Stochastic Partial Di⁄erential Equations III

Stochastic Partial Di¤erential Equations III

Franco Flandoli, Scuola Normale Superiore

Bologna 2019

Plan of the lectures

The method of compactness Kolmogorov equations

Open problems of regularization by noise

Kolmogorov equations in Hilbert spaces

Consider a separable Hilbert space H, a linear operator

A : D(A) H !H, a bounded Borel measurable operator B : H !^H, and a selfadjoint bounded operator Q : H !H. We investigate the equation

∂tU(t, x) = ¹

2Tr QD²U(t, x) + h^Ax+B(x), DU(t, x)i U(0, x) = U₀(x).

The backward form of this equation corresponds to the SDE in the Hilbert space H

dXt = (AXt+B(Xt))dt+p QdWt

which may be the abstract formulation of an SPDE.

The main idea (Da Prato) to investigate

∂tU(t, x) = ¹

2Tr QD²U(t, x) + h^Ax+B(x), DU(t, x)i U(0, x) = U0(x).

is to solve the "linear" case (Ornstein-Uhlenbeck)

∂tV(t, x) = ¹

2Tr QD²V(t, x) + h^{Ax, DU}(t, x)i V(0, x) = V₀(x)

using Probability (Gaussian stochastic analysis) and the nonlinear case by perturbation, analytically.

Called (StV0) (x)the solution of

∂tV(t, x) = ¹

2Tr QD²V(t, x) + h^{Ax, DU}(t, x)i V(0, x) = V0(x)

we write the original equation in the perturbative form U(t, x) = (S_tU₀) (x) +

Z _t

0 (S_{t s}h^B( ), DU(s, )i) (^x)ds.

Linear case

The "linear" case (Ornstein-Uhlenbeck)

∂tV(t, x) = ¹

2Tr QD²V(t, x) + h^{Ax, DU}(t, x)i V(0, x) = V₀(x)

can be solved explicitly. We do not enter the nontrivial question of the meaning of the terms and in which sense it is satis…ed, but simply state, by analogy with the …nite dimensional case, that the solution to this equation is given by

V (t, x) =E[V0(X_t^x)]

where

dX_t^x =AX_t^xdt+p

QdWt, X₀^x =x or explicitly (we use semigroup theory here)

X^x =e^tAx+

Z t

e^{(t s)A}p QdW .

Consider for simplicity the diagonal case, A=A with compact resolvent, Ae_k = λke_k:

X_t^x =

∑

k

e ^{t λk}xk+

Z _t

0

e ^{(t s)λk}σkd β_s ek. It is well de…ned when

∑

Z T 0

e ^{2(t s)λk}σ²_kds <∞ namely

∑

1 e ^{2T λk} 2λk

σ²_k <∞ namely when

∑

σ²_k λk

< _∞.

Example

On the torus T^d =_R^d_/Z^d take H =L² T^d with zero mean (scalar valued, for simplicity; heat equation instead of Navier-Stokes), A=∆,

ek(x) =exp(2πik x), k 2Z^d Ae_k = 4π²j^kj^{2ek (namely λk} =4π²j^kj^2).

If we want to deal with space-time white noise, namely σ²_k =1, we need

∑

σ²_k λk

= ¹

4π²

∑

k2Z^dnf⁰g

1 j^kj² < _∞ hence it is true only in

d =1.

Summary of notations, equations, assumptions:

U(t, x) = (S_tU₀) (x) +

Z t

0 (S_{t s}h^B( ), DU(s, )i) (^x)ds (StV0) (x) =_E[V0(X_t^x)]

X_t^x =

∑

k

e ^{t λk}x_k+

Z t 0

e ^{(t s)λk}σkd β_s e_k

∑

σ²_k λk

<∞, B 2 B^b(H, H).

Clearly the di¢ culty in using iteratively the equation U(t, x) = (StU0) (x) +

Z t

0 (St sh^B( ), DU(s, )i) (^x)ds

is in the derivative DU(s, )on the RHS. The problem can be solved if we have an estimate of the form

k^DStV0k ^C(t) k^V0k where k kis some kind of uniform or L^p norm.

For general problems, Malliavin calculus or Bismut-Elworthy-Li formula are good tools. Here the process behind St is Gaussian and thus we have an explicit formula for DS_tV₀ which does not involve derivatives of V₀.

Introduce

Q(t) =

Z t 0

e^sAQe^sA ds.

It is the covariance operator, in the "space variable", of the process X_t^x =e^tAx+

Z _t

0

e^{(t s)A}p QdW_s EhD

X_t^x e^tAx, hE D

X_t^x e^tAx, kEi

= h^Q(t)h, ki^. If Q(t)is injective, we deduce

EhD

X_t^x e^tAx, Q(t) ^1/2fE D

X_t^x e^tAx, Q(t) ^1/2gEi

= h^{f , g}i^. The following fact is true: when Q(t)is injective, there is a rigorous de…nition, as an L²-limit, for the random variable formally denoted by

D

Q(t) ^1/2 X_t^x e^tAx , hE and it is centered Gaussian:

N 0,k^hk²H .

Theorem

Assume R ^etA R ^Q(t)^1/2 for t>0, and set Λ(t) =Q(t) ^1/2e^tA. Then

h^{h, D}(S_tV₀) (x)i = ^EhV0(X_t^x)^DΛ(t)h, Q(t) ^1/2 X_t^x e^tAx Ei . Proof.

S_t⁽ⁿ⁾V₀ (x) =E[V₀(X_t^{x ,n})]

X_t^x =

∑

e ^{t λk}x_k+

Z t 0

e ^{(t s)λk}σkd β_s e_k N e ^{t λk}xk,

Z t 0

e ^{2(t s)λk}σ²_kds = N mk(t, xk), σ²_k(t)

With the notation y = (y₁, ..., y_n),

S_t⁽ⁿ⁾V₀ (x)

=

Z

RⁿV₀(y)C(n, t)exp 1 2

∑

(y_k m_k(t, x_k))² σ²_k(t)

! dy

∂i S_t⁽ⁿ⁾V₀ (x)

=

Z

Rⁿ

(y_i m_i(t, x_k))e ^{t λi}

σ²_i (t) ^V0(y)C(n, t)

exp 1

2

∑

(yk mk(t, xk))² σ²_k(t)

! dy

= E V0(X_t^{x ,n})((X_t^{x ,n})_i mi(t, xk))e ^{t λi} σ²_i (t)

Hence

∂i S_t⁽ⁿ⁾V0 (x)

=

Z

Rⁿ

(yi mi(t, xk))e ^{t λi}

σ²_i (t) ^V0(y)C(n, t)

exp 1

2

∑

(y_k m_k(t, x_k))² σ²_k(t)

! dy

= _{E V}0(X_t^{x ,n})((X_t^{x ,n})_i m_i(t, x_k))e ^{t λi} σ²_i (t)

Hence

h^{h, D}(S_tV₀) (x)i = ^E

"

V₀(X_t^x)

∑

((X_t^x)_i mi(t, x_k))e ^{t λi} σ²_i (t) ^hi

#

= _E^hV₀(X_t^x)^DQ(t) ¹e^tAh, X_t^x e^tAxEi .

Now the question is how to use the formula

h^{h, D}(StV0) (x)i = ^EhV0(X_t^x)^DΛ(t)h, Q(t) ^1/2 X_t^x e^tAx Ei We know that

DΛ(t)h, Q(t) ^1/2 X_t^x e^tAx E

N 0,kΛ(t)hk²H . Hence we simply have

Theorem

jh^{h, D}(S_tV₀) (x)ij k^Λ(t)hkH Eh

j^V0(X_t^x)j^2i1/2. In particular,

k^D(StV0) (x)kH k^Λ(t)k_L(H ,H)k^V0k_∞

Z

H k^D(StV0) (x)k²H µ(dx) kΛ(t)k²_L(H ,H) Z

H Eh

j^V0(X_t^x)j^2iµ(dx)

The …rst inequality

k^D(StV0) (x)kH k^Λ(t)k_L(H ,H)k^V0k_∞ will be used in the C⁰-theory. The second one

Z

H k^D(S_tV₀) (x)k²H µ(dx) k^Λ(t)k²_L(H ,H) Z

H Eh

j^V0(X_t^x)j^2iµ(dx) in the L²-theory, taking as µ an invariant measure of X_t^x, because in such

a case _Z

H j^V0(X_t^x)j^2µ(dx) =

Z

H j^V0(x)j^2µ(dx) hence

Z

H k^D(StV0) (x)k²H µ(dx) k^Λ(t)k²_L(H ,H)k^V0k²L²(H ,µ).

The problem, as said above, is to have a good control of k^Λ(t)k_L(H ,H)

for t !^0,Λ(t) =Q(t) ^1/2e^tA, Q(t) =Rt

0 e^sAQe^sA ds.

If the noise is "regular in the space variable" then Q(t) ¹ is a very

"irregular" operator. Consequence: this approach works for cylindrical noise (space-time white noise) or little modi…cations but not in general.

Example: Q =I , A=A , t 2 [^{0, T}]: Q(t)ek =

Z t 0

e^2sAekds= ^{1 e}

2t λk

2λk

Λ(t)e_k =

p2λke ^{t λk} p1 e ^{2t λk} = p¹

tg(tλk), g(s) 1 k^Λ(t)k_L(H ,H)

p1 t.

g(s) =

p2se ^s

p1 e ^2s, g(s) 1

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5

x y

The analogous result for second derivatives is similar but more lengthy and we omit the proof.

Theorem

k, D²(S_tV₀) (x)h

= Eh

V₀(X_t^x) ^nDΛ(t)h, Q(t) ^1/2 X_t^x e^tAx E DΛ(t)k, Q(t) ^1/2 X_t^x e^tAx E

hΛ(t)h,Λ(t)ki^oi Theorem

D²(S_tV₀) (x) p

2kΛ(t)k²_L(H ,H)k^V0k_∞^.

With these results in our hands we approach U(t, x) = (StU0) (x) +

Z _t

0

(St sh^B( ), DU(s, )i) (^x)ds in the space

U 2^C [0, T]; C¹(H,R) under the assumption

B 2^C([0, T]; C(H, H)).

Theorem If limt!⁰Rt

0 k^Λ(s)k_L(H ,H)ds =0, given U0 2^C1(H,R)there is a unique solution U 2^C [0, T]; C¹(H,R) of the Kolmogorov equation.

The idea of proof is simply a …xed point in the space C [0, T]; C¹(H,R) . Contraction of the map

Λ : C [0, T]; C¹(H,R) !^C [0, T]; C¹(H,R) Λ(U) (t) = (StU0) (x) +

Z t

0 (St sh^B( ), DU(s, )i) (^x)ds is guaranteed by the estimate

k^DΛ(U₁) (t) DΛ(U₂) (t)k_∞

Z t

0 k^{DSt s}h^B( ), DU1(s, ) DU2(s, )ik_∞^ds

Z _t

0 k^Λ(t s)k_L(H ,H)kh^B( ), DU₁(s, ) DU₂(s, )ik_∞^ds where we have used

k^D(S_tV₀) (x)kH k^Λ(t)k_L(H ,H)k^V0k_∞^.

Putting together all the previous pieces plus some additional detail we get:

Theorem

Consider the SDE in Hilbert space H

dXt = (AXt +B(Xt))dt+dWt, X0 =x0

where W_t is cylindrical, A=A , A ¹ is trace class, B : H !^{H is} continuous and bounded. Then there exists a unique solution in law.

With some more e¤ort based on the second derivatives and the assumption B 2^Cb^α(H, H)

it is possible to prove pathwise uniqueness (Da Prato-F. JFA 2010). With more e¤ort one can work in L²(H, µ)where µ is the invariant measure of the linear equation and prove a similar result when

B 2^L∞(H, H)

(Da Prato-F.-Priola-Röckner, AoP 2013). However, the result in this case has been proved only for µ-a.e. x0 2^H.

Open problem 1

The main open problem is to extend previous results to drifts B relevant for Mathematical Physics.

An example like Navier-Stokes has the spaces and operators:

H L² (precisely vector valued, divergence free with b.c.) A ∆ (up to projection to divergence free …elds)

B(u) = u ru (as above)

du = (Au+B(u))dt+p QdWt.

The operator B is just "polynomial" in the variables (not irregular as a function only of class C^α(H, H)) but:

quadratically unbounded, opposite to k^B(u)kH C

de…ned only on subspaces (B : W^{s ,2} !H for s large enough), or with range in larger spaces (B : H !^{W r ,2).}

Think to the calculation

k^DΛ(U1) (t) DΛ(U2) (t)k_∞

Z _t

0 k^{DSt s}h^B( ), DU₁(s, ) DU₂(s, )ik_∞^ds

Z t

0 k^Λ(t s)k_L(H ,H)kh^B( ), DU₁(s, ) DU₂(s, )ik_∞^ds.

When B is not bounded, how to close the estimate?

When B is not de…ned over H but only on a smaller space, what kind of properties are needed to make estimates?

Da Prato and Debussche JMPA 2003 succeded to investigate precisely the case of 3D Navier-Stokes equations but the di¢ culty mentioned above re‡ected into poor estimates on the derivatives of Kolmogorov equations.

These estimates are not su¢ cient to prove uniqueness.

Thus the question of good results on Kolmogorov equations when B is like those of ‡uid mechanics (or other applications in Mathematical Physics) remains open.

Open problem 2

Another question is about degenerate Kolmogorov equations. This is motivated by the transport noise used for Euler equations:

d ω+u r^ωdt+

∑

σne_n r^{ω d β}n =0.

In abstract terms: the di¤usion coe¢ cient is state-dependent and vanishes when r^ω=0.

The associated Kolmogorov equation is

∂tU(t, ω) + h^u(ω) r^{ω, DU}(t, ω)i

= ¹ 2

∑

σ²_nh^en r^{ω, D}h^en r^{ω, DU}(t, ω)ii^.

It is a fully open problem how to study this equation in uniform (Hölder) topologies. We have done some preliminary work in L²(H, µ)spaces (F.-Luo, arXiv).

In …nite dimensions it would be dXt =b(Xt) +

∑

CnXt dWn

∂tU =b(x) r^U+¹ 2

∑

Cnx r (^Cnx r^U).

Open problem 3

Finally let us mention an open problem related to applications.

Assume an equation of the form

du= (Au+B(u))dt+p QdW_t

is used in some application, like weather or climate prediction.

Given an uncertain initial condition, namely a random variable u0, maybe our theorems guarantee well posedness of the SPDE.

Which is the probability to have, at the future time T , a value of some quantity larger than a thresold?

P(Φ(u(T)) >λ) =?

A classical method is to simulate several times the SPDE, namely for several realizations of the i.c. and the noise. Very expensive! Monte Carlo has a rate of convergence of order p¹

N.

In principle, the law of u(t), call it µ_t, satis…es the Fokker-Planck equation, in weak form

Z

H

F(t, x)µ_t(dx)

Z

H

F(0, x)µ₀(dx) =

Z t

0 (∂tF + L^F)µ_s(dx)ds where LF is the Kolmogorov operator. Can we simulate the

Fokker-Planck equation?

P(_Φ(u(T)) >λ) =µ_T f^y 2^{H :}Φ(y) >λg^. Or alternatively, can we simulate the Kolmogorov equation (it gives expected values)?

Since numerical simulation of parabolic PDEs is limited in practice to small dimension (usually 3), we have no chance to apply a trivial …nite dimensional approximation.

New ideas?

Maybe it is possible to design e¢ cient schemes based on U(t, x) = (S_tU₀) (x) +

Z _t

0 (S_{t s}h^B( ), DU(s, )i) (^x)ds.

Spectral methods based on Fourier analysis of L²(H, µ)? (cf.

Delgado-F., IDAQP 2016).

The author expresses his gratitude to the organizers of the School for this opportunity, to many participants for useful discussions and to Luigi Amedeo Bianchi who indicated several missprints.