not Markovian, nor a semimartingale; both Itocalculus and PDE techniques break down. Thus alternative methodsmust be developed to handle singular drifts.

it is known, after the pioneering work of Catellier and Gubinelli (seelater) that fBm has a strong regularising effect on SDEs.

FBm can also be seen as a case study of what should happen for morecomplicated choices (mixed fBm, fractional processes, etc).

Our goal: establish wellposedness for (MKV) for W sampled as fBm, fora suitable class of B, which must at least include the case B = K ∗ µ+ Vfor non-Lipschitz coefficients K , V (possibly very singular).


Our main result

Here d1 = 1-Wasserstein distance on P1; ‖ · ‖α is the Cα = Bα∞,∞ norm.

Theorem

Let H ∈ (0, 1) and assume K ,V ∈ Cα for α > 1− 1/(2H).

Wellposedness: for any µ0 ∈ P1 there exist a strong solution X tothe DDSDE, which is unique in law; set µt = L(Xt);

Stability: given K i ,V i with i = 1, 2, denoting by X i , µi theassociated solutions, it holds

supt∈[0,T ]

d1(µ1t , µ

2t ) . d1(µ0, ν0) + ‖K 1 − K 2‖α + ‖V 1 − V 2‖α.

If X i are solutions defined on the same probability space, then

E[‖X 1· − X 2

· ‖CT

]. E[|X 1

0 − X 20 |] + ‖K 1 − K 2‖α + ‖V 1 − V 2‖α.

The constants can be taken uniform over ‖K i‖α, ‖V i‖α ≤ M, for anyfixed M > 0.

The statement can be given for more general B, see below.Lucio Galeati Distribution dependent SDEs driven by additive fBm 7 / 22

Recap: fBm and transport distances

Fractional Brownian motion of Hurst parameter H ∈ (0, 1) on R is theunique centered Gaussian process W with covariance

E[WtWs ] =1

2

(|t|2H + |s|2H − |t − s|2H

);

fBm on Rd is given by i.i.d. components distributed as above.

It satisfies a strong form of local nondeterminism (LND), related toits regularising properties and its local time.

There is a version of Girsanov theorem for fBm.

Let P1(Rd) be the set probability measures on Rd with finite firstmoment; metric space endowed with the 1-Wasserstein distance

d1(µ, ν) := infm∈Π(µ,ν)

∫|x − y |m(dx ,dy)

where Π(µ, ν) denotes the set of all couplings of (µ, ν).


Recap: study of (MKV) in the Lipschitz setting

Assume B satisfies the Lipschitz-type condition

|B(x , µ)− B(y , ν)| ≤ C (|x − y |+ d1(µ, ν)).

Existence and uniqueness for (MKV) follows if the map I given by

µ 7→ b(t, ·) := B(·, µt) 7→ I(µ)t := L(Xt)

can be shown to be a contraction from C ([0,T ];P) to itself.

Let X i be solutions associated to µit = L(X it ), bi (t, x) = B(x , µit), then bi

are Lipschitz with constant C ; stability estimates for ODEs give

supt∈[0,T ]

|X 1t − X 2

t | ≤ eCT(|X 1

0 − X 20 |+ T sup

t,x|b1(t, x)− b2(t, x)|

)and so applying the hypothesis on B, taking expectation

supt∈[0,T ]

d1(I(µ)t , I(ν)t) .C ,T d1(µ0, ν0) + supt∈[0,T ]

d1(µt , νt)

Working iteratively on sufficiently short intervals allows to conclude.Lucio Galeati Distribution dependent SDEs driven by additive fBm 9 / 22

Key features of the argument

The Lipschitz could be recast as follows:

‖B(·, µ)‖W 1,∞ ≤ C , ‖B(·, µ)− B(·, ν)‖∞ ≤ C d1(µ, ν).

The proof basically requires the following ingredients:

a) For each µ, the drift B(·, µ) belongs to a Banach space E , in which asolution theory for the SDE holds (here E = W 1,∞).

b) Stability estimates for the SDE are in a weaker norm ‖ · ‖F (hereF = C 0) and µ 7→ B(·, µ) is Lipschitz w.r.t ‖ · ‖F .

Tipically, F corresponds to 1 degree less than E ; this is a general fact: inDiPerna-Lions theory, E ∼W 1,p, F ∼ Lp; for RDEs with level-n roughpaths, E ∼ Cn+1, F ∼ Cn.

On the other hand, Wasserstein distance ”eats one derivative”: by K-Rduality, ‖b ∗ (µ− ν)‖C0 ≤ ‖b‖W 1,∞d1(µ, ν).

Conclusion: for such a strategy to work, we need stability estimates forsingular SDEs driven by fBm.


Singular SDEs driven by fBm

Consider an SDE driven by additive fBm (Cauchy problem in integral form)

Xt = x +

∫ t

0b(Xs)ds + Wt . (1)

The drift b is assumed not be Lipschitz (nor W 1,p); if H < 1/2, b can bepurely distributional, in which case the integral above is only formal.

Theorem (Catellier, Gubinelli, 2016)

If b ∈ Cα with α > 1− 1/(2H), then the SDE admits a pathwisemeaningful interpretation; for any x ∈ Rd , strong existence andpath-by-path uniqueness holds for the Cauchy problem (1).

The above also implies uniqueness both pathwise and in law.The result can be extended to time dependent drifts (see also [GG20]):

if H > 1/2, one can take b ∈ CαHt C 0x ∩ C 0

t Cαx for α as above;

if H < 1/2, can take b ∈ Lqt Cαx for α− 1

Hq > 1− 12H .

It can also be easily extended to random initial data.Lucio Galeati Distribution dependent SDEs driven by additive fBm 11 / 22

Meaning of the SDE

The Cauchy problem (1) has a pathwise interpretation in terms ofnonlinear Young integration. If b ∈ Cα, α > 1− 1/(2H), the random field

(t, x) 7→ TW b(x , t) :=

∫ t

0b(x + Ws)ds

is well-defined (even when α < 0!) and enjoys space-time regularity: P-a.s.

|Tws,tb(x)− Tw

s,tb(x , s)− Twb(y , t) + Twb(y , s)| . |t − s|γ |x − y |.

In turn, this allows to define rigorously the process t 7→∫ t

0 b(Xs)ds as alimit of Riemann-Stjeltes sums whenever X = W + θ with θ ∈ Cγt .

Heuristic : if X “looks like W up to a more regular remainder”, then it hassimilar regularising properties, allowing to define TXb as well.

No adaptability condition whatsoever is required on θ nor X fixed TW (ω)b, can carry pathwise analysis on Eω := W (ω) + Cγt .

For b as above, Girsanov transform can be applied, thus a posteriorithe unique solution X has the same pathwise properties as W .


Regularizing properties of fBm paths

The statement for TW b is one istance of a wider class of estimates:

Proposition

Let H ∈ (0, 1), bi ∈ L∞t Cαx for α > −1/(2H). There exists γ > 1/2 s.t.

E[∣∣∣ ∫ t

s(b1 − b2)(s,Ws)ds

∣∣∣p]1/p

. |t − s|γ‖b1 − b2‖L∞t Cαx .

notation∫ t

0 b(s,Ws)ds is purely formal: it’s not a well-definedLebesgue integral, rather the unique limit of smooth approximations;

Kolmogorov continuity criterion implies that the random process∫ ·0 b(s,Ws)ds has almost γ-Holder trajectories;

exponential integrability can also be shown;

the estimate holds for a more general class of Gaussian processessatisfying a (LND) condition; see [GG20], [HP20].

As mentioned before, once Girsanov is available, similar bounds hold forW replaced by X solution to the SDE.


Stability estimates for singular SDEs

Assumption

Let b ∈ E for E = CαHt C 0x ∩ C 0

t Cαx if H > 1/2; E = Lqt C

αx if H < 1/2,

parameters (α, q) as before. Shorthand notation ‖b‖E .

Theorem (G., Harang, Mayorcas)

Let T <∞, M > 0; let bi be as above with ‖bi‖E ≤ M and denote by X i

the associated solutions; there exists γ > 1/2 s.t. for any p ∈ [1,∞)

E[‖X 1 − X 2‖pγ;[0,τ ]

]1/p.p E

[‖X 1

0 − X 20 ‖p]1/p

+ supt∈[0,τ ]

‖b1t − b2

t ‖α−1

uniformly in τ ∈ [0,T ].

these solutions are the only possible limit of smooth approximations;

stability is in Cα−1x while b ∈ Cαx , as desired;

can be used in order to numerically simulate singular SDEs.


Heuristic proof

Let X i be solutions associated to bi ; we know that they both have thesame sample path properties of W by Girsanov. Set Y = X 1 − X 2, then

Yt = Y0 +

∫ t

0[b1(s,X 1

s )− b1(s,X 2s )]ds +

∫ t

0[b1 − b2](s,X 2

s )ds.

Formally, we expect the first integral to behave like∫ t

0Db1(s,X 1

s ) · (X 1s − X 2

s )ds ∼∫ t

0Ads · Ysds, At :=

∫ t

0Db1(s,Ws)ds

while the second one should behave instead like

ψt =

∫ t

0[b1 − b2](s,Ws)ds.

The equation for Y corresponds heuristically to dYt = AdtYt + dψt , whereby the results for W we expect A, ψ ∈ Cγt with γ > 1/2 equation for Y meaningful in the Young sense, pathwise estimates areavailable. Take expectation from these to conclude.


Proof of the main result (sketch)

With stability estimates for singular SDEs at hand, we can run a similarargument to the Lipschitz case. Let X i be solutions for the same initialdata X i

0 = ξ, µi = L(X it ), then:

for bit(·) = B(·, µit) = K ∗ µit + V , the weaker norm Cα−1x nicely

balances the 1-Wasserstein distance:

‖b1t − b2

t ‖Cα−1x. ‖K‖Cαx d1(µ1

t , µ2t )

taking p = 1 in the stability estimate, for any τ ∈ [0,T ] we get

supt∈[0,τ ]

d1(µ1t , µ

2t ) . τγ sup

t∈[0,τ ]‖b1

t − b2t ‖Cα−1

x. τγ sup

t∈[0,τ ]d1(µ1

t , µ2t )

which provides contractivity and the conclusion.

Solutions X i defined on different probability spaces can be dealt by acoupling argument; the case of different K i ,V i follows a similar procedure.


Regularity of the law of solutions

Our analysis provides some auxiliary results for SDEs driven by fBm:

Proposition

Let X be a solution to the SDE for b as above. Then L(X·) ∈ Lqt Lpx for

any (q, p) ∈ (1,∞)2 satisfying

1

q+

Hd

p> Hd .

If Hd < 1, then L(X·) ∈ Lqt L∞x for all q < (Hd)−1.

Proposition

Let X be the solution to the SDE for B as above. Then L(X·) ∈ L1tB

α1,1 for

all α < 1/(2H). Moreover for any such α there exists γ > 1/2 such that∥∥∥∫ t

sL(Xr )dr

∥∥∥Bα1,1

. |t − s|γ .

Proofs rely on Girsanov and duality arguments, no use of Malliavin calculus.


Sketch of proof

Fix −α > −1/(2H), let f = f (t, x) be a smooth function, then∣∣∣ ∫ T

0〈fs ,L(Xs)〉ds

∣∣∣ ≤ E[∣∣∣ ∫ T

0fs(Xs)ds

∣∣∣]≤ E

[( dPdQ

)2]1/2

E[∣∣∣ ∫ T

0fs(Ws)ds

∣∣∣2]1/2

. ‖f ‖L∞t C−αx

where in the second apssage we used Girsanov with new measure Q underwhich X is distributed as W . Identifying L1

tBα1,1 with the predual of

L∞t C−αx (and with a bit of work) the estimate then follows.

The proof of the statement in Lqt Lpx follows similar lines.

Even without relying on Girsanov, a similar proof (based on nonlinearYoung integration) allows to deduce bounds for the law of X = θ + W

with θ ∈ Cβt with β ≥ 1/2. In this case however L(X·) ∈ L1tB

α1,1 for

α < 12H −

12β . Observe that θ does not need to be adapted!

If θ is adapted, further improvement using stochastic sewing lemma.Lucio Galeati Distribution dependent SDEs driven by additive fBm 18 / 22

General assumptions on B

Consider now a general measurable B : [0,T ]× P1 → Cα.

Assumption

Case H ≤ 1/2: assume there exists h ∈ Lqt such that for all t, µ, ν:

i) ‖B(t, µ)‖α ≤ ht ;ii) ‖B(t, µ)− B(t, ν)‖α−1 ≤ htd1(µ, ν).

Moreover the parameters (α, q,H) satisfy α− 1Hq > 1− 1

2H .

Case H > 1/2: assume α > 1− 1/(2H) > 0 and that there exists aconstant C > 0 such that for all s, t, µ, ν:

i) ‖B(t, µ)‖α ≤ C ;ii) ‖B(t, µ)− B(s, ν)‖0 ≤ C (|t − s|αH + d1(µ, ν)α);iii) ‖B(t, µ)− B(t, ν)‖α−1 ≤ C d1(µ, ν).

Under the above assumption, it’s possible to prove wellposedness for theDDSDE and stability estimates for solutions associated to different B i .

Moreover d1 can be replaced by any other Wasserstein distance dp.


More refined results

Consider the purely convolutional case B = K ∗ µ; if the initial law µ0 isassumed to be more regular, we can slightly weaken the conditions on K .

Proposition

Let H ∈ (0, 1), K ∈ Bαp,p with divK ∈ L∞, µ0 ∈ P1 ∩ Lq; assume

α > 1− 1

2H,

1

p+

1

q≤ 1.

Then there exists a strong solution X to (MKV); moreover uniquenessholds in the class of solutions satisfying L(X ) ∈ L∞t Lq.

Proposition

Let H ∈ (0, 1/2), K ∈ Lp, µ0 ∈ P1 ∩ Lq; assume

Hd

p<

1

2,

1

q< 1− 1

d.

Then there exists a strong solution X to (MKV); moreover uniquenessholds in the class of solutions satisfying L(X ) ∈ L∞t Lq.


Open problems and future directions

Can one show that (MKV) is the mean field limit of (IPS)?If B is not Lipschitz, we currently don’t have a proof of that.

Can one improve estimates for L(X )?In analogy with BM, one expects L(X ) ∈ L1

tBα1,1 for α < 1/H.

Estimates between different solutions in the total variation norm?The classical settin relies on the maximum principle for the PDE.

Instead of the DDSDE, one could consider a singular SDE in thepresence of a control. What changes in this case?


Essential bibliography

P. E. Jabin, Z. Wang: Mean field limit for stochastic particle systems.Active Particles, Vol. 1

J. Hoeksema, T. Holding, M. Maurelli, O. Tse: Large deviations forsingularly interacting diffusions. arXiv:2002.01295 (2020).

A.S. Sznitman. Topics in propagation of chaos. In Ecole d’ete deprobabilites de Saint-Flour XIX (1989)

M. Coghi, J. D. Deuschel, P. Friz, M. Maurelli: PathwiseMcKean-Vlasov theory with additive noise. Annals of AppliedProbability (2020).

R. Catellier, M. Gubinelli: Averaging along irregular curves andregularisation of ODEs. Stochastic Processes and Applications (2016)

L. Galeati, G. Gubinelli, Noiseless regularisation by noise.arXiv:2003.14264 (2020).

F. A. Harang, N. Perkowski. C∞-regularization of ODEs perturbed bynoise. arXiv:2003.05816 (2020)


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