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Maurelli - Regularisation by noise

The lectures of Mario Maurelli at the second workshop of the Bernoulli Center Program “New developments and challenges in Stochastic Partial Differential Equations”

Maurelli | Regularisation by noise | Lecture 1 | Monday July 22, 9:00–10:30

Joint work with Bagnara, Galeati.

I. Introduction

Regularisation by noise (RbN): possibly ill-posed ODE/PDE becomes well-posed or gains regularity by addition of a suitable noise.

Motivations: 1) surprising phenomenon (things become better adding noise); 2) noise may model disorder at small scales; 3) better understanding of properties of noise.

Main aim: show regularisation effect for 2d Euler with unbounded vorticity by non-smooth Kraichnan noise.

Plan: (today) II. Regularisation by noise, for ODE/linear PDEs; III. Kraichnan noise; (Lecture 2) IV. 2d Euler equation; V. Regularisation by noise of 2D Euler; (Lecture 3) VI. Further properties and results on reg. by Kraichnan.

II. RbN for ODEs and linear PDEs.

1. ODEs

ODE on $[0, T] \times \mathbb{R}^d$:

$\displaystyle \dot{X} (t) = b (t, X (t)), X (0) = x_0 .$

Counterexample for non-Lipshitz $b$:

$\displaystyle b (x) = | x |^{\alpha} \frac{x}{| x |} \mathbb{1}_{| x | \neq 0}, \qquad \alpha < 1, \qquad x_0 = 0.$

with multiple solutions:

$\displaystyle X (t) = c_{\alpha} (t - t_0)^{1 / (1 - \alpha)} \mathbb{1}_{t > t_0} v, \qquad v \in \mathbb{S}^{d - 1}, \qquad \forall t_0 \in [0, \infty] . $

(1)

SDE on $[0, T] \times \mathbb{R}^d$:

$\displaystyle \mathrm{d} X_t = b (t, X_t) \mathrm{d} t + \mathrm{d} W_t, \qquad X_0 = x_0$

where $b : [0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ given, $W$ $d$-dim Bronwian motion.

Literature: Zwokin '74, Veretennikov '80, Krylov-Röckner 2005, Flandoli–Gubinelli–Priola 2010, Catellier–Gubinelli 2016.

Theorem. (Krylov-Röckner) $b \in L^q_t L^p_x$ $p, q \in (2, + \infty)$ $2 / q + d / p < 1$ imples strong existence and pathwise uniqueness for all $x_0 \in \mathbb{R}^d$.

Under the same conditions, there exists also a stochastic flow of homeomeorphisms. $(t, x_0) \mapsto X_t^{x_0, \omega}$, i.e. a modification of this map which is a flow of homeomorphisms for $\mathbb{P}$-a.e. $\omega \in \Omega$.

Intuition on the mechanism of regularisation: the noise is “much stronger” than the drift. An heuristic computation:

$\displaystyle \mathrm{d} X_t = | X_t |^{\alpha} \frac{X_t}{| X_t |} \mathrm{d} t + \mathrm{d} W_t, \qquad X_0 = 0.$

Drift contribution is given by (1), i.e. $\mathrm{d} Y_t = | Y_t |^{\alpha} \frac{Y_t}{| Y_t |} \mathrm{d} t$ so $| Y_t | \lesssim t^{1 / (1 - \alpha)}$ while $| W_t | \approx t^{1 / 2}$ so

$\displaystyle | Y_t | \ll | W_t | \Leftrightarrow (1 - \alpha)^{- 1} > 1 / 2, \Leftrightarrow \alpha > - 1 \Leftrightarrow | x |^{\alpha} \frac{x}{| x |} \in L^p$

with $p > d$ (the Krylov-Röckner condition)

2. Stochastic continuity/transport equations

Deterministic case (formally). Consider the ODE

$\displaystyle \left\{ \begin{array}{l} \dot{X}^{x_0}_t = b (t, X^{x_0}_t)\\ \dot{X}^{x_0}_0 = x_0 \end{array} \right.$

with a corresponding flow of homeo/diffeomorphisms $(t, x) \mapsto X_t^x$.

Continuity equation ($\operatorname{div}b = 0$) :

$\displaystyle \partial_t \mu +\operatorname{div} (b \mu) = 0 \qquad \text{on $[0, T] \times \mathbb{R}^d$}$

$\mu = \mu (t, x)$, with $\mu (t, x) \mathrm{d} x = \mu (t, \mathrm{d} x)$. Equation for the evolution of the mass associated with the ODE given an initial mass $\mu_0$. The mass $\mu (t)$ at time $t$ will be given by $\mu (t) = (X_t)_{\#} \mu_0$ where $X_t : \mathbb{R}^d \rightarrow \mathbb{R}^d$ is the map sending initial conditions to the position of particles at time $t$.

For all test functions $\varphi \in C^{\infty}_c$, by the chain rule

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mu_t (\varphi) = \frac{\mathrm{d}}{\mathrm{d} t} \mu_0 (\varphi \circ X_t) = \mu_0 ((b (t, \cdot) \nabla \varphi) \circ X_t) = \mu_t (b (t, \cdot) \nabla \varphi)$

which is the distributional form of the continuity equation.

Transport equation:

$\displaystyle \partial_t \theta + b \cdot \nabla \theta = 0,$

the evolution of a passive scalar advected with the ODE. This is equivalent to the continuity equation in the case where $\operatorname{div}b = 0$. Note that $\theta$ is constant along the characteristics: $\theta (t, X^x_t) = \theta (0, x)$ for all $t \geqslant 0$.

Stochastic case:

$\displaystyle \left\{ \begin{array}{l} \mathrm{d} X_t^x = b (t, X^x) \mathrm{d} t + \sum_a \sigma_a (t, X^x) \mathrm{d} W_t^a, \qquad t \geqslant 0\\ X_0^x = x \end{array} \right.$

For amost all $\omega \in \Omega$, $X_t : \mathbb{R}^d \rightarrow \mathbb{R}^d$, it gives rise to a stochastic flow of homeo/diffeos: random evolution of particles. Letting $\mu (t) = (X_t)_{\#} \mu_0$ we get the stochastic continuity equation:

$\displaystyle \mathrm{d} \mu +\operatorname{div} (b \mu) \mathrm{d} t + \sum_a \operatorname{div} (\sigma_a \mu) \circ \mathrm{d} W^a = 0, \qquad \mu = \mu (t, x, \omega)$

it expresses the random evolution of mass $\mu_t^{\omega}$ associated with the SDE. Analogously we have the stochastic transport equation:

$\displaystyle \mathrm{d} \theta + b \cdot \nabla \theta \mathrm{d} t + \sum_a \sigma_a \cdot \nabla \theta \circ \mathrm{d} W^a = 0,$

expressing the random evolution of a passive scalar.

How do we prove these properties: test with test function and Ito formula for the Stratonovich integration (which is a first order expression), there is a corresponding Ito form of the equations but it involves second derivatives of the solution.

3. RbN for linear stochastic transport equation

$\displaystyle \mathrm{d} \theta + b \cdot \nabla \theta \mathrm{d} t + \sum_{a = 1}^d \nabla_a \theta \circ \mathrm{d} W^a = 0,$

where $W$ is a $d$-dimensional Brownian motion.

Theorem. (Flandoli–Gubinelli–Priola 2010) $b \in L^{\infty}_t (C^{0 +}_{x, \operatorname{bounded}})$, $\operatorname{div}b \in L^{2 +}_{t, x}$ then there exists strong and pathwise unique transport equation in $L^{\infty}_{t, x}$ for all $\theta_0 \in L^{\infty}_x$.

The strategy of proof uses the diffeomorphic flow of characteristics. Uniqueness follows from the relation $\theta (t, X^x_t) = \theta (0, x)$.

III. Non-smooth Kraichnan model of passive scalars

1. Isotropic structure of noise

$\dot{W}$ random velocity on $[0, T] \times \mathbb{R}^d$ field: Gaussian, centered, white in time and isotropic (i.e. translation and rotation invariant) in space and divergence free:

$\displaystyle \mathbb{E} [\dot{W} (t, x) \dot{W} (s, y)] = \delta (t - s) Q (x, y),$

where the covariance is translation invariant:

$\displaystyle Q (x, y) = Q (x - y)$

with Fourier transform

$\displaystyle \hat{Q} (\xi) = \langle \xi \rangle^{- d - 2 \alpha} \underbrace{\left( \mathbb{I}- \frac{\xi \xi^T}{| \xi |^2} \right)}_{\text{Leray projection giving div$\dot{V}$=0}}$

with $\langle \xi \rangle = (1 + | \xi |^2)^{1 / 2}$ and $\alpha \in (0, 1)$ (non-smooth regime). Fact: $Q$ satisfies:

$\displaystyle Q (G z) = G Q (z) G^T$

for all $G \in O (d)$. This implies that the law of $\dot{W} (t, x)$ is translation invariant and invariant under orthogonal transformation.

Structure of $Q$: due to the symmetries, it decomposes into parallel and orthogonal tensors (to $x$).

$\displaystyle Q (x) = Q_L (x) \frac{x x^T}{| x |^2} + Q_N (x) \left( \mathbb{I}+ \frac{x x^T}{| x |^2} \right),$

with scalar quantities $Q_L (x) = Q_L (| x |) = Q_{i i} (| x | e_i)$, the logitudinal component and $Q_N (x) = Q_N (| x |) = Q_{i i} (| x | e_j)$ (with $i \neq j$), the orthogonal component (see the book of Baxendale–Harris).

2. Kraichnan passive scalar model

Model of stochastic transport/continuity equation with no-drift and noise $W$

$\displaystyle \mathrm{d} \mu +\operatorname{div} (\mu \circ \mathrm{d} W) = 0$

on $[0, T] \times \mathbb{R}^d$ with $d \geqslant 2$. We can represent the noise as an infinite series of Brownian contributions:

$\displaystyle W (t, x) = \sum_{k = 1}^{\infty} \sigma_k (x) \mathrm{d} B^k_t,$

with $(B^k)_k$ i.i.d. Brownian motions and $(\sigma_k (x))_k$ appropriate vector fields. In this case we have

$\displaystyle \mathrm{d} \mu + \sum_{k = 1}^{\infty} \operatorname{div} (\sigma_k \mu) \circ \mathrm{d} B^k = 0.$

Lagrangian viewpoint: SDE

$\displaystyle \left\{ \begin{array}{l} \mathrm{d} X_t^x = \sum_k \sigma_k (t, X^x) \circ \mathrm{d} B_t^k = \circ \mathrm{d} W (t, X^x_t) \xequal[\operatorname{div}W = 0]{} \mathrm{d} W (t, X^x_t), \qquad t \geqslant 0,\\ X_0^x = x. \end{array} \right.$

Feature of this model (rigorously for $\alpha > 1$ where $\dot{W}$ is $C^1$ in space). Using the divergence

1. The one point motion $X^x_t$ is actually just a $d$-dimensional Brownian motion. $(X^x_t)_t$ is a martingale and

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} [X^x]_t = \sum_k \sigma_k (X^x_t) \sigma_k (X^x_t) = Q (0) = c\mathbb{I}.$

2. The two point motion (Baxendale–Harris in the smooth case). Closed SDE for the difference $V = X^x - X^y$ of two points. $V$ is a martingale and

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} [V]_t = \sum_k [\sigma_k (X^x_t) - \sigma_k (X^y_t)] [\sigma_k (X^x_t) - \sigma_k (X^y_t)]$ $\displaystyle = 2 Q (0) - 2 Q (X^x_t - X^y_t) = 2 Q (0) - 2 Q (V_t)$

so $V$ satisfies in law the following SDE:

$\displaystyle \mathrm{d} V = g (V) \mathrm{d} B$

where $B$ is a $d$-dimensional Brownian motion and $g : \mathbb{R}^d \rightarrow \mathbb{R}^d \times \mathbb{R}^d$ such that $g (x)^2 = 2 Q (0) - 2 Q (x)$, and

$\displaystyle g (x) = [2 (Q_L (0) - Q_L (| x |))]^{1 / 2} \frac{x x^T}{| x |^2} + [2 (Q_N (0) - Q_N (| x |))]^{1 / 2} \left( \mathbb{I}- \frac{x x^T}{| x |^2} \right)$

Maurelli | Regularisation by noise | Lecture 2 | Thursday July 25, 11:00–12:30

Recall the setting:

Noise

Gaussian, divergence free, centered

$\displaystyle \mathbb{E} [\dot{W} (t, x) \dot{W} (s, y)] = \delta (t - s) Q (x - y)$

with

$\displaystyle \hat{Q} (n) = \langle n \rangle^{- d - 2 \alpha} \left( \mathbb{I}- \frac{n n^T}{| n |^2} \right), \qquad W (t, x) = \sum_k \sigma_k (x) W^k (t)$

$\alpha > 1$ the noise is $C^1$ in space, and $\alpha \in (0, 1)$ where the noise is $C^{\alpha -}$ in space. Isotropic structure

$\displaystyle Q (x) \quad = \quad \underbrace{Q_L (| x |) \frac{x x^T}{| x |^2}}_{\text{longitudinal component}} \quad + \quad \underbrace{Q_N (| x |) \left( \mathbb{I}- \frac{x x^T}{| x |^2} \right)}_{\text{normal component}}$

Kraichnan model of passive scalar

(Kraichnan 1968)

$\displaystyle \mathrm{d} \mu +\operatorname{div} (\mu \circ \mathrm{d} W) = 0,$

with noise $W$ as above.

Lagrangian viewpoint: (formally since the vectorfields are not necessarily smooth)

$\displaystyle \mathrm{d} X^x_t = \circ \mathrm{d} W_t (t, X^x_t) = \sum_k \sigma_k (X^x_t) \circ \mathrm{d} W^k_t, \qquad X^x_t = x.$

And $\mu$ represent the motion of the mass of an ensemble of particle following this dynamics.

$\displaystyle \mu^{\omega}_t (x) \mathrm{d} x = \mu^{\omega}_t (\mathrm{d} x) = (X^{\omega}_t)_{\#} \mu_0 .$

Closed equation for $V = X^x - X^y$, in law (in Ito sense)

$\displaystyle \mathrm{d} V = \sqrt{2} (Q (0) - Q (V))^{1 / 2} \mathrm{d} B$

where $B$ is a $d$-dimensional BM. Closed $1 d$ SDE for the distance $| V | = | X^x - X^y |$, by Ito formula

$\displaystyle \mathrm{d} | V | = \frac{V}{| V |} \sqrt{2} (Q (0) - Q (V))^{1 / 2} \mathrm{d} B + \frac{d - 1}{| V |} (Q_N (0) - Q_N (| V |)) \mathrm{d} t$ $\displaystyle d [| V |]_t = \frac{V}{| V |} g (V) g (V)^T \frac{V^T}{| V |} \mathrm{d} t = 2 (Q_L (0) - Q_L (| V |)) \mathrm{d} t$

so in the end we have that $| V |$ is a one dimensional diffusion with generator $A$ and satisfying the SDE:

$\displaystyle \mathrm{d} | V | = \sqrt{(2 (Q_L (0) - Q_L (| V |)))} \mathrm{d} \tilde{B} + \frac{d - 1}{| V |} (Q_N (0) - Q_N (| V |)) \mathrm{d} t.$

Eulerian counterpart of this computation: we have

$\displaystyle \int \int \varphi (| X^x_t - X^y_t |) \mu_0 (\mathrm{d} x) \mu_0 (\mathrm{d} y) = \int \int \varphi (| x - y |) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y)$

and therefore

$\displaystyle \mathrm{d} \int \int \varphi (| x - y |) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) = \int \int A \varphi (| x - y |) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) \mathrm{d} t + \mathrm{d} \text{(local martingale)} .$

III. Nonsmooth Kraichnan model $(\alpha \in (0, 1))$

Form of $Q$. As a consequence of the formula

$\displaystyle \hat{Q} (n) = \langle n \rangle^{- d - 2 \alpha} \left( \mathbb{I}- \frac{n n^T}{| n |^2} \right)$

we have (see e.g. Le Jan–Raimond 2002)

$\displaystyle Q_L (0) - Q_L (| x |) = \beta_L | x |^{2 \alpha} + O_{| x | \ll 1} (| x |^2)$ $\displaystyle Q_N (0) - Q_N (| x |) = \beta_N | x |^{2 \alpha} + O_{| x | \ll 1} (| x |^2)$

We have $\beta_L, \beta_N$ and we have

$\displaystyle \beta_N = \left( 1 + \frac{2 \alpha}{d - 1} \right) \beta_L$

by the divergence free condition.

Main features of the non-smooth Kraichnan model.

1) Spontaneous stochasticity / particle splitting (Bernard–Gawedzi–Kupiainen 1998)

In the self-similar case, i.e. forgetting the full form of $Q$ and keeping only the leading term. Heuristically, if we start $\mu_0 = \delta_x$ in the smooth case we have $\mu_t = \delta_{X^x_t}$, however in the non-smooth case $\mu_t$ becomes diffuse immediately for $t > 0$. Unexpected for a continuity equation. “Formal proof”:

$\displaystyle \mathrm{d} | V | = (d - 1) \beta_N | V |^{2 \alpha - 1} \mathrm{d} t + \sqrt{2 \beta_L | V |^{2 \alpha}} \mathrm{d} B_t$

and

$\displaystyle \delta | V |^{1 - 2 \alpha} = (1 - \alpha) (d - 1 + \alpha) \beta_L \frac{1}{| V |^{1 - \alpha}} \mathrm{d} t + (1 - \alpha) \sqrt{2 \beta_L} \mathrm{d} B_t$

and this is a Bessel process of dimension $d / (1 - \alpha) > 2$. Therefore if $V_0 = 0$ then $V_t > 0$ for all $t > 0$ (i.e. there exists a solution which exits from $0$).

2) Wellposedness of the Eulerian model (from Le Jan–Raimond 2002)

Let's transform the equation in Ito form

$\displaystyle \mathrm{d} \mu +\operatorname{div} (\mu \mathrm{d} W) = \frac{c}{2} \Delta \mu \mathrm{d} t$

with $c I = Q (0)$.

Theorem. (LJ-R) There exists a unique solution adapted to the Brownian filtration in the class $L^{\infty}_{t, \omega} L_x^2$.

The proof goes via Wiener chaos decomposition, proving that the various projections are well-posed.

Let's make connection with the two-point motion.

Define $(X^x, X^y)$ as a weak solution to the two-point motion equation which is well defined as long as $x \neq y$. Then

$\displaystyle \int \int \operatorname{Law} ((X^x, X^y) |W) \mu_0 (\mathrm{d} x) \mu_0 (\mathrm{d} y) = (\mu_t \otimes \mu_t)$

and the SDE for $| V | = | X^x - X^y |$ holds rigorously and in particular there is spontaneous stochasticity.

There exists a unique selection of mass evolution which exhibiths splitting.

3) Anomalous regularisation of the two-point motion $(X^x, X^y)$

Eyink–Xin (1996+), Hakulinen 2003 : regularisation properties for the two-point motion.

4) Anomalous dissipation of the $L^2$-norm

qualitative Eyink–Drivas 2017, quantitative Rowan 2023+.

IV. 2d Euler Equation

1. Deterministic

$\displaystyle \left\{ \begin{array}{l} \partial_t \theta + u \cdot \nabla \theta = 0\\ u = K \ast \theta \end{array} \right.$

with vorticity field $\theta : [0, T] \times \mathbb{R}^2 \rightarrow \mathbb{R}$, velocity field $u : [0, T] \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$, $\theta =\operatorname{curl}u \Leftrightarrow u = K \ast \theta$,

$\displaystyle K (x) = \frac{x^{\perp}}{| x |^2} = \nabla^{\perp} G (x), \qquad G (x) = - \log | x | .$

Features: transport/continuity equation with divergence-free velocity field; non-linear, non-local; singular for “mass concentration”, $K$ is singular in $0$;

$\displaystyle \theta \in L^p, p \in (1, \infty) \Rightarrow G \ast \theta \in W^{2, p} \Rightarrow u \in W^{1, p} .$

Formally conserved quantities: 1) $L^p$ norms of $\theta$. $p = 2$ this is the enstrophy. 2) the $\dot{H}^{- 1}$ norm of $\theta$, i.e. $L^2$ norm of $u$:

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \| \theta \|^2_{\dot{H}^{- 1}} = \frac{\mathrm{d}}{\mathrm{d} t} \langle \theta, G \ast \theta \rangle = 2 \int \theta (\nabla^{\perp} G \ast \theta) \cdot (\nabla G \ast \theta) \mathrm{d} x = 0.$

Results:

$\theta_0 \in L^{\infty} \Rightarrow \exists ! \theta \in L^{\infty}_{t, x}$ (Yudovich). However to model vorticity concentrated configurations (vortex-sheets, etc…) we would like to free ourselves from the boundedness condition.
For $\theta_0 \in L^p$, $p \in (1, + \infty)$. We have existence (Di Perna–Majda 1987, and maybe Kato–Ponce) via compactness arguments. What is not know is uniqueness. The are counterexamples for forced Euler equation (Vishik 2018+) and other via convex integration on the torus. Overall is believed that there should not be uniqueness for these initial condition.
For $\theta_0 \in \dot{H}^{- 1}$ existence? Wiedemann 2011 via convex integration.
For $\theta_0 \in \dot{H}^{- 1}$ and positive measure existence has been established by Delort 1991.

V. Regularisation by nonsmooth Kraichnan noise for 2d Euler

We consider 2d Euler + non-smooth Kraichnan isotropic noise $W$ of index $\alpha \in (0, 1)$:

$\displaystyle \mathrm{d} \theta + u \cdot \nabla \theta + \nabla \theta \circ \mathrm{d} W = 0, \qquad u = K \ast \theta .$

Theorem. (Coghi-Maurelli 2023+) Existence of $\dot{H}^{- 1}$ solutions. For $\alpha \in (0, 1)$ and $\omega_0 \in \dot{H}^{- 1}$ then there exists probabilistically weak solutions $\theta \in L^{\infty}_t \dot{H}^{- 1}_x \cap C_t (H^{- 4})$ $\mathbb{P}- a.s$ satisfying in addition

$\displaystyle \theta \in L^2_{t, \omega} (\dot{H}^{- \alpha}_x) .$

There is an anomalous regularisation of the solutions which live for almost every time and almost surely in $\dot{H}^{- \alpha}_x$.

Theorem. (Coghi-Maurelli 2023+) Uniqueness in $L^p$. For $\theta \in L^p \cap L^1 \cap \dot{H}^{- 1}$ and

$\displaystyle \frac{3}{2} < p < \infty, \qquad 0 \vee \left( \frac{2}{p} - 1 \right) < \alpha < \frac{1}{2} \wedge \left( 1 - \frac{1}{p} \right)$

then there is pathwise uniqueness (and strong existence) in $L^{\infty}_{t, \omega} (L^p_x \cap L^1_x) \cap L^{\infty}_t L^2_{\omega} \dot{H}^{- 1}_x \cap L^2_{t, \omega} (\dot{H}^{- \alpha})$.

[end of second lecture]

Maurelli | Regularisation by noise | Lecture 3 | Friday July 26, 9:00–10:30

V. Reg. by nonsmooth Kraichnan nois for 2d Euler

Recall the equation

$\displaystyle \mathrm{d} \theta + u \cdot \nabla \theta + \nabla \theta \circ \mathrm{d} W = 0, \qquad u = K \ast \theta . $

(2)

with a Kraichnan isotropic noise of index $\alpha \in (0, 1)$. Let's talk about motivations for the noise:

Preserves the transport nature of the equation
Can be viewed as a representation of small scale behaviour of a turbulent fluid
Why $\alpha \in (0, 1) ?$ The equation has problems when the mass concentrates, and the splitting nature of the noise gives an hope that it could improve the behaviour of the equation by preventing mass concentration.
Physicist predict is that $\alpha = 2 / 3$, since in this case we reproduce Obukov theory of scalar turbulence in $d = 2$. Unfortunately uniqueness result that we have of Euler+Kraichnan does not hold for this value of the exponent.

Lagrangian counterpart of the SPDE:

$\displaystyle \mathrm{d} X^x_t = \int K (X^x_t - X^y_t) \omega_0 (\mathrm{d} y) \mathrm{d} t + \circ \mathrm{d} W (t, X^x_t)$

if we add just a constant (or smooth) noise $+ \mathrm{d} B$, we can remove it via a Galileian transformation and should not have hope to regularise the equation. In order to hope to have some regularisation we need some noise which has a substantial effect when $| X^x_t - X^y_t | \ll 1$ which is the region where the Biot–Savart kernel is singular.

$\displaystyle \| \theta \|_{\dot{H}^{- s}}^2 := \int | n |^{- 2 s} | \hat{\theta} (n) | \mathrm{d} n$

Definition. $\theta$ is an $\dot{H}^{- 1}$ solution to the equation (2), there exists a $(\mathcal{F}_t)_t$-progressively measurable $\theta \in L^{\infty}_t (\dot{H}^{- 1}_x) \cap C_t (H_x^{- 4})$ $\mathbb{P}- a.s.$ and

$\displaystyle \theta_t = \theta_0 - \int_0^t \underbrace{(K \ast \theta_r) \cdot \nabla \theta_r}_{=\operatorname{curl}\operatorname{div} (u_r u_r^{\mathrm{T}})} \mathrm{d} r - \int_0^t \nabla \omega_r \cdot \mathrm{d} W_r + \int_0^t \frac{c}{2} \Delta \theta_r \mathrm{d} r.$

The basic results we would like to discuss are the following about existence and uniqueness

Theorem 1. $(\exists)$ $\alpha \in (0, 1)$, $\theta_0 \in \dot{H}^{- 1}$ then $\exists \theta$ $\dot{H}^{- 1}$ solution to the equation (2) such that

$\displaystyle \theta \in L^2_{x, t} (\dot{H}^{- \alpha}_x) .$

Theorem 2. $(!)$  For $\theta \in L^p \cap L^1 \cap \dot{H}^{- 1}$ and

$\displaystyle \frac{3}{2} < p < \infty, \qquad 0 \vee \left( \frac{2}{p} - 1 \right) < \alpha < \frac{1}{2} \wedge \left( 1 - \frac{1}{p} \right)$

then there exists a unique strong solution in $L^{\infty}_{t, \omega} (L^p_x \cap L^1_x)$.

Galeati–Luo (2023+) weak uniqueness for log-Euler 2d via Girsanov's theorem (a nice paper)

Let's discuss the proofs now.

Step 1. Anomalous regularisation in negative Sobolev for Kraichnan

Consider

$\displaystyle \mathrm{d} \mu +\operatorname{div} (\mu \mathrm{d} W) = \frac{c}{2} \Delta \mu, \qquad \operatorname{on} [0, T] \times \mathbb{R}^2$

Idea: particle splitting brings anomalous regularisation in negative Sobolev spaces. Formally, two particular $X^x$ and $X^y$ if $x \rightarrow y$ we still have that $X^x_t \neq X^y_t$ for positive times, and it is reasonable to expect that $\left| X^x_t {- X^y_t} \right|^{- 1}$ should get smaller on average, and so gain integrability in space.

This can be verified by using the explicit SDE satisfied by $\left| X^x_t {- X^y_t} \right|$. In the Eulerian picture

$\displaystyle \int \int \varphi (x - y) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) = \int \int \varphi (X^x_t - X^y_t) \mu_0 (\mathrm{d} x) \mu_0 (\mathrm{d} y)$

and we take $\varphi (x - y) = \log | x - y |^{- 1} = G (x - y)$. Applying Ito formula:

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\langle G \ast \mu, \mu \rangle] =\mathbb{E} [\langle \operatorname{Tr} [Q (0) \mathrm{D}^2 \mu], G \ast \mu \rangle] +\mathbb{E} \left[ \sum_k \langle \sigma_k \cdot \nabla \mu {,} G \ast (\sigma_k \cdot \nabla \mu) \rangle \right]$ $\displaystyle =\mathbb{E} [\langle \operatorname{Tr} [Q (0) \mathrm{D}^2 \mu], G \ast \mu \rangle] +\mathbb{E} \left[ \sum_k \langle \operatorname{div} (\sigma_k \mu) {,} G \ast (\sigma_k \cdot \nabla \mu) \rangle \right]$ $\displaystyle =\mathbb{E} [\langle \operatorname{Tr} [Q (0) \mathrm{D}^2 \mu], G \ast \mu \rangle] -\mathbb{E} \left[ \left\langle \left( \left( \sum_k \sigma_k \sigma_k^{\mathrm{T}} \right) \mathrm{D}^2 G \right) \ast \mu, \mu \right\rangle \right]$ $\displaystyle =\mathbb{E} \int \int \operatorname{Tr} [(Q (0) - Q (x - y)) \mathrm{D}^2 G (x - y)] \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y)$

Now

$\displaystyle \nabla G (x) = - \frac{x}{| x |^2}, \qquad \nabla^2 G (x) = - \frac{1}{| x |^2} \left( -\mathbb{I}+ \frac{2 x x^T}{| x |^2} \right),$

and

$\displaystyle Q (0) - Q (x) \approx \beta_L | x |^{2 \alpha} \frac{x x^T}{| x |^2} + \beta_N | x |^{2 \alpha} \left( \mathbb{I}- \frac{x x^T}{| x |^2} \right), \qquad \beta_N = (1 + 2 \alpha) \beta_L > \beta_N$ $\displaystyle \operatorname{Tr} [(Q (0) - Q (x)) \mathrm{D}^2 G (x)] = \frac{1}{| x |^2} (\beta_L | x |^{2 \alpha} - \beta_N | x |^{2 \alpha}) + O (| x |^{2 \alpha - 2})$ $\displaystyle = \underbrace{(\beta_L - \beta_N)}_{< 0 (!)} | x |^{- 2 + 2 \alpha} + O (| x |^{- 2})$

with a negative sign! Therefore

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} \left[ \int \int \log (| x - y |^{- 1}) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) \right] = - C\mathbb{E} \int \int | x - y |^{2 \alpha - 2} \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) + \text{l.o.t.}$

Note that

$\displaystyle \int \int \log (| x - y |^{- 1}) \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) = \langle \mu, (- \Delta)^{- 1} \mu \rangle = \| \mu \|_{\dot{H}^{- 1}}^2$

and

$\displaystyle \int \int | x - y |^{2 \alpha - 2} \mu_t (\mathrm{d} x) \mu_t (\mathrm{d} y) = \langle \mu, (- \Delta)^{- \alpha} \mu \rangle = \| \mu \|_{\dot{H}^{- \alpha}}^2$

from which we have a gain of $1 - \alpha$ regularity since

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} \| \mu \|_{\dot{H}^{- 1}}^2 = - C\mathbb{E} \| \mu \|_{\dot{H}^{- \alpha}}^2 + \text{l.o.t.}$

Step 2. Anomalous regularisation for Euler+Kraichnan

Wanted regularisation for the full equation. The idea is that the non-linear drift preserves the $\dot{H}^{- 1}$ norm.

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\langle \theta, G \ast \theta \rangle] = -\mathbb{E} [\underbrace{\langle u \cdot \nabla \theta, G \ast \theta \rangle}_{= 0}] \underbrace{- c\mathbb{E} [\| \theta \|_{\dot{H}^{- \alpha}}^2] + \text{l.o.t.}}_{\text{terms from the noise}}$

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\| \theta \|_{\dot{H}^{- 1}}^2] = - c\mathbb{E} [\| \theta \|_{\dot{H}^{- \alpha}}^2] + \text{l.o.t.}$

See also recent paper of Coti-Zelati, Drival, Gwalani (2024).

Step 3. Uniqueness

Take $\theta^1, \theta^2$ for E+K, let $\theta = \theta^1 - \theta^2$, then

$\displaystyle \mathrm{d} \theta = (K \ast \theta^1) \nabla \theta + (K \ast \theta) \nabla \theta^2 + \nabla \theta \mathrm{d} W = \frac{c}{2} \Delta \theta .$

Idea: get the $\dot{H}^{- \alpha}$ norm from the noise, for the drift use conservation of $\dot{H}^{- 1}$, conservation of $L^p$ and control with $\dot{H}^{- \alpha}$.

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\| \theta \|_{\dot{H}^{- 1}}^2] = - C\mathbb{E} [\| \theta \|_{\dot{H}^{- \alpha}}^2] - 2\mathbb{E} [\langle G \ast \theta, (K \ast \theta^1) \nabla \theta \rangle] - 2\mathbb{E} [\langle G \ast \theta, (K \ast \theta) \nabla \theta^2 \rangle] + \text{l.o.t.}$

the conservation of $\dot{H}^{- 1}$ norm we have $\langle G \ast \theta, (K \ast \theta) \nabla \theta^2 \rangle = 0$, while the other trilinar term needs some control (actually we do not care much since we can anyway bound it).

$\displaystyle | \langle G \ast \theta, (K \ast \theta^1) \nabla \theta \rangle | = | \langle \underbrace{(K \ast \theta^1) \nabla G \ast \theta}_{\dot{H}^{\alpha}}, \underbrace{\theta}_{\dot{H}^{- \alpha}} \rangle |$

and using the inequality $\| f g \|_{\dot{H}^{\alpha}} \lesssim \| f \|_{\dot{H}^{2 \alpha}} \| g \|_{\dot{H}^{1 - \alpha}}$ if $\alpha < 1 / 2$ we have, by apriori estimates and if $\alpha \leqslant 1 - 1 / p, \alpha < 1 / 2$:

$\displaystyle \lesssim \| (K \ast \theta^1) \|_{\dot{H}^{2 \alpha}} \underbrace{\| \nabla G \ast \theta \|_{\dot{H}^{1 - \alpha}}}_{\lesssim \| \theta \|_{\dot{H}^{- \alpha}}} \| \theta \|_{\dot{H}^{- \alpha}} \lesssim \| \theta \|_{\dot{H}^{- \alpha}}^2 \| \theta^1 \|_{\dot{H}^{2 \alpha - 1}} \lesssim \| \theta \|_{\dot{H}^{- \alpha}}^2 \| \theta^1 \|_{L^P \cap L^1} \| \theta \|_{\dot{H}^{- \alpha}}^2 \| \theta_0 \|_{L^P \cap L^1}$

however if $\alpha < 1 - 1 / p$ we can play with usual tricks and interpolation to obtain a slightly better bound and close the estimate via Young's inequality and obtain

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\| \theta \|_{\dot{H}^{- 1}}^2] + c\mathbb{E} [\| \theta \|_{\dot{H}^{- \alpha}}^2] \lesssim \varepsilon \| \theta \|_{\dot{H}^{- \alpha}}^2 + C_{\varepsilon} \| \theta^1 \|_{L^P \cap L^1}^2 \| \theta \|_{\dot{H}^{- 1}}^2$

which gives uniqueness.

In the last 15 minutes we will talk about other topics. The problem in Euler is to control the non-linear term, apriori estimates give easily compactness of smooth approximations in $L^2$. With Kraichnan we get estimates in $\dot{H}^{1 - \alpha}$ which gives strong compactness in $L^2$ and we can pass to the limit easily in the equation.

Theorem. (Galeati–Grotto–Maurelli, 2024+) Linear Kraichnan with $\alpha \in (0, 1)$

$\displaystyle \partial_t \mu +\operatorname{div} (\mu \circ \mathrm{d} W) = 0,$

then, for all $s \in (0, d / 2)$,

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} \mathbb{E} [\| \mu \|_{\dot{H}^{- s}}^2] = - C\mathbb{E} [\| \mu \|_{\dot{H}^{- s + 1 - \alpha}}^2] + \text{l.o.t.}$

Theorem. (Jian–Luo, 2024+ & Bagnara–Galeati–M. 2024+) Consider generalized SQG, the strong uniqueness for solutions in $L^p \cap L^1$ with $p \geqslant 2$.