Sample-path large deviation principle for a 2-D stochastic interacting vortex dynamics with singular kernel

The 2-D vortex dynamics (1.2) is a weakly interacting stochastic particle model with independent noise. It is called ”weak” because when the number of particles tends to infinity, the force between any two particles tends to zero.

For the mean-field limit, the case in the presence of smooth interaction kernels has been well studied since McKean’s work [38]. But there is only few references in the case of singular interactions for a long time. Recently, based on the estimation for relative entropy, Jabin and Wang [28] proved the mean-field limit and obtained the optimal convergent rate of distributions for a general class of singular kernels including the Biot-Savart law. Meanwhile, based on modulated energy, Serfaty [47] and Duernckx [16] proved the mean-field limit for deterministic systems with more singular kernels. Combining the two methods, Bresch, Jabin and Wang [6] derived the mean-field limit for the Patlak-Keller-Segel kernel and gave the corresponding convergent rate.

To quantify the difference between empirical measure processes and mean-field limit more precisely, some considered the fluctuation in path space, usually called the central limit theorem (CLT) for Gaussian fluctuations. Results of this type were firstly obtained by Tanaka and Hitsuda [50] and by Tanaka [49] for one dimensional non-singular interacting diffusions. For more general case with regular enough coefficients, it was proved by Fernandez and Méléard [22]. Recently, Wang, Zhao, and Zhu [53] proved CLT for a special class of singular interaction including (1.2).

A further topic is the sample-path large deviation principle for empirical measures. For non-singular interaction kernels only in drift coefficients, Dawson and Gärtner [11] proved the sample-path LDP for general weakly interacting stochastic particles model using measure transformation from independent diffusions. Budhiraja, Dupuis and Fischer [8] further developed this result, allowing the non-singular interaction kernels in both drift and diffusion terms. The method by Feng and Kurtz, based on Hamiltonian-Jocabi theory, also works for smooth kernel (see Section 13.3 of [20]).

For singular interaction kernels, Fontbona [23] proved the sample-path LDP for diffusing particles with electrostatic repulsion in one dimension. Applying the method in [20], at least formally, Feng and Świech [21] gave the rate function of sample-path LDP for empirical measures of (1.2), starting from initial data with finite entropy. However, making this approach rigorous is really subtle, requiring an uniqueness theory for infinite-dimensional Hamiltonian-Jocabi equation, especially when only having initial data with finite energy in hand.

In the present paper, we aim to study the sample-path large deviation principle of (1.2) starting from more rough initial data only with finite energy. We follow the standard approach used in [11, 33, 14], i.e., applying an exponential martingale inequality to prove the upper bound and approximating trajectories with finite action (rate function) by a series of mildly perturbed systems for the proof of lower bound. The main difficulty here comes from the term in the rate function containing the singular kernel $\mathcal{K}$.

We use the idea from [44, 45, 5], introducing an auxiliary functional as a modification of the rate function to make the approximating strategy for the proof of lower bound work. Moreover, during the proof of large deviation upper bound, this auxiliary functional ensures the continuity of singular terms in the Markov generator with respect to weak topology, which is not a problem for the case of non-singular kernels.

As a price, we need a sharp prior estimation for the auxiliary functional for the upper bound establishment. We finally choose the supremum of energy functional and the integral of $L^{2}$ norm along time as the auxiliary functional in our paper. Both in the procedure of nice trajectory approximation during the proof of lower bound, and in the prior estimation of auxiliary functional during the proof of upper bound, the singular terms are shown to be bound by the integral of $L^{2}$ norm over time, which is our key observation and main tools throughout this paper.

We write $C^{\infty}(\Omega)$ and $\mathcal{D}^{\prime}(\Omega)$ for the collection of infinitely differentiable functions on $\Omega$ and generalized functions on $\Omega$.

We use $\left\langle\cdot,\cdot\right\rangle$ to represent pairing of linear space and its dual space, such as $L^{2}$ inner product, continuous function integral with respect to a measure, and so on.

Denote by $\mathcal{P}(\mathbb{T}^{2})$ the space of probability measures on $\mathbb{T}^{2}$. For $\gamma,\eta\in\mathcal{P}(\mathbb{T}^{2}),$ we define the Wasserstein$-2$ metric

where

Then $(\mathcal{P}(\mathbb{T}^{2}),d)$ is a complete separable metric space. See Chapter 7 of [1] or Chapter 7 of [52] for properties of this metric. Given $\mu\in\mathcal{P}(\mathbb{T}^{2})$, for any $m\in\mathcal{D}^{\prime}(\mathbb{T}^{2})$ we define

Let $AC((s,t);\mathcal{P}(\mathbb{T}^{2}))$ be the set of all $\rho\in C([s,t];\mathcal{P}(\mathbb{T}^{2}))$ satisfying that there exists $m\in L^{1}[s,t]$ such that

We refer to smooth mollifier throughout this paper as a non-negative even function that is smooth and its integral over $\mathbb{T}^{2}$ is one.

We introduce the state space

As shown by [15], for the case $X_{i}(0)=x_{i}$, the system (1.2) is well-posed for almost every $(x_{i})_{i=1,\cdots,n}$, but without an explicit characterization. Actually, we need a well-posed result like Theorem 2.10 in [24], and [48] gives a probabilistic proof which can be directly applied to the case of torus and lead to the following lemma.

Now we introduce the energy functional for empirical measures

where $\boldsymbol{D}=\{(x,y)\in(\mathbb{T}^{2})^{2}:x=y\}$ is the diagonal set. We impose the following superexponential finite energy condition for initial data.

The removal of diagonal set is not convenient for estimations, so we also introduce the energy functional containing the diagonal set,

It’s always well defined since $\mathcal{N}$ is bounded below. By Fatou’s Lemma, $e$ is lower semi-continuous functional under the weak topology, and we left more properties of $e$ into Appendix C.

Let $Q_{T}:C([0,T];\mathcal{P}(\mathbb{T}^{2}))\mapsto\mathbb{R}$ be defined by

and for $\rho\in AC((0,T);\mathcal{P}(\mathbb{T}^{2}))$ and $Q_{T}(\rho)<\infty$, we define

To check $\mathbb{A}_{T}(\rho)$ is well defined, by Theorem 8.3.1 of [1], for $\rho\in AC((0,T);\mathcal{P}(\mathbb{T}^{2}))$, we can define $\partial_{t}\rho\in\mathcal{D}^{\prime}(\mathbb{T}^{2})$ for almost every $t\in[0,T]$. In addition, $Q_{T}(\rho)<\infty$ implies that $\rho(t)\in L^{2}(\mathbb{T}^{2})$ for almost every $t$. As a consequence of (1.5) and Young’s inequality, if $\gamma\in L^{2}(\mathbb{T}^{2})$, then $\mathcal{K}*\gamma\in L^{2}(\mathbb{T}^{2})$, so

is well defined for any smooth function $\phi$. Thus ${\rm div}\,[\rho(t)(\mathcal{K}*\rho(t))]\in D^{\prime}(\mathbb{T}^{2})$ for almost every $t$.

Our main result is

As in the seminar works [11, 33, 14], the sample-path large deviation upper bound can be obtained through an exponential martingale estimation and exponential tightness. Additional treatment is required in the presence of singular terms, i.e. a prior energy estimation to make sure the generator is continuous under weak typology.

The basic strategy in [33, 14] to prove lower bound is to compute the relative entropy between a model with regular perturbation and the original one, which actually provides a lower bound on the probability of the neighborhood of the mean-field limit path of the perturbed process. However, this strategy for lower bound estimation would fail in general when the rate function is not convex, let alone in the presence of singular interaction kernels, unless there is an additional modification for the rate function, as have been done in [44, 45, 5]. As we have stated in the main theorem, we choose $Q_{T}<\infty$ as the modification, which implies finite energy and finite integral of $L^{2}$ norm of the distributions over time.

Besides the modification by $Q_{T}<\infty$, the proof for the lower bound still needs several new inequalities, which can bound the singular terms such as ${\rm div}\,[\rho(\mathcal{K}*\rho)]$ by the integral of $L^{2}$ norms along sample path. These inequalities are originated from an energy dissipation structure similar to the classical result for 2D Navier-Stokes equation (e.g. see (3.20) in [51]), in which the integral of $L^{2}$ norms along sample path naturally emerges. We can not expect the $L^{2}$ norms to be bounded from above since the initial $L^{2}$ norm can be infinite, but the integral of $L^{2}$ norm along sample path with finite action is proven to be finite, utilizing such an energy dissipation structure. As one may expect, the finite energy condition is required for the energy dissipation structure, and as far as we know, this is the weakest condition that makes the mean-field limit holds for (1.2) with general $\nu>0$.

As stated in [5], the rate function with modification makes it harder to prove the upper bound, requiring a prior estimation of $Q_{T}$. However, $Q_{T}$ always takes $\infty$ on the empirical measure $\rho_{n}\in\mathcal{X}_{n}$, even if their limit as $n\to\infty$ has finite $Q_{T}$. Thus we make use of the convolution method with a sequence of smooth mollifiers. By taking specific mollifiers $\zeta_{n}$ converging to Dirac measure with a suitable slow rate, we yield an energy dissipation structure for $\zeta_{n}*\rho_{n}$ assuming $e(\zeta_{n}*\rho_{n}(0))$ is finite, which gives the control for $Q_{T}(\zeta_{n}*\rho_{n})$. The key ingredient in the proof is a sharp estimation for the singular term in order to show its vanishing in the energy dissipation structure. Moreover, the energy dissipation structure, as well as related inequalities, can also help to provide a more explicit formula of the rate function than the variational one directly derived from the exponential martingale inequality.

Furthermore, in order to characterize the limit of infintesimal operators of the SDEs with singular terms, we utilize a symmetrization technique to first represent the limit operator defined on arbitrary probability measure, then perform the regularity analysis, and finally come back to the original representation without symmetrization.

In more detail, take $\zeta:\mathbb{R}\mapsto\mathbb{R}^{+}:$

where the constant is to make $\int_{0}^{\infty}2\pi r\zeta(r)dr=1$. $\zeta(x)$ is a non-negative smooth function on $\mathbb{R}$ with a compact support. Then let $\zeta_{n}:\mathbb{R}^{2}\mapsto\mathbb{R}$ be the specific smooth mollifiers generated by $\zeta$ and defined by

where $m_{n}\uparrow\infty,nm_{n}^{-2}\to\infty$. Note that the slow convergent rate of $m_{n}$ to infinity is crucial for the prior estimation that we will show later.

In Section 2 we will illustrate the proof for the main theorem under the following superexponential finite energy condition for $\zeta_{n}*\rho_{n}$ that we will show later.

To conclude the proof, we will also show that finite $e_{0}(\rho(0))$ is stronger than finite $e(\zeta_{n}*\rho(0))$, i.e. the Condition 1.1 implies the Condition 1.2, in Section 3.

In Section 3, we also prove several crucial inequalities for bounding the singular term ${\rm div}\,\rho(\mathcal{K}*\rho)$ under convolution and give the prior estimation of $Q_{T}$ for the upper bound proof. Section 4 is about regularities of trajectories with finite rate function, which helps us to express the rate function in a more explicit way. In Section 5, we prove the law of large number for the model with certain regular perturbation and perform the ”nice” trajectory approximation strategy for the lower bound proof.

Our proof of large deviation upper bound is based on the analyses of the exponential martingale problem.

Recall that the dynamics of $X(t)$ are determined by the stochastic differential equations

By Ito’s formula, for each smooth function $\phi$ on $\mathbb{T}^{2}$, we have

To write the generator of $\rho_{n}$, we need to write the singular term $\frac{1}{n^{2}}\sum_{i\neq j}\nabla\phi(X_{i}(t))\cdot\mathcal{K}(X_{i}(t)-X_{j}(t))$ as a $\rho_{n}(t)$ dependent term. Here we use the observation from Delort [12], proposing an alternative representation for singular term which can be easily extended to general measure.

Recalling $\boldsymbol{D}$ is the diagonal set, noting $\mathcal{K}(x)=-\mathcal{K}(-x),$

is only dependent on $\nabla\phi$ and $\rho_{n}(t)$ and linear of $\nabla\phi.$ In view of (1.5), $w(x,y)=r(x,y)\mathcal{K}(x-y)$ is a bounded function so that for each $\varphi\in C^{\infty}(\mathbb{T}^{2};\mathbb{R}^{2})$,

is bounded. Hence the final expression of (2.2) is always well defined and can be extended to any $\gamma\in\mathcal{P}(\mathbb{T}^{2})$. Now we introduce the symmetrization operator $\mathbf{R}:\mathcal{P}(\mathbb{T}^{2})\mapsto\mathcal{D}^{\prime}(\mathbb{T}^{2})$ satisfying

As stated above, (2.4) is always well-defined and

In fact $\mathbf{R}(\gamma)$ is an extension for $\gamma(\mathcal{K}*\gamma).$ If $\gamma\in L^{2}(\mathbb{T}^{2}),$ as stated before Theorem 1.2, $\mathcal{K}*\gamma\in L^{2}(\mathbb{T}^{2})$. Then for each $\varphi\in C^{1}(\mathbb{T}^{2};\mathbb{R}^{2})$,

Hence

With this notation, after further calculations, we can write the generator for $\rho_{n}$ as

for each $f\in D_{0}:=\{f:\mathcal{P}(\mathbb{T}^{2})\mapsto\mathbb{R},f(\gamma)=\psi(\left\langle\varphi_{1},\gamma\right\rangle,\cdots,\left\langle\varphi_{k},\gamma\right\rangle),\psi\in C^{2}(\mathbb{R}^{k}),\varphi_{i}\in C^{\infty}(\mathbb{T}^{2})\}$ such that

Then we define

in which $\Delta\gamma$ as well as ${\rm div}\,\mathbf{R}(\gamma)$ is defined in distribution sense and the variational derivative

is a smooth function on $\mathbb{T}^{2}$.

At least formally by Ito’s formula, for $(f_{t})_{t\geq 0}$ such that $f_{t}\in D_{0}$, the exponential form

is always considered to be a martingale. In fact, the proof of [33, 14] only use the case when $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{2})$ and $f_{t}(\rho)=\left\langle\phi(t),\rho\right\rangle.$ To adapt their proof to the more general initial data, we shall use a bigger domain $D_{0}$, which is sufficient to determine the rate function (see Section 3.2 of [20]).

First, as a standard step to prove large deviation principle, we prove the exponential tightness of $\rho_{n}$, with which we only need to prove the upper bound for compact sets.

Formally, there are three steps to prove the large deviation upper bound for compact sets. First, by Ito’s formula, for each $f\in D_{0}$, $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{2})$ and $t_{0}\in(0,T)$,

is a non-negative martingale, hence for each measurable set $B\subset C([0,T];\mathcal{P}(\mathbb{T}^{2}))$,

where $\chi$ is the indicator function and