¹

1 Introduction↩︎

The subject of this paper are interacting particle systems that are closely related or modelled after the following system: \[\label{ks} dX_t^i= -\frac{\nu}{N}\sum_{j=1, j \neq i}^N \frac{X_t^i-X_t^j}{|X_t^i-X_t^j|^2}dt + \sqrt{2}dB_t^i.\tag{1}\] Here \(X_t^i\) is the position of the \(i\)-th particle in \(\mathbb{R}^2\) at time \(t\), \(1 \leq i \leq N\), and \(\{B_t^i\}_{t \geq 0}\) are independent 2-dimensional Brownian motions. This system is a finite particle approximation of the famous Keller-Segel model of chemotaxis 5 . The constant \(\nu \geq 0\) measures the strength of attraction between the particles. In the absence of noise, 1 is a system of ordinary differential equations, and so whenever \(\nu>0\) the particles collide and stay glued up due to the strong attraction between them, i.e. there is a blow up. Introducing Brownian noise (or thermal excitation) moves the blow-up threshold: \[\nu_\star=0 \quad \longrightarrow \quad \nu_\star=4.\] i.e. now for every \(\nu <\nu_\star:=4\) the evolution of the particles continues indefinitely provided that \(N\) is large and the initial distribution has no atoms. More precisely, under these assumptions the particle system 1 has a global in time weak solution in the sense of stochastic differential equations (SDEs). If, on the other hand, \(\nu \geq \nu_\star\), then all the particles collide a.s. and stay glued up; this can be seen upon noting that \(R_t:=\frac{1}{4N}\sum_{i,j=1}^N|X_t^i-X_t^j|^2\) is a local squared Bessel process of dimension \((N-1)(2-\frac{\nu}{2})\). See [1] for detailed discussion and references (we comment on the existing literature further below).

Our goal in this paper is to obtain an upper bound on the density of the law of 1 (\(=:\) heat kernel). The singularities of the drift in 1 make invalid any Gaussian heat kernel upper bound. Along the way we will need to establish some regularity results for solutions of the Kolmogorov backward and forward equations behind 1 that are interesting on their own, given that the standard regularity theory does not apply to these equations.

The Keller-Segel finite system 1 exhibits critical behaviour in two important ways. First, there are blow-ups. The blow-ups, however, also occur in the higher-dimensional counterpart of 1 that we consider in Section 3. The two-dimensionality of 1 is the other reason that makes it difficult to handle. Namely, the drift in the Kolmogorov backward operator corresponding to 1 \[L=-\Delta + \frac{\nu}{N}\sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|^2} \cdot \nabla_{x_i}\] is not in \(L^2_{{\rm loc}}=L^2_{{\rm loc}}(\mathbb{R}^{2N})\), the Cauchy problem for the corresponding parabolic equation is not well-posed in the standard Hilbert triple of Sobolev spaces \(W^{1,2}(\mathbb{R}^{2N}) \rightarrow L^2(\mathbb{R}^{2N}) \rightarrow W^{-1,2}(\mathbb{R}^{2N})\), and the use of De Giorgi’s or Moser’s methods (including Moser’s iterations run in the setting of Dirichlet forms) is problematic. All these difficulties, however, do not appear in the higher-dimensional counterparts of 1 .

The source of these analytic difficulties is, one can argue, the lack of the Hardy inequality in \(\mathbb{R}^2\). Let \(\langle \, \rangle\) denote the integration over \(\mathbb{R}^{d}\), \(d \geq 2\). If \(d \geq 3\), then the (usual) Hardy inequality \[\label{hardy0} \frac{(d-2)^2}{4}\big\langle \frac{f^2}{|x|^2}\big\rangle \leq \big\langle |\nabla f|^2 \big\rangle, \quad f \in W^{1,2}(\mathbb{R}^{d}),\tag{2}\] allows us to control the drift term in \(L\) in terms of the quadratic form of the Laplacian, and thus allows to prove the energy inequality, which is the point of departure for De Giorgi’s and Moser’s methods. There is no non-trivial Hardy inequality in dimension \(d=2\). There is, however, a non-trivial fractional Hardy inequality in \(\mathbb{R}^2\): \[\label{frac1} \biggl(\frac{1}{2} \frac{\Gamma\left( \frac{1}{4}\right)^{2}}{\Gamma\left( \frac{3}{4}\right)^{2}}\biggr)^{-1} \big\langle\frac{f^2}{|x|}\big\rangle \leq \big\langle |(-\Delta)^{\frac{1}{4}} f|^2 \big\rangle, \quad f \in \mathcal{W}^{\frac{1}{2},2}(\mathbb{R}^{2}).\tag{3}\] We will exploit that in order to estimate the heat kernel \(p(t,x,y)\) of 1 (or, rather, of a very closely related system), even though a priori there is nothing non-local about the Keller-Segel system 1 . Furthermore, 3 will provide us with the weak well-posedness of Cauchy problem for the parabolic equation corresponding to 1 in the shifted Hilbert triple of Bessel potential spaces \(\mathcal{W}^{-\frac{1}{2},2}(\mathbb{R}^{2N}) \rightarrow \mathcal{W}^{\frac{1}{2},2}(\mathbb{R}^{2N}) \rightarrow \mathcal{W}^{\frac{3}{2},2}(\mathbb{R}^{2N})\).

Our main instrument in dimension \(d=2\) is an abstract desingularization theorem (Theorem A) obtained earlier in [2] for different purposes. We are going to use some ideas of Nash [3]. We are also going to use some old ideas of Semënov [4] (Step 3 in the proof of Theorem 1).

The upper heat kernel bound for 1 that we will obtain (Theorem 1) has form \[p(t,x,y) \leq Ct^{-N}\varphi(y)\] for weight \(\varphi\) that explodes at appropriate rate along “collision hyperplanes” \(x^i=x^j\). In dimensions \(d \geq 3\) we will improve this upper bound: replace \(t^{-N}\) with Gaussian factor and make weight \(\varphi\) time dependent in order to recover the delta-function at time \(t=0\) (Theorem 2). See also our comment regarding the lower heat kernel bound after Theorem 2. That said, in dimension \(d=2\), if we only use 3 , then we arrive at the upper heat kernel bound valid under the condition \(\nu<\frac{C}{N}\), i.e. the assumption on \(\nu\) degenerates quickly as the number of particles \(N\) goes to infinity. In fact, 3 underexploits the regularity of the interaction kernel in 1 , i.e. we can actually apply the fractional Hardy inequality \[\label{frac2} \biggl(\frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}\biggr)^{-1} \big\langle\frac{f^2}{|x|^\alpha}\big\rangle \leq \big\langle |(-\Delta)^{\frac{\alpha}{4}} f|^2 \big\rangle,\tag{4}\] provided that \(1 \leq \alpha<2\). It turns out that, with the proper choice of \(\alpha\), we can improve (slow down) quite substantially the rate of degeneration of the condition on \(\nu\). For example, if we use 4 with appropriately chose \(\alpha\), then the maximal admissible \(\nu\) for \(N=10^9\) is only two times smaller than the maximal admissible \(\nu\) for \(N=10^3\), see Figure 1. The same argument, applied to the higher dimensional analogue 6 of 1 in \(\mathbb{R}^{dN}\) for \(d \geq 3\), produces a constraint on \(\nu\) that is essentially independent of \(N\) (Theorem 3, see also Theorem 2 – its proof uses a different approach). It would be interesting to understand how our upper heat kernel bound behaves under the mean field limit \(N \rightarrow \infty\), see [5], [6] and references therein, at least for \(d \geq 3\). This would need to take into account the fact that if at time \(t=0\) the particles are, say, i.i.d. with sufficiently regular density, then the nonlinear drift in the McKean-Vlasov equation is regular enough so that the mean field density satisfies two-sided Gaussian bounds integrated against the initial distribution; informally, there is no more singular weight of type \(\varphi\) in the mean field limit. We plan to address this in a subsequent paper.

1.1 Outline↩︎

1. In Section 2 we state Theorem 1 for the Keller-Segel finite particles 1 , first in a priori form, i.e. for regularized interaction kernel. We also describe there the abstract desingularization theorem that allows to prove upper heat kernel bounds of the type discussed in Theorem 1.

2. In Section 3 we consider a higher-dimensional counterpart of 1 and refine the upper bound rather substantially (Theorem 2). We also comment there on the sharpness of our upper bound.

3. In Appendix 7 we describe an alternative approach for proving an upper heat kernel bound for 1 in dimensions \(d \geq 3\), in Appendix 10 we discuss adding a divergence-free drift. Appendix 12 specifies Theorems 1 and 2 to the two particle case.

1.2 Literature↩︎

Regarding heat kernels of singular particle systems, we refer to the results of Graczyk-Sawyer [7], [8] and Dziubański-Hejna [9] on Dunkl Laplacian. Their situation is different, i.e. the focus is on the repulsing interactions, moreover, one has explicit, albeit an non-elementary formula for the heat kernel. We also mention the work of Giunti-Gu-Mourrat [10] on the upper heat kernel bound for the symmetric simple exclusion process, although this particle system is quite different from ours.

There is quite rich literature on heat kernel bounds for local and non-local operators with coefficients having critical polar singularities that make the standard heat kernel bounds invalid, see Milman-Semënov [11], Metafune-Sobajima-Spina [12], Kinzebulatov-Semënov-Szczypkowski [2]; this list is far from being exhaustive. Generally speaking, non-standard heat kernel bounds appear in many other settings, see e.g. Boutiah-Rhandi-Tacelli [13] and references therein.

Regarding the weak well-posedness of the Keller-Segel finite particle system 1 , we refer to Cattiaux-Pédèches [14] and Fournier-Jourdain [6] who proved detailed and many ways optimal or close to optimal results, see also recent advances in Fournier-Tardy [15] and Tardy [16]. The point of departure for Cattiaux-Pédèches [14] is the setting of Dirichlet forms with test functions having support outside of a measure zero “pairwise collisions” set in \(\mathbb{R}^{2N}\), to address, in particular, the lack of higher integrability of the drift in 1 , see discussion above. They solve the martingale problem with such test functions, and construct the heat kernel for 1 , among many other results. The argument of Fournier-Jourdain [6] appeals directly to the corresponding SDEs. It exploits in an essential manner the special form of the interaction kernel in 1 . The analysis of collisions due to [6] allows [14] to further solve the classical martingale problem, i.e. without cutting out the singular locus of the drift. See also recent developments in Cattiaux [17] where, among many results, the author develops an approach to taming the singularities of the Keller-Segel system based on Orlicz spaces.

Regarding well-posedness of particle systems with other singular interaction kernels (e.g. of Bessel process type), see Graczyk-Malecki [18] and Hufnagel-Andraus [19] and references therein.

We also mention that the theory of SDEs with general singular drifts that can, in particular, introduce strong attraction to submanifolds, was brought up to the task only recently, i.e. one can now handle the Keller-Segel finite particle system 1 as a special case of what is now known about general SDEs [20], albeit with losses in the assumptions on \(\nu\) compared to [6], [14]–[16] whose methods are tailored to 1 . But, on the other hand, there is no loss in higher dimensions \(d \geq 3\) [21], [22]. One now has flexible means for modifying e.g. the interaction kernel in SDE 1 .

There is a rich literature on the Keller-Segel model of chemotaxis, described by a distribution-dependent SDE \[\label{ks95chem} dY_t=(K \star \eta)(Y_t) dt+\sqrt{2} dB_t \quad \text{ in } \mathbb{R}^2,\tag{5}\] where \(K(y)=\nu |y|^{-2}y\), \(y \in \mathbb{R}^2\), is the interaction kernel in 1 , \(\eta(t,y)\) is the law of \(Y_t\) and \(\star\) is the convolution in the spatial variables, see, in particular, [17], [23] and references therein. If \((X_t^{1,N},\dots,X_t^{N,N})\) denotes the solution of 1 then, under the exchangeability hypothesis on the initial condition, the process \(Y_t\) is obtained as the limit of (sub-)sequence \(X_t^{1,N}\) as \(N \rightarrow \infty\), see [6], i.e. \(Y_t\) describes the behaviour of a typical particle in the limit as the number of particles goes to infinity. Regarding the propagation of chaos in general critical settings, see Bresch-Jabin-Wang [5], Jabin-Wang [24] and Hao-Röckner-Zhang [25], see also references therein.

Assuming that some mild regularity conditions are imposed on the density of \(Y_0\), the regularizing effect of the convolution in 5 makes the drift in 5 more regular than the drift in the finite particle system 5 and thus opens up a way for the use of other methods such as De Giorgi method [26]; in this setting, the problem with the lack of the Hardy inequality in dimension \(2\) does not arise.

1.3 Notations↩︎

Given a sequence \(\{T_n\}\) of bounded linear operators \(X \rightarrow Y\) between Banach spaces \(X\), \(Y\), endowed with the operator norm \(\|\cdot\|_{X \rightarrow Y}\), we write \[T=s- Y -\lim_n T_n\] if \[\lim_n\|Tf- T_nf\|_Y=0 \quad \text{ for every f \in X}.\]

Let \(L^p=L^p(\mathbb{R}^{dN},dx)\), \(W^{1,p}=W^{1,p}(\mathbb{R}^{dN},dx)\) denote the usual Lebesgue and Sobolev spaces, respectively.

Set \(\|\cdot\|_p:=\|\cdot\|_{L^p}\) and denote operator norm \(\|\cdot\|_{p \rightarrow q}:=\|\cdot\|_{L^p \rightarrow L^q}\).

Given \(1 < p < \infty\), we set \(p':=\frac{p}{p-1}\).

We denote by \(\mathcal{S}\) the Schwartz space, and by \(\mathcal{S}'\) the space of tempered distributions on \(\mathbb{R}^d\).

Let \(\mathcal{W}^{\alpha,p}\) (\(\alpha>0\)) denote the Bessel potential space endowed with norm \(\|u\|_{p,\alpha}:=\|g\|_p\), \(u=(1-\Delta)^{-\frac{\alpha}{2}}g\), \(g \in L^p\), and \(\mathcal{W}^{-\alpha,p'}\), \(p'=p/(p-1)\), the anti-dual of \(\mathcal{W}^{\alpha,p}\).

Put \[\langle f,g\rangle = \langle f g\rangle :=\int_{\mathbb{R}^{dN}}f g dx\] (all functions considered in this paper are real-valued). For vector fields \(b\), \(\mathsf{f}\), we put \[\langle b,\mathsf{f}\rangle:=\langle b \cdot \mathsf{f}\rangle \qquad \text{(\cdot is the scalar product)}.\]

2 Two-dimensional particles↩︎

The Keller-Segel system 1 can be written as an SDE in \(\mathbb{R}^{2N}\): \[dX_t=\frac{\nabla \psi(X_t)}{\psi(X_t)}dt + \sqrt{2}dB_t, \quad \quad X_t=(X_t^1,\dots,X_t^N),\] where \(B_t=(B_t^1,\dots,B_t^N)\) is a Brownian motion in \(\mathbb{R}^{2N}\), and \[\psi(x):=\prod_{1 \leq i<j \leq N}|x^i-x^j|^{-\frac{\nu}{N}}\] is a Lyapunov function for 1 , i.e. we have, at least at the level of formal calculations, \(L^\ast \psi=0\) for \(L^\ast\) the Kolmogorov forward operator for 1 : \[L^\ast =-\Delta - \frac{\nu}{N}\sum_{i=1}^N \sum_{j=1, j \neq i}^N \nabla_{x_i} \cdot \frac{x^i-x^j}{|x^i-x^j|^2}.\] The former is seen right away once one rewrites operator \(L\) in the form \[L=-\Delta - \frac{\nabla \psi}{\psi} \cdot \nabla \quad \text{ on } \mathbb{R}^{2N}.\] Let us note that \(\psi\) is locally in \(L^1\) if and only if \(\nu<\nu_\star=4\).

In this paper we will consider a particle system that has the same singular behaviour around the collision hyperplanes \(x^i=x^j\) as 1 , but that will make our calculations somewhat simpler. This system corresponds to the Lyapunov function \[\varphi(x):=\psi(x)+1,\] i.e. we replace drift \(\frac{\nabla \psi}{\psi}\) with \(\frac{\nabla \varphi}{\varphi}\) and consider from now on the following modified Keller-Segel type SDE \[\label{ks2} dX_t=\frac{\nabla \varphi(X_t)}{\varphi(X_t)}dt + \sqrt{2}dB_t,\tag{6}\] and the backward Kolmogorov operator \[\Lambda:=-\Delta - \frac{\nabla \varphi}{\varphi} \cdot \nabla \quad \text{ in } \mathbb{R}^{2N}.\] We will use the following regularization of \(\varphi\) and \(\Lambda\): \[\label{varphi95eps} \varphi_\varepsilon(x):= \prod_{1 \leq i<j \leq N}|x^i-x^j|_\varepsilon^{-\frac{\nu}{N}}+1,\tag{7}\] where \[\label{eps95reg} |x^i-x^j|_\varepsilon:=\sqrt{|x^i-x^j|^2+\varepsilon}, \quad \varepsilon>0,\tag{8}\] and \[\Lambda_\varepsilon :=-\Delta - \frac{\nabla \varphi_\varepsilon}{\varphi_\varepsilon} \cdot \nabla.\] The latter has, for every \(\varepsilon>0\), bounded smooth drift, so by the classical theory, \(e^{-t\Lambda_\varepsilon}\) is a strongly continuous semigroup of integral operators in \(L^r\) for every \(1 \leq r<\infty\). We denote their integral kernel by \(p_\varepsilon(t,x,y)\), a smooth function for every \(\varepsilon>0\). One has \[\mathbb{E}_{X_0^\varepsilon=x}[f(X_t^\varepsilon)]=\int_{\mathbb{R}^{2N}}p_\varepsilon(t,x,y)f(y)dy,\] where \(X_t^\varepsilon\) solves SDE \[dX_t=\frac{\nabla \varphi_\varepsilon(X_t)}{\varphi_\varepsilon(X_t)}dt + \sqrt{2}dB_t \quad \text{ in } \mathbb{R}^{2N}.\]

2.1 Comments↩︎

1. We expect the upper heat kernel bound in Theorem 1 to be optimal at \(t=1\) around the “collision hyperplanes” \(x^i=x^j\). An improved upper bound must take into account that it takes time for the singularities around the collision hyperplanes to propagate, where the tradeoff between the distance in time is determined by the parabolic scaling. More precisely, we expect a sharper upper bound to have form \[p(t,x,y)\;\leq\; C t^{-N}\varphi_t(y),\] where \(\varphi_t(y)=\varphi(\frac{y}{\sqrt{t}})\). In fact, we will introduce the proper time-dependence in the weight, and will also include a Gaussian factor, in Theorem 2, that deals with higher dimensions \(d \geq 3\). It should be added that there already exist heat kernel bounds for the Kolmogorov operator with polar drift, which can be viewed as corresponding to the two-particle system, i.e. \(L=-\Delta + \nu |x|^{-2}x\cdot \nabla\) in \(\mathbb{R}^d\): \[\label{bd95w} c_1\Gamma_{c_2 t}(x-y)\hat{\varphi}_t(y) \leq e^{-tL}(x,y) \leq c_3\Gamma_{c_4 t}(x-y)\hat{\varphi}_t(y),\tag{10}\] for all \(t>0\) and a.e. \(x,y \in \mathbb{R}^d\), where \[\Gamma_{t}(x):=(4\pi t)^{-\frac{d}{2}}e^{-\frac{|x|^2}{4t}} \quad \text{ and } \quad \hat{\varphi}_t(y)=\left\{ \begin{array}{ll} \big(\frac{|y|}{\sqrt{t}}\big)^{-\nu}, & |y| \leq \sqrt{t}, \\ \frac{1}{2}, & |y| \geq 2\sqrt{t}. \end{array} \right.\] This result is essentially contained in [11] and is a special case of the result in [12], although the construction of the a posteriori heat kernels in [12] is quite different.

Note that, with this dependence on \(t\), the weight disappears as \(t \downarrow 0\), and so at time \(t=0\) one recovers the delta-function at \(x=y\), as one would expect.

2. The a priori upper heat kernel bound Theorem 1(i) is a consequence of the following general desingularization theorem. Let \(X\) be a locally compact topological space, let \(\mu\) be a \(\sigma\)-finite Borel measure on \(X\). Let \(L^\infty_{{\rm com}}\) denote bounded functions with compact support.

(The constant \(C\) is given explicitly in terms of \(c_0\), \(c_1\) and \(a\), see Appendix 8.)

Theorem A was introduced in [2] for different purposes, i.e. to prove an upper bound on the heat kernel of the fractional Kolmogorov operator with singular polar drift in \(\mathbb{R}^d\), \(d \geq 3\): \[\label{frac95kolm} (-\Delta)^{\frac{a}{2}} + \kappa \frac{x}{|x|^a}\cdot \nabla, \quad 1<a<2.\tag{11}\] It is interesting to note that in [2] the verification of condition ?? was relatively easy, i.e. via the standard Sobolev inequality and a variant of the classical Nash’s argument in \(L^r\). The main difficulty in [2] is in the verification of ?? due to the non-locality of \((-\Delta)^{\frac{a}{2}}\). In the present paper, however, the main difficulty is in the verification of ?? , while verifying ?? is easy since our Kolmogorov operator is local.

The proof of Theorem A uses a weighted variant of the Coulhon-Raynaud extrapolation theorem which is interesting on its own. To make the paper self-contained, we included the proof in Appendix 8.

3. The process \[R_t:=\frac{1}{4N}\sum_{i,j=1}^N|X_t^i-X_t^j|^2\] is a local squared Bessel process of dimension \(\delta=(N-1)(2-\frac{\nu}{2})\), i.e. \(R_t=R_0+2\int_0^t \sqrt{R_s}dW_s + \delta t,\) see [1]. In turn, the corresponding Bessel process \(X_t=\sqrt{R_t}\), i.e. \[X_t=X_0+\frac{\delta-1}{2}\int_0^t X_s^{-1}ds + W_t, \quad X_0 \geq 0,\] has explicit heat kernel \[\label{bessel95heat} p(t,x,y)=\left\{ \begin{array}{ll} \frac{y}{t}\big(\frac{y}{x}\big)^{\frac{\delta}{2}-1}e^{-\frac{x^2+y^2}{2t}}I_{\frac{\delta}{2}-1}\big(\frac{xy}{t}\big), & x,y>0 \\ \frac{2^{-\frac{\delta}{2}+1}}{\Gamma(\frac{\delta}{2})}t^{-\frac{\delta}{2}}y^{\delta-1}e^{-\frac{y^2}{2t}}, & y>0,\; x=0, \end{array} \right.\tag{12}\] where \(I_{\frac{\delta}{2}-1}\) is the modified Bessel function (see e.g. [27]). We are not aware of any explicit formulas for the heat kernel of the Keller-Segel particle system 1 . (It should be added that, as is well known, having an explicit formula for a heat kernel does not necessarily mean that it is easy to derive from it some practical elementary estimates, such as the one in Theorem 1.) Still, the question arises whether the methods that we use in the present paper is an overkill. To some extent, they are, i.e. they allow to prove more than what we are asking for in this paper (hence more restrictive than one would hope conditions on the strength of attraction between the particles). To illustrate this, in the end of the next section we discuss extending the upper heat kernel bound in Theorem 2 to the particle system additionally “immersed” in a turbulent flow.

4.  Along the proof of Theorem 1 we show that the many-particle drift \(b:=\frac{\nabla \varphi}{\varphi}:\mathbb{R}^{2N} \rightarrow \mathbb{R}^{2N}\) in the operator \(\Lambda\) belongs to the class \(\mathbf{F}^{\scriptscriptstyle 1/2}_\delta\) of weakly form-bounded vector fields, i.e. \(b \in L^1_{{\rm loc}}(\mathbb{R}^{2N})\) and \[\||b|^{\frac{1}{2}}(-\Delta)^{-\frac{1}{4}}\|_{L^2(\mathbb{R}^{2N}) \rightarrow L^2(\mathbb{R}^{2N})} \leq \delta\] for \(\delta=C \sqrt{\nu}\). This includes many classes of singular vector fields found in the literature, such as essentially the largest Morrey class: for an \(\varepsilon>0\) fixed arbitrarily small, the integral of \(|b|^{1+\varepsilon}\) over a ball of radius \(r>0\) in \(\mathbb{R}^{2N}\) must satisfy \(\int_{B_r(x)} |b|^{1+\varepsilon} \leq cr^{2N-1-\varepsilon}\) for constant \(c\) independent of \(r\) or the centre \(x \in \mathbb{R}^{2N}.\) There is a detailed Sobolev regularity theory of parabolic equations with weakly form-bounded drifts, and a weak solution theory of the corresponding stochastic differential equations, see [20], [28], [29] (regarding SDEs with Morrey class drifts, see also Krylov [30] and references therein, although he deals with a different part of the Morrey scale since he also allows irregular diffusion coefficients). All these results require, naturally, the weak form-bound \(\delta\) to be sufficiently small, otherwise one runs into blow up effects. By considering \(b:=\frac{\nabla \varphi}{\varphi}\) as a weakly form-bounded drift one thus has in the setting of Theorem 1:

1. By running Lions’ variational approach in the “non-standard” Hilbert triple \[\mathcal{W}^{-\frac{1}{2},2}(\mathbb{R}^{2N}) \rightarrow \mathcal{W}^{\frac{1}{2},2}(\mathbb{R}^{2N}) \rightarrow \mathcal{W}^{\frac{3}{2},2}(\mathbb{R}^{2N})\] of Bessel potential spaces, one can show that the Cauchy problem \((\partial_t + \Lambda)u=0\), \(u|_{t=0}=u_0,\) has a unique (appropriately defined) weak solution, see [29].

2. For the solutions of the corresponding elliptic equation one has the following Sobolev regularity result: \[(\mu+\Lambda)^{-1} \text{ is a bounded operator from } \mathcal{W}^{-\frac{1}{r'},p}(\mathbb{R}^{2N}) \text{ to } \mathcal{W}^{1+\frac{1}{q},p}(\mathbb{R}^{2N})\] \[\text{for 1 \leq r and q<\infty satisfying r<p<q}\] (generally speaking, fixed close to \(p\) since it gives the strongest regularity result) and \(p\) that can be chosen arbitrarily large at the expense of requiring \(\delta\) (and thus the strength of attraction \(\nu\)) to be sufficiently small; this allows to further construct a realization of \(\Lambda\) as a Feller generator using the above embedding into \(\mathcal{W}^{1+\frac{1}{q},p}(\mathbb{R}^{2N})\) and applying the Sobolev embedding theorem [31].

Considering \(b=\frac{\nabla \varphi}{\varphi}\) as a weakly form-bounded drift produces a quite restrictive constraint on the strength of attraction between the particles \(\nu\), i.e. the one that corresponds to the choice \(\alpha=1\) in 9 ; it is important to be able to choose appropriate \(1<\alpha<2\). In the proof of Theorem 1 we introduce the class \(\mathbf{F}^{\scriptscriptstyle \alpha/2}_\delta\) of \(\alpha\)-weakly form-bounded drifts and develop some aspects of its theory that are needed to prove Theorem 1. The extensions of the Sobolev regularity results from [31] and of the Lions’ variational approach from [29] to the \(\alpha\)-weakly form-bounded drifts are possible, but we will not pursue them here.

3 Higher dimensions↩︎

Let \(d \geq 3\). The following counterpart of system 1 is of interest since it exhibits blow up effects analogous to the ones discussed in the beginning of the introduction (with \(\nu_\star=\nu_\star(d)\), see [22] for details): \[\label{ks4} dX_t^i = - \frac{\nu}{N}\sum_{j=1, j \neq i}^N \frac{X_t^i-X_t^j}{|X_t^i-X_t^j|^2}dt + \sqrt{2}dB_t^i,\tag{13}\] where \(\{B_t\}_{t \geq 0}\) is a Brownian motion in \(\mathbb{R}^{dN}\). The Kolmogorov operator behind this particle system is operator \(L\) defined in Section 2: \[L=-\Delta + \frac{\nu}{N}\sum_{i=1}^N\sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|^2} \cdot \nabla_i.\] Compared to Theorem 1, we will additionally include the Gaussian factor in the upper bound and will introduce a proper dependence of the weight on time which will allow us, in particular, to recover in the heat kernel the Dirac delta-function at \(x=y\) as \(t \downarrow 0\).

Define \[\psi(x)=\prod_{1 \leq i<j \leq N} |x^i-x^j|^{-\frac{\nu}{N}}.\] Then 13 and \(L\) take form \[dX_t=\frac{\nabla \psi(X_t)}{\psi(X_t)}+\sqrt{2}dB_t, \quad X_t=(X_t^1,\dots,X_t^N),\] and \(L=-\Delta - \frac{\nabla \psi}{\psi} \cdot \nabla.\)

Case \(d \geq 4\). Recalling that \(|x^i-x^j|_\varepsilon^2:=|x^i-x^j|^2+\varepsilon\), we define a regularization of \(\psi\), \[\psi_\varepsilon(x)=\prod_{1 \leq i<j \leq N} |x^i-x^j|_\varepsilon^{-\frac{\nu}{N}},\] and consider the approximating operators \(L_\varepsilon=-\Delta - \frac{\nabla \psi_\varepsilon}{\psi_\varepsilon} \cdot \nabla\), i.e. \[\label{L95eps95} L_\varepsilon=-\Delta + \frac{\nu}{N}\sum_{i=1}^N\sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|_\varepsilon^2} \cdot \nabla_i.\tag{14}\] The desingularizing weights will now be defined in a somewhat different manner compared to Theorem 1. Instead of \(\varphi_\varepsilon\), \(\varphi\), we consider \[\hat{\varphi}_{\varepsilon}(y):=\prod_{1 \leq i<j \leq N} \eta(|y^i-y^j|_\varepsilon),\] \[\hat{\varphi}(y):=\prod_{1 \leq i<j \leq N} \eta(|y^i-y^j|),\] where \(0< c \leq \eta \in W_{loc}^{2,\infty}(]0,\infty[)\) is a cutoff function: \[\label{eta95def} \eta(r)=\left\{ \begin{array}{ll} r^{-\frac{\nu}{N}} & 0<r<1, \\ e^{\frac{\nu}{N}(r-1)}r^{-2\frac{\nu}{N}} & 1 \leq r \leq 2, \\ e^{\frac{\nu}{N}} 2^{-2\frac{\nu}{N}}, & r>2. \end{array} \right.\tag{15}\] Let us emphasize that, taking into account \(N \geq 2\), \(\nu<2(d-2)\), in the region \(r>2\) the cutoff function \(\eta\) is bounded from below by a positive \(N\)-independent constant \(e^{d-2} 2^{-2(d-2)}>0\). With this choice of \(\eta\), we have on the interval \([1,2]\) the following inequality that will be used in the proof of Lemma 2: \[\label{lem295ineq} \frac{\eta'(r)}{\eta(r)}+\frac{\nu}{N}\frac{1}{r} \geq 0.\tag{16}\] The continuity of the first derivatives of \(\eta\) and the boundedness of the second derivatives will also be used in the proof of Lemma 2.

It is easily seen that the weights \(\hat{\varphi}_\varepsilon\), \(\hat{\varphi}\) are still pointwise comparable to the weights \(\varphi_\varepsilon\), \(\varphi\) in Theorem 1. Next, we introduce a proper time-dependence in the weight, i.e. apply the parabolic scaling: \[\hat{\varphi}_{t,\varepsilon}(y):=\hat{\varphi}_\varepsilon(t^{-\frac{1}{2}}y),\] \[\hat{\varphi}_{t}(y):=\hat{\varphi}(t^{-\frac{1}{2}}y).\] Let \(k_\varepsilon(t,x,y)\) denote the heat kernel of \(L_\varepsilon\). We estimate it from above in Theorem 2. Before stating the theorem, we discuss what in the previous definitions needs to be adjusted in the case \(d=3\).

Case \(d=3\). We regularize \(\psi\) differently: \[\psi_\varepsilon(x)=\prod_{1 \leq i<j \leq N} |x^i-x^j|_{[\varepsilon]}^{-\frac{\nu}{N}},\] where \[\label{eps95reg2} |x^i-x^j|_{[\varepsilon]}:=|x^i-x^j|+\varepsilon.\tag{17}\] (If we were to use the previous regularization of \(\psi\) for \(d=3\), we would have to impose more restrictive condition on \(\nu\) in the proof of Lemma 2 used in the proof of the weighted Sobolev embedding in Lemma 3, and therefore in Theorem 2.) The other definitions do not change, up to replacing \(|x^i-x^j|_\varepsilon\) by \(|x^i-x^j|_{[\varepsilon]}\). We emphasize that by dealing with this regularization of the Euclidean norm, we lose smoothness of the regularized drift and only have boundedness. Namely, using \(\nabla_i\,|x^i-x^j|_{[\varepsilon]}^{-\frac{\nu}{N}}=-\frac{\nu}{N}|x^i-x^j|_{[\varepsilon]}^{-\frac{\nu}{N}-1}\frac{x^i-x^j}{|x^i-x^j|}\), we evaluate: \[L_\varepsilon=-\Delta + \frac{\nu}{N}\sum_{i=1}^N\sum_{j=1, j \neq i}^N \frac{1}{|x^i-x^j|+\varepsilon}\frac{x^i-x^j}{|x^i-x^j|} \cdot \nabla_i\] (cf.@eq:L95eps95 ). Even with an \(L^\infty\) drift, our manipulations with the solution \(u_\varepsilon\) of the \((\partial_t+L_\varepsilon)u_\varepsilon=0\) in the proof of Theorem 2 are justified. (The only potentially subtle situation that we face is when we deal with singular potential \(-{\rm div\,}\frac{\nabla \psi_\varepsilon}{\psi_\varepsilon}\) in the proof of the Sobolev embedding in Lemma 3, but we will be careful there.)

3.1 Comments↩︎

1. Regarding the optimality of the upper bound in Theorem 2, we invoke the “two-particle” two-sided heat kernel bounds due to [11], [12] discussed after Theorem 1, see also Appendix 12. In fact, it is possible to extend the proof of the lower bound from [11] via weighted Nash’s method to the particle system of Theorem 2 under the condition \(\nu<\frac{C}{N}\); this amounts to proving weighted Spectral gap inequality differently: the proof in [11] exploits the fact that, in the “two-particle case” the unbounded part of the desingularizing weight \(\hat{\varphi}\) has compact support. We plan to address the lower heat kernel bound in a subsequent paper.

2. Since \(d \geq 3\), we can use in the proof of Theorem 2 the many-particle Hardy inequality due to [32], \[\label{hardy} \frac{(d-2)^2}{N}\sum_{1 \leq i<j \leq N}\int_{\mathbb{R}^{dN}}\frac{|f(x)|^2}{|x^i-x^j|^2}dx \leq \int_{\mathbb{R}^{dN}}|\nabla f(x)|^2 dx, \quad f \in W^{1,2}(\mathbb{R}^{dN}),\tag{19}\] which allows us to work in the standard framework of Dirichlet forms, prove the corresponding weighted Sobolev inequality and thus obtain an improved, compared to Theorem 1, upper heat kernel bound using Moser’s iterations.

One can obtain 19 , albeit with a constant that is two times smaller, by adding up the ordinary Hardy inequalities, each in its own copy of \(\mathbb{R}^d\). Note that there is no non-trivial analogue of 19 when \(d=2\), see [32]. The constant in 19 is not the best possible. In fact, in dimensions \(3 \leq d \leq 6\) [32] obtains, using an additional geometric argument, a larger constant: \[\label{multi95hardy} C_{d,N}\sum_{1 \leq i<j \leq N}\int_{\mathbb{R}^{dN}}\frac{|f(x)|^2}{|x^i-x^j|^2}dx \leq \int_{\mathbb{R}^{dN}}|\nabla f(x)|^2 dx,\tag{20}\] with \[\label{HHLT95est} C_{d,N} \geq (d-2)^2 \max\bigg\{\frac{1}{N},\frac{1}{1+\sqrt{1+\frac{3(d-2)^2}{2(d-1)^2}(N-1)(N-2)}} \bigg\}.\tag{21}\] To our knowledge, the question of what is the best possible constant in 19 is still open in all dimensions \(d \geq 3\). (There is, however, an upper bound \(C_{d,N} \leq \frac{d(d-2)}{N}\), see [22], so for large \(d\) 19 is actually close to the best possible constant.)

3.  The proof of Theorem 2 extends to the operator \[\label{ks7} \Lambda=- \nabla \cdot (I+C) \cdot \nabla - \frac{\nabla \psi}{\psi}\cdot (I+C)\cdot \nabla,\tag{22}\] where \(C \in [L^\infty(\mathbb{R}^{dN})]^{d \times d}\) is a skew-symmetric matrix (Appendix 10). In the case there is no interaction between the particles, i.e. \(\nu=0\) and so \(\nabla \psi=0\), this operator was treated by Osada [33] who has obtained two-sided Gaussian bounds on its heat kernel. This operator arises in the linear theory connected to the Navier-Stokes equations: due to the skew-symmetry of \(C\), one can write \[- \nabla \cdot (I+C) \cdot \nabla = -\Delta - (\nabla C)\cdot \nabla,\] where \(\nabla C\) is a distributional divergence-free drift (\(=\) velocity field). (Qian-Xi [34] extended the result of Osada to the case when the stream matrix \(C\) has entries in the space \({\rm BMO}\).) The particle system determined by 22 can be viewed as 13 immersed in a turbulent flow. In contrast to 13 , in 22 there is an additional drift \(\nabla C\) that is not only singular, but also lacks any particular structure. So, there is no hope of obtaining an explicit formula for the heat kernel of 22 similar to 12 for the Bessel process.

4. To handle 13 , we could also repeat the proof of Theorem 1 for operator \(\Lambda\) in dimensions \(d \geq 3\). Higher dimensions \(d \geq 3\) give us the following condition on the admissible strength of attraction between the particles \(\nu\): \[\label{alpha95max2} \nu<\max_{1 \leq \alpha \leq 2} \biggl[ \frac{N^{\frac{3}{2}\alpha-1}}{(N-1)^{1+\frac{\alpha}{2}}}2^\alpha \frac{\Gamma(\frac{d}{4}+\frac{\alpha}{4})^2}{\Gamma(\frac{d}{4}-\frac{\alpha}{4})^2}\biggr]^{\frac{1}{\alpha}}.\tag{23}\] Now, in dimensions greater or equal to three, \(\alpha=2\) is admissible, moreover, \(\alpha=2\) is optimal, i.e. it gives the least restrictive constraint on \(\nu\), and this constraint essentially does not depend on \(N \rightarrow \infty\). So, 23 should be read as \[\label{alpha95max3} \nu<2\frac{N}{N-1}\frac{\Gamma(\frac{d}{4}+\frac{1}{2})}{\Gamma(\frac{d}{4}-\frac{1}{2})}.\tag{24}\] (Taking \(\alpha=2\) means that we are using the usual Hardy inequality.)

4 Proof of Theorem 1↩︎

(i) We apply Theorem A to \(\varphi:=\varphi_\varepsilon\), where \(\varphi_\varepsilon\) is defined by 7 , and \(\Lambda:=\Lambda_\varepsilon\), where, recall, \[\Lambda_\varepsilon=-\Delta - b_\varepsilon \cdot \nabla,\] \[b_\varepsilon=\frac{\nabla \varphi_\varepsilon}{\varphi_\varepsilon}=\frac{\nabla \psi_\varepsilon}{\psi_\varepsilon+1}, \quad \psi_\varepsilon(x)= \prod_{1 \leq i<j \leq N}|x^i-x^j|_\varepsilon^{-\frac{\nu}{N}}.\]

The conditions ?? and ?? are, obviously, satisfied for \(\varphi_\varepsilon\) with \(c_0=1\) (here, crucially, \(c_0\) does not depend on \(\varepsilon\)).

Let us verify condition ?? . Here we are dealing with operators with bounded smooth coefficients (i.e. \(\Lambda_\varepsilon\)) and with bounded smooth weights (i.e. \(\varphi_\varepsilon\)), so it is easily seen that \(\varphi_\varepsilon \Lambda_\varepsilon \varphi_\varepsilon^{-1}\) is the generator of \(\varphi_\varepsilon e^{-t\Lambda_\varepsilon} \varphi_\varepsilon^{-1}\) in \(L^1\). Let us compute \(\varphi_\varepsilon \Lambda_\varepsilon \varphi_\varepsilon^{-1}\): \[\begin{align} \varphi_\varepsilon \Lambda_\varepsilon \varphi_\varepsilon^{-1} v = -\Delta v + \nabla \cdot \bigg(\frac{\nabla \varphi_\varepsilon}{\varphi_\varepsilon}v\bigg), \end{align}\] so \(\varphi_\varepsilon \Lambda_\varepsilon \varphi_\varepsilon^{-1}\) generates an \(L^1\) contraction semigroup (\(\Leftrightarrow\) ?? holds).

Next, let us verify ?? for \(r=2\) and \(a=2N\) (i.e. the dimension of the Euclidean space \(\mathbb{R}^{2N}\) where our particle system exists). That is, our goal is to prove \[\label{disp95est} \|e^{-t\Lambda_\varepsilon}\|_{2 \rightarrow \infty} \leq ct^{-\frac{2N}{2}}, \quad t \in ]0,T],\tag{25}\] with \(c\) independent of \(\varepsilon\). We will prove 25 in three steps:

Step 1. First, we show that, for every \(1 \leq \alpha<2\), the vector field \(b_{\varepsilon}\) is \(\alpha\)-form-bounded uniformly in \(\varepsilon\), i.e. the following operator norm inequality holds \[\label{b95g0} \||b_\varepsilon|^{\frac{\alpha}{2}}(-\Delta)^{-\frac{\alpha}{4}}\|_{2 \rightarrow 2} \leq \sqrt{\delta}\tag{26}\] or, equivalently, \[\label{b95g} \langle |b_\varepsilon|^\alpha g,g\rangle \leq \delta \langle (-\Delta)^{\frac{\alpha}{4}}g,(-\Delta)^{\frac{\alpha}{4}}g\rangle \quad \forall\,g \in \mathcal{S},\tag{27}\] with the constant \(\delta\) (measuring the strength of singularities of \(b_\varepsilon\) and called the \(\alpha\)-form-bound) given by \[\delta=\nu^\alpha \frac{(N-1)^{1+\frac{\alpha}{2}}}{N^{\frac{3\alpha}{2}-1}} \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}.\] This will be abbreviated as \(b_{\varepsilon} \in \mathbf{F}_\delta^{\frac{\alpha}{2}}=\mathbf{F}_\delta^{\frac{\alpha}{2}}(\mathbb{R}^{2N})\). (At Step 2 we will need \(\delta<1\), hence the condition on \(\nu\). So, in order to arrive to the least restrictive conditions on \(\nu\) that can be obtained using this argument, later we will choose \(\alpha\) that minimizes the value of \(\delta\).)

The proof of 27 will use the fractional Hardy inequality in \(\mathbb{R}^2\), applied consecutively in each variable \(x^i\), \(i=1,\dots,N\). That is, if we denote by \(b^i_\varepsilon\) the \(i\)-th component of \(b\), i.e. \[b^i_\varepsilon=\frac{\nabla_{i} \varphi_\varepsilon}{\varphi_\varepsilon}=\frac{\nabla_i \psi_\varepsilon}{\psi_\varepsilon+1},\] then \[\label{majorant} |b^i_\varepsilon| \leq \frac{|\nabla_i \psi_\varepsilon|}{\psi_\varepsilon} \leq \frac{\nu}{N}\sum_{j=1, j \neq i}^N \frac{|x^i-x^j|}{|x^i-x^j|^2+\varepsilon} \leq \frac{\nu}{N}\sum_{j=1, j \neq i}^N \frac{1}{|x^i-x^j|}.\tag{28}\] Hence, applying Cauchy-Schwarz once, we can estimate \[\begin{align} |b_{\varepsilon}|^{\alpha} & = \big( \sum_{i=1}^N |b_{\varepsilon}^i|^{2} \big)^{\alpha/2} \leq \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \left( \sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{1}{|x_i-x_j|^{2}} \right)^{\alpha/2}. \end{align}\] Since \(\frac{\alpha}{2}<1\), we further obtain \[|b_{\varepsilon}|^{\alpha} \le \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{1}{|x_i-x_j|^{\alpha}}.\] Therefore, returning to our task of estimating the LHS of 27 , we write \[\begin{align} \langle |b_\varepsilon|^{\alpha}g, g\rangle & \leq \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \sum_{i=1}^N \sum_{j=1,j \neq i}^N \left\langle \frac{1}{|x_i-x_j|_{}^{\alpha}}g, g\right\rangle, \end{align}\] where, denoting by \(\bar{x}\) vector \(x\) with component \(x^i\) removed, we further estimate \[\begin{align} \left\langle \frac{1}{|x_i-x_j|_{}^{\alpha}}g,g \right\rangle & = \int_{\mathbb{R}^{(N-1)2}}\int_{\mathbb{R}^2}\frac{1}{|x_i-x_j|_{}^{\alpha}} g^{2}(x^i,\bar{x})dx^i d\bar{x}, \\ & (\text{we apply the fractional Hardy inequality \cite{KPS} in x^i \in \mathbb{R}^2}) \\ & \le \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}} \int_{\mathbb{R}^{(N-1)2}}\int_{\mathbb{R}^2} | (-\Delta_{x_i})^{\frac{{\alpha}}{4}}g(x_i,\bar{x})|^2dx_i d\bar{x} \\ & \equiv \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}\|(-\Delta_{x_i})^{\frac{{\alpha}}{4}}g\|_2^2. \end{align}\] Thus, \[\begin{align} \langle |b_\varepsilon|^{\alpha}g, g\rangle & \leq \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}\sum_{i=1}^N \sum_{j=1,j \neq i}^N\|(-\Delta_{x_i})^{\frac{{\alpha}}{4}}g\|_2^2 \\ & = \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}(N-1)\sum_{i=1}^N\|(-\Delta_{x_i})^{\frac{{\alpha}}{4}}g\|_2^2. \end{align}\] Denoting by \(\hat{g}(\xi)\) the Fourier transform of \(g(x)\) and using \[\sum_{i=1}^N \langle |\xi_i|^\alpha \hat{g},\hat{g}\rangle \leq N^{1-\frac{\alpha}{2}} \langle |\xi|^\alpha \hat{g},\hat{g}\rangle,\] we finally obtain \[\langle |b_\varepsilon|^{\alpha}g, g\rangle \leq \big(\frac{\nu}{N}\big)^\alpha(N-1)^{\frac{\alpha}{2}} \frac{1}{2^\alpha} \frac{\Gamma\left( \frac{1}{2} -\frac{\alpha}{4} \right)^{2}}{\Gamma\left( \frac{1}{2} + \frac{\alpha}{4} \right)^{2}}(N-1) N^{1-\frac{\alpha}{2}}\|(-\Delta)^{\frac{{\alpha}}{4}}g\|_2^2,\] i.e. we have arrived at 27 .

Step 2. Having bound 26 at hand, we can establish the following resolvent representation for \(\Lambda_\varepsilon\) in the complex half-plane; it will play a crucial role in what follows. That is, for all \({\rm Re}\,\,\eta > 0\), \[\label{lambda95repr} (\eta+\Lambda_\varepsilon)^{-1}=(\eta-\Delta)^{-1}-(\eta-\Delta)^{-\frac{1}{2}-\frac{\alpha}{4}}Q(1+T)^{-1}R(\eta-\Delta)^{-\frac{1}{2}+\frac{\alpha}{4}},\tag{29}\] where³ \[\begin{align} & R:=b_\varepsilon^{\frac{\alpha}{2}}\cdot \nabla (\eta-\Delta)^{-\frac{1}{2}-\frac{\alpha}{4}}, \\[2mm] & Q:=(\eta-\Delta)^{-\frac{1}{2}+\frac{\alpha}{4}}|b_\varepsilon|^{1-\frac{\alpha}{2}} \end{align}\] and \[T:=R\,Q.\] The operators \(R\), \(Q\) and \(T\) are uniformly in \(\varepsilon\) bounded on \(L^2\), and \(\|T\|_{2 \rightarrow 2}<1\), so the geometric series in 29 converges; once this is established, one can see right away that 29 is the Neumann series for \((\eta+\Lambda_\varepsilon)^{-1}\). In detail, \[\begin{align} \|R\|_{2 \rightarrow 2} & \leq \||b_\varepsilon|^{\frac{\alpha}{2}}(\eta-\Delta)^{-\frac{\alpha}{4}}\|_{2 \rightarrow 2}\|\nabla(\eta-\Delta)^{-\frac{1}{2}}\|_{2 \rightarrow 2} \\ & \leq \||b_\varepsilon|^{\frac{\alpha}{2}}(-\Delta)^{-\frac{\alpha}{4}}\|_{2 \rightarrow 2} \leq \sqrt{\delta}, \end{align}\] where we have used the boundedness of the Riesz transform and estimate 27 combined with \(|(\eta-\Delta)^{-\frac{\alpha}{4}} (x,y)| \leq (-\Delta)^{-\frac{\alpha}{4}} (x,y),\) which is immediate from \[(\eta-\Delta)^{-\frac{\alpha}{4}} (x,y) = \frac{1}{\Gamma\bigl(\frac{\alpha}{4}\bigr)}\int_0^\infty e^{-\eta t} t^{\frac{\alpha}{4}-1} (4\pi t)^{-N}e^{-\frac{|x-y|^2}{4t}} dt.\] Next, by duality, \[\|Q\|_{2 \rightarrow 2} = \||b|^{1-\frac{\alpha}{2}}(\eta-\Delta)^{-\frac{1}{2}+\frac{\alpha}{4}}\|_{2 \rightarrow 2} \leq \delta^{\frac{2-\alpha}{2\alpha}},\] where at the last step we have applied the Heinz inequality to 26 , which amounts to raising both operators under the operator norm in 26 to power \(\frac{2-\alpha}{\alpha}<1\). It now follows that \[\|T\|_{2 \rightarrow 2} \leq \|R\|_{2\rightarrow 2}\|Q\|_{2 \rightarrow 2} \leq \delta^{\frac{1}{\alpha}}.\] Now, our condition on the strength of attraction between the particles \(\nu\) ensures that \(\delta<1\), so \(\|T\|_{2 \rightarrow 2} <1\) and thus the geometric series in 29 converges.

Crucially, the estimates on the operator norms of \(R\), \(Q\) and \(T\) are independent of \(\varepsilon\).

As an immediate consequence of 29 , we obtain \[\label{eta95est} \|(\eta+\Lambda_\varepsilon)^{-1}\|_{2 \rightarrow 2} \leq \frac{C}{|\eta|}, \quad {\rm Re}\,\,\eta>0,\tag{30}\] where \(C>1\) is independent of \(\varepsilon\), so \[\label{bdd95est} \|e^{-t\Lambda_\varepsilon}\|_{2 \rightarrow 2} \leq K e^{\omega t}, \quad t \geq 0\tag{31}\] for \(K>1\) and \(\omega\) independent of \(\varepsilon\), see [36].

Step 3. We are in position to establish the dispersion estimate 25 . The following argument, up to a few minor modifications, is due to Semënov [4]. The plan is as follows: we will prove that, for some \(r \in ]2, \infty[,\) \[\| e^{-t \Lambda_\varepsilon} \|_{2 \rightarrow r} \leq C t^{-N \big(\frac{1}{2} - \frac{1}{r} \big)} \quad \quad t \in ]0,T]. \] Then, using the extrapolation (Theorem 4 with \(\psi=1\)) between the previous estimate and the straightforward a priori bound \(\|e^{-t\Lambda_\varepsilon} f \|_\infty \leq \|f\|_\infty\), we will arrive at the sought estimate 25 .

Let us prove \((\star)\). Let \(\lambda \geq 1.\) Set \[\Gamma_0 = \lambda - \Delta,\quad \Gamma = \lambda + \Lambda_\varepsilon.\] It follows from 29 that, for every \(\mu \geq 0\), \[\begin{align} \|(\mu + \Gamma)^{-1} \|_{2 \rightarrow 2} & \leq (1-\delta^{\frac{1}{\alpha}})^{-1} \|(\mu + \Gamma_0)^{-\frac{1}{2}-\frac{\alpha}{4}} \|_{2 \rightarrow 2} \|(\mu + \Gamma_0)^{-\frac{1}{2}+\frac{\alpha}{4}} \|_{2 \rightarrow 2} \notag \\ & \leq (1-\delta^{\frac{1}{\alpha}})^{-1} (1+ \mu)^{-1}. \label{gamma951} \end{align}\tag{32}\] Hence, by interpolation, for any \(r \in ]2, \infty [\), \[\|(\mu + \Gamma)^{-1} \|_{r \rightarrow r} \leq c (1+\mu)^{-1}.\] Next, given \(0<\beta<1\), we have well-defined fractional powers \[\Gamma^{-\beta} = \frac{\sin \pi \beta}{\pi} \frac{1}{1-\beta} \int_0^\infty \mu^{1-\beta} (\mu + \Gamma)^{-2} d\mu\] and \(\Gamma^\beta := \big( \Gamma_r^{-\beta} \big)^{-1}\). We have \[\label{e1} \|\Gamma^\beta (\mu + \Gamma)^{-1} \|_{2 \rightarrow 2} \leq C(1+ \mu)^{-1 +\beta}\tag{33}\] and \[\label{e2} \|\Gamma^\beta e^{-t \Gamma} \|_{2 \rightarrow 2} \leq C t^{-\beta},\tag{34}\] (for the proof see e.g. [37]).

Now, fix \(\beta = \frac{1}{4}\) and \(r = 2\frac{2 N}{2N-1}.\) Let \[F_t := \Gamma^{2 \beta} e^{-t \Gamma} f, \quad f \in L^2 \cap L^r.\] We have \[\label{gamma952} \|e^{-t \Gamma} f \|_r = \| \Gamma^{-2 \beta} F_t \|_r \leq \frac{2}{\pi} \int_0^\infty \mu^{1-2 \beta} \|(\mu + \Gamma_r)^{-2} F_t \|_r d \mu.\tag{35}\] Let us estimate the right-hand side. Using the embedding \((\mu +\Gamma_0)^{-\beta} L^2 \subset L^r\), we obtain from 29 \[\begin{align} \| (\mu + \Gamma)^{-2} F_t \|_r \leq c_1\|(\mu + \Gamma_0)^{-\frac{1}{4}-\frac{\alpha}{4}} ( 1 + T)^{-1} (\mu + \Gamma_0)^{-\frac{1}{2}+\frac{\alpha}{4}} (\mu + \Gamma)^{-1} F_t \|_2. \end{align}\] Therefore, by 32 , \[\|(\mu + \Gamma)^{-2} F_t \|_r \leq c_1(1-\delta^{\frac{1}{\alpha}})^{-1} \mu^{-3 \beta} \|(\mu + \Gamma)^{-1} F_t \|_2 .\] Thus, returning to 35 and using 33 , 34 and \(\|e^{-t\Lambda_\varepsilon}\|_{2 \rightarrow 2} \leq Ke^{\omega t}\), \[\begin{align} \| e^{-t \Gamma} f \|_r & \leq C \int_0^\infty \mu^{-\beta} \|(\mu + \Gamma)^{-1} F_t \|_2 \; d \mu\\ &\leq C_1 \bigg ( \int_0^\frac{1}{t} \mu^{-\beta} (1 + \mu)^{-1+2 \beta} d \mu \;\|f\|_2 + \int_\frac{1}{t}^\infty \mu^{-\beta-1} d \mu \; \|\Gamma^{2 \beta} e^{-t \Gamma} f \|_2 \bigg) \\ & \leq C_2 \bigg ( \int_0^\frac{1}{t} \mu^{-\frac{3}{4}} d \mu + \int_\frac{1}{t}^\infty \mu^{-\frac{5}{4}} d \mu \; t^{-2 \beta} \bigg) \|f\|_2 = 4 C_2 \; t^{-\beta} \|f\|_2. \end{align}\] Note that, with our choice of \(\beta\) and \(r\), we have \(\beta=N(\frac{1}{2}-\frac{1}{r})\). This ends the proof of \((\star)\).

In view of the discussion in the beginning of Step 3, \((\star)\) gives us 25 , i.e. we have verified ?? . Now Theorem A, i.e. the upper bound ?? , yields assertion (i).

(ii) The convergence of the semigroups \(e^{-t\Lambda_\varepsilon}\) to a strongly continuous semigroup \(e^{-t\Lambda}\) will follow from the Trotter approximation theorem (Appendix 9). Its conditions:

\(1^\circ\)) \(\sup_{\varepsilon>0}\|(\zeta+\Lambda_\varepsilon)^{-1}\|_{2 \rightarrow 2} \leqslant C|\zeta|^{-1}\), \({\rm Re}\,\zeta>0\).

\(2^\circ\)) \(\mu (\mu+\Lambda_\varepsilon)^{-1} \overset{s}{\rightarrow} 1 \text{ in L^2 as } \mu\uparrow \infty \text{ uniformly in \varepsilon}.\)

\(3^\circ\)) There exists \({\text{\small s-L^2-}}\lim_{\varepsilon \downarrow 0} (\zeta+\Lambda_\varepsilon)^{-1}\) for some \({\rm Re}\,\zeta>0\).

\(1^\circ\)) is the content of 30 . \(2^\circ\)) is proved in the same way as in [31] (see also [38]), using the resolvent representation 29 where the gradient in the last occurrence of \(R\) is placed on the function on which the resolvent acts (it suffices to verify \(2^\circ\)) e.g. on \(C_c^\infty\)). This gives us an extra \(\mu^{-\frac{1}{2}}\), which allows us to prove the convergence in \(2^\circ\)). \(3^\circ\)) is also proved in the same way as in [31] or [38], that is, we apply in the resolvent representation 29 the convergence \[R(b_\varepsilon) \rightarrow R(b), \quad Q(b_\varepsilon) \rightarrow Q(b) \quad \text{ strongly in }L^2\] as \(\varepsilon \downarrow 0\). The latter is proved using the Dominated convergence theorem: we have a.e. convergence \(b_\varepsilon \rightarrow b\), which is immediate from the definition of these vector fields, and appropriate majorant given by 28 .

We now pass to the limit \(\varepsilon \downarrow 0\) in assertion (i). The latter is equivalent to \[\|e^{-t\Lambda_\varepsilon }f\|_{\infty} \leq ct^{-N}\|\varphi_\varepsilon f\|_1, \quad t \in ]0,T], \quad f \in C_c(\mathbb{R}^{2N}).\] We can pass to the limit \(\varepsilon \downarrow 0\) in both sides using convergence (ii). Now the Dunford-Pettis theorem yields the existence of the integral kernel of \(e^{-t\Lambda}\) (\(=:\) heat kernel \(p(t,x,y)\)), and the sought upper heat kernel bound follows e.g. using the Lebesgue differentiation theorem.

5 Proof of Theorem 2↩︎

5.0.1 Some notations↩︎

1. We are going to use the many-particle Hardy inequality 19 , so it will be convenient to re-normalize the strength of attraction between the particles \(\nu\) as follows: \[\nu=\sqrt{\kappa}\frac{d-2}{2}.\] where \(\kappa>0\). The condition of Theorem 2 now takes form: \[\kappa<16\] in all dimensions \(d \geq 3\).

2. In addition to the weights defined before Theorem 2, we define a time-dependent version of \(\psi_\varepsilon\): \[\psi_{s,\varepsilon}(y):=\prod_{1 \leq i<j \leq N} (s^{-\frac{1}{2}}|y^i-y^j|_\varepsilon)^{-\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}}\] if \(d \geq 4\); if \(d=3\), replace \(|y^i-y^j|_\varepsilon\) by \(|y^i-y^j|_{[\varepsilon]}\).

3. It will be convenient to introduce notation for the drift in the approximating operator \(L_\varepsilon\): \[b_\varepsilon(y)=-\frac{\nabla \psi_{s,\varepsilon}}{\psi_{s,\varepsilon}}\] Notice that \(s\) gets cancelled out, so \(b_\varepsilon\) does not depend on \(s\). But it is convenient to keep \(s\) because we are going to compare drift \(b_\varepsilon\) that actually interests us with the drift that will have to deal with at the next stage when we plunge in the Dirichlet forms setting: \[\bar{b}_\varepsilon(y):=-\frac{\nabla \hat{\varphi}_{s,\varepsilon}}{\hat{\varphi}_{s,\varepsilon}}.\] The vector field on the left-hand side depends on \(s\).

5.0.2 Reduction to the Dirichlet form setting↩︎

Lemma 1 allows us to compare the heat kernel \(k_\varepsilon(t,x,y)\) of operator \(L_\varepsilon\) with the heat kernel \(\bar{k}_\varepsilon(t,x,y)\) of the intermediate operator \(-\Delta + \bar{b}_\varepsilon \cdot \nabla\): \[\label{p95bar95p} k_\varepsilon(t,x,y) \approx e^{\frac{C^2}{s}\frac{(N-1)^2}{N}t}\bar{k}_\varepsilon(t,x,y).\tag{38}\] Thus, it suffices for us to prove the following upper bound, for each fixed \(0<s \leq t\): \[\label{p2} \bar{k}_\varepsilon(t,x,y)\;\leq\; c_0 e^{\frac{c t}{s}}\Gamma_{c_2}(t,x-y)\hat{\varphi}_{s,\varepsilon}(y),\tag{39}\] (with constant independent of \(\varepsilon\)), and then in the end take \(s=t\), thus rendering the exponential terms both in 38 and 39 constant in \(t\); then in Theorem 2 \(c_1=c_0e^c e^{C^2\frac{(N-1)^2}{N}}\).

In the rest of the proof of (i), we put for brevity \[\label{varphi95def} \hat{\varphi}:=\hat{\varphi}_{s,\varepsilon}.\tag{40}\] Let \(A_\varepsilon\) denote the self-adjoint operator associated with the sequilinear form \(a_\varepsilon[u,w]=\langle\nabla u, \nabla w\rangle_{\hat{\varphi}}\) on \(W^{1,2}(\mathbb{R}^{dN})\): \[\label{form} a_\varepsilon[u,w]=\langle A_\varepsilon u,w\rangle_{\hat{\varphi}} \quad u \in D(A)=W^{2,2}(\mathbb{R}^{dN}), \quad w \in D(a),\tag{41}\] where \[\langle f \rangle:=\int_{\mathbb{R}^{dN}} f \hat{\varphi}, \quad \langle f,g \rangle_{\hat{\varphi}}:=\langle fg \rangle_{\hat{\varphi}}.\] We have \[\label{A95def} A_\varepsilon=(-\nabla + \bar{b}_\varepsilon)\cdot \nabla.\tag{42}\] Let \(q_{\varepsilon}(t,x,y):=e^{-t A_\varepsilon}(x,y)\) denote the heat kernel of the semigroup \(e^{-t A_\varepsilon}\) acting in \(L^2_{\hat{\varphi}}\), so that \[\label{q95def} q_\varepsilon(t,x,y)\hat{\varphi}(y)=\bar{k}_\varepsilon(t,x,y).\tag{43}\] Combining 38 and 43 , we see that the proof of the upper bound 39 reduces to establishing Gaussian upper bound \[\label{q95est} q_\varepsilon(t,x,y) \leq c_0 e^{\frac{c t}{s}}\Gamma_{c_2}(t,x-y).\tag{44}\]

In what follows, all constants are independent of \(\varepsilon>0\). We omit index \(\varepsilon>0\) where possible.

5.0.3 Sobolev embedding↩︎

Set \[\|u\|_{p,\hat{\varphi}}:=\langle |u|^p\hat{\varphi}\rangle^{1/p}.\] The Sobolev exponent in \(\mathbb{R}^{dN}\): \[\ell:=\frac{dN}{dN-2}.\] The next estimate will be used in the proof of the weighted Sobolev embedding (Lemma 3).

We prove Lemma 2, arguing similarly to the proof of Lemma 1, in Appendix 6.

The next lemma is proved under somewhat less restrictive conditions on \(\kappa\) compared to Theorem 2.

5.0.4 Proof of the Gaussian upper bound on \(q(t,x,y)\)↩︎

We now apply the “Davies device” and Moser’s iterations in the weighted setting. Define \[\phi _\alpha(x):=e^{\alpha \cdot x}, \quad x \in \mathbb{R}^{dN}\] and put \[A_\alpha:= \phi _\alpha A \phi _{-\alpha} \biggl(= A + 2 (\alpha \cdot \nabla) + \alpha \cdot \frac{\nabla \hat{\varphi}}{\hat{\varphi}} - |\alpha|^2\biggr).\] It is clear that the quadratic form of the operator \(A_\alpha\) is \[\label{A95exp} \langle A_\alpha u,v\rangle_{\hat{\varphi}}=a[ \phi _{-\alpha}u,\phi_\alpha v].\tag{50}\]

From Lemma 4 and the dual estimate \(\|e^{-tA_\alpha}\|_{L^1_{\hat{\varphi}} \rightarrow L^2_{\hat{\varphi}}} \leq Ct^{-\frac{dN}{4}} e^{\frac{C_2 }{s} t }e^{C_3|\alpha|^2t}\) (use that \((A_\alpha)^\ast=A_{-\alpha}\) in the weighted space) we obtain, using the semigroup property, \[\|e^{-tA_\alpha}\|_{L^1_{\hat{\varphi}} \rightarrow L^\infty} \leq C_2^2 t^{-\frac{dN}{2}} e^{2\frac{C_2 }{s} t } e^{2C_3|\alpha|^2t},\] Therefore, recalling the definition of weights \(\phi_\alpha\), \(\phi_{-\alpha}\) in \(e^{-tA_\alpha}\), the integral kernel \(q(t, x, y)\) of \(e^{-t A}(x,y)\) satisfies \[\label{q-32Gaussian32upper32bound} q(t,x,y) \le C t^{-\frac{dN}{2}} e^{2 \frac{C_2}{s}t} e^{\alpha\cdot (x-y)+ 2C_3\vert \alpha \vert^2 t}, \quad t>0,\, x,y \in \mathbb{R}^{dN}.\tag{56}\] Selecting \(\alpha = \frac{y-x}{2 C_3 t}\), we get \[\label{q-esti} q(t,x,y) \le C t^{-\frac{dN}{2}} e^{2 \frac{C_2}{s}t} e^{-\frac{\vert y-x \vert^2}{2C_3 t}},\tag{57}\] This gives us the sought upper Gaussian bound on \(q (= q_\varepsilon)\). Now, we trace back the chain of equivalences 45 , and select \(s= t\) in the resulting upper bound on \(k_{\varepsilon}(t,x,y)\). This ends the proof of assertion (i).

(ii) The construction of the semigroup is the content of Proposition 1 (Appendix 11). The fact that we have a semigroup of integral operators follows from the Dunford-Pettis theorem, via the \(L^r \rightarrow L^q\) embedding property for \(e^{-tL}\). The latter is proved e.g. by repeating the proof of [38] supplemented with Proposition 6), i.e. a realization of Nash’s classical argument. Alternatively, we can simply use the a priori upper bound established in (i).

This a priori upper bound yields a posteriori upper bound on the integral kernel of \(e^{-tL}\) via the Lebesgue differentiation theorem. 0◻

6 Proof of Lemma 2↩︎

Step 1: At this step we compare \(-{\rm div\,}\frac{\nabla \hat{\varphi}}{\hat{\varphi}}\) to \(-{\rm div\,} \frac{\nabla \psi}{\psi}\), i.e. we first prove a preliminary variant of 46 : \[\label{prelim95phi95calc} -{\rm div\,}\frac{\nabla \hat{\varphi}}{\hat{\varphi}} \leq -{\rm div\,} \frac{\nabla \psi}{\psi} + \frac{b_0}{s}.\tag{58}\]

The function \(\psi=\psi_{s,\varepsilon}\), defined before Theorem 2, is easier to do calculations with. Once the previous inequality is established, we will estimate \(-{\rm div\,} \frac{\nabla \psi}{\psi}\) at Step 2. Note that \(s\) gets cancelled out in \(-{\rm div\,} \frac{\nabla \psi}{\psi}\).

Since the definition of \(\psi\) uses different regularization of the Euclidean norm depending on whether \(d=3\) or \(d \geq 4\), we consider these cases separately:

(a) Case \(d\geq 4\). Recalling that \(|x^i-x^j|_{\varepsilon}^2 := |x^i-x^j|^2+\varepsilon\), we have

\[-{\rm div\,}\frac{\nabla \hat{\varphi}}{\hat{\varphi}} = -\Delta \log \hat{\varphi} = -2\sum_{1\le i<j\le N} \Delta_{x^i} \log \eta\left( \frac{|x^i-x^j|_{\varepsilon}}{\sqrt{s}} \right),\] and \[-{\rm div\,} \frac{\nabla \psi}{\psi} = - \Delta \log \psi.\] For brevity, define \[G(r):=\log \eta(r)+\frac{\sqrt{\kappa}}{N}\frac{d-2}{2}\log r, \qquad r>0,\] \(G\) is identically zero in the singular region \(r<1\), while on the transition region \(1\le r\le 2\) it is smooth and all its derivatives \(G'\), \(G''\) are bounded.

For every pair \(1\leq i<j\leq N\), \[\log \eta\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right) = -\frac{\sqrt{\kappa}}{N} \frac{d-2}{2}\log |x^i-x^j|_\varepsilon + \frac{1}{2} \frac{\sqrt{\kappa}}{N} \frac{d-2}{2}\log s + G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right).\] Summing over all pairs gives \[\begin{align} \log \hat{\varphi}_{} & = -\frac{\sqrt{\kappa}}{N} \frac{d-2}{2}\sum_{1\leq i<j\leq N}\log |x^i-x^j|_\varepsilon + \frac{1}{2}\frac{\sqrt{\kappa}}{N} \frac{d-2}{2} \frac{N(N-1)}{2} \log s + \sum_{1\leq i<j\leq N} G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right), \\ & = \log \psi + \frac{1}{2}\frac{\sqrt{\kappa}}{N} \frac{d-2}{2} \frac{N(N-1)}{2} \log s + \sum_{1\leq i<j\leq N} G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right). \end{align}\] Since the term involving \(\log s\) is independent of \(x\), we obtain \[\label{err95G} -\Delta \log \hat{\varphi}_{} = -\Delta \log \psi - \Delta R_s(x),\tag{59}\] where \[R_s(x) := \sum_{1\leq i<j\leq N} G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right),\] It remains to estimate the last term in 59 . We claim that there exists a constant \(C>0\), independent of \(\varepsilon\), such that for every \(x \in\mathbb{R}^{dN}\) \[\label{estim95rest} -\Delta R_s(x)\leq \frac{C}{s} \sum_{1\leq i<j\leq N} \mathbf{1}_{\{ \sqrt{s}\leq |x^i-x^j|_\varepsilon\leq 2\sqrt{s} \}}.\tag{60}\] Indeed, fix a pair \(1\le i<j \le N\) and set \[z=x^i-x^j\] Since \(G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right)\) depends only on the difference \(z\), only the variables \(x^i\) and \(x^j\) contribute to the full Laplacian \(-\Delta\) on \(\mathbb{R}^{dN}\). So, we have \[\Delta_{x^i}G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right) = \Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right), \quad \text{and} \quad \Delta_{x^j}G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right) = \Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right).\] Thus \[\Delta G\left(\frac{|x^i-x^j|_\varepsilon}{\sqrt{s}}\right) = 2\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right),\] here \(\Delta_z\) denotes the Laplacian on \(\mathbb{R}^d\). Now, summing over all pairs \(i<j\), we obtain \[-\Delta R_s(x) = -2\sum_{1\leq i<j\leq N} \Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) \bigg|_{z=x^i-x^j}.\] From the definition of \(\eta\) , we have :

1. If \(|z|_\varepsilon<\sqrt{s}\), then \(\eta(r)=r^{-\frac{\sqrt{\kappa}}{N}\frac{d-2}{2} }\). Hence \(G(r)= 0\) and \(\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right)= 0\).

2. If \(|z|_\varepsilon>2\sqrt{s}\), then \(\eta\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right)=e^{\frac{\nu}{N}} 2^{-2\frac{\nu}{N}}\). So \[G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) = \log (e^{\frac{\nu}{N}} 2^{-2\frac{\nu}{N}}) + \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \log |z|_\varepsilon - \frac{1}{2}\frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \log s.\] Hence, \[\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) = \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \Delta_z\log |z|_\varepsilon = \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \frac{(d-2)|z|^2+d\varepsilon}{\left(|z|^2+\varepsilon \right)^2}.\] Since \(d\geq 4\), we have \[-2\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) = -2\frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \frac{(d-2)|z|^2+d\varepsilon}{\left(|z|^2+\varepsilon \right)^2} \le 0.\]

3. If \(\sqrt{s}\leq |z|_\varepsilon\leq 2\sqrt{s}\), then on this interval \(G\in C^2([1,2])\), so its first and second derivatives are bounded. Put \[\zeta(z):=\frac{|z|_\varepsilon}{\sqrt{s}}.\] Moreover, we have \(|\nabla_z \zeta |^2\leq \frac{1}{s}\) and \(|\Delta_z \zeta| \leq \frac{C }{\sqrt{s}\,|z|_\varepsilon}.\) Since \(|z|_\varepsilon\geq \sqrt{s}\), this yields \(|\Delta_z \zeta| \leq \frac{C}{s}\). Then, we get \[\label{chain95rule} \Delta_z G(\zeta) = G''|\nabla_z \zeta|^2 + G'\Delta_z \zeta.\tag{61}\] Therefore, \[\left| \Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) \right| \leq \frac{C}{s},\] where \(C\) is independent of \(\varepsilon\). Combining the estimates in the three regions, we obtain the pointwise bound \[-2\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) \leq \frac{C}{s} \mathbf{1}_{\{\sqrt{s}\leq |z|_\varepsilon\leq 2\sqrt{s}\}}.\] Therefore, \[-\Delta R_s(x) \leq \frac{C}{s} \sum_{1\leq i<j\leq N} \mathbf{1}_{\{\sqrt{s}\leq |x^i-x^j|_\varepsilon\leq 2\sqrt{s}\}}.\] Thus \[-\Delta \log \hat{\varphi} \leq -\Delta\log\psi + \frac{b_0(x)}{s},\] where \[b_0(x) := C \sum_{1\leq i<j\leq N} \mathbf{1}_{\{\sqrt{s}\leq |x^i-x^j|_\varepsilon\leq 2\sqrt{s}\}}.\]

(b) Case \(d= 3\). Recalling that \(|x^i - x^j|_\varepsilon := |x^i - x^j|+\varepsilon\), and set \(z=x^i-x^j\). The proof follows the same strategy as above. In detail, first, assume that \(|z|_{[\varepsilon]}<\sqrt{s}\). Then, by the definition of \(\eta\), \(\eta(r)=r^{-\frac{\sqrt{\kappa}}{N}\frac{d-2}{2} }\). Therefore \(G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right)= 0\), so \(\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right)= 0\). Next, assume that \(|z|_{[\varepsilon]}>2\sqrt{s}\). In this region, \(\eta\left(\frac{|z|_{[\varepsilon]}}{\sqrt{s}}\right)=e^{\frac{\nu}{N}} 2^{-2\frac{\nu}{N}}\), and hence \[G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) = \log (e^{\frac{\nu}{N}} 2^{-2\frac{\nu}{N}}) + \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \log \left( |z|+\varepsilon \right) - \frac{1}{2}\frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \log s.\] It follows that, for \(z\neq 0\), \[\Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) = \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \Delta_z \log(|z|+\varepsilon),\] we have \[\begin{align} \Delta_z \log(|z|+\varepsilon) & = -\frac{1}{(|z|+\varepsilon)^2} + \frac{d-1}{|z|(|z|+\varepsilon)} \\ &= \frac{(d-2)|z|+(d-1)\varepsilon}{|z|(|z|+\varepsilon)^2}. \end{align}\] Consequently, \[\begin{align} -2 \Delta_z G\left(\frac{|z|_\varepsilon}{\sqrt{s}}\right) & = -2 \frac{\sqrt{\kappa}}{N}\frac{d-2}{2} \frac{(d-2)|z|+(d-1)\varepsilon}{|z|(|z|+\varepsilon)^2} \le 0. \end{align}\] It remains to consider the transition region \(\sqrt{s}\leq |z|_{[\varepsilon]}\leq2\sqrt{s}\). Put \[\zeta(z):=\frac{|z|_{[\varepsilon]}}{\sqrt{s}} = \frac{|z|+\varepsilon}{\sqrt{s}}.\] Then \(1\leq\zeta\leq2\). On this interval, \(G\in C^2([1,2])\), and \(G'\), \(G''\) are bounded.

For \(z\neq 0\), we have \(\nabla_z \zeta = \frac{1}{\sqrt{s}}\frac{z}{|z|}\), and therefore \(|\nabla_z \zeta|^2=\frac{1}{s}\). Furthermore, \[\Delta_z \zeta = \frac{1}{\sqrt{s}}\Delta_z |z| = \frac{d-1}{\sqrt{s}\,|z|} > 0.\] We obtain \[-2\Delta_zG(\zeta) = -2G''(\zeta)|\nabla_z\zeta|^2 -2G'(\zeta)\Delta_z\zeta.\] The second term in the RHS gives, \[\begin{align} -2G'(\zeta)\Delta_z\zeta &= -\frac{d-1}{\sqrt{s}|z|} \left( \frac{\eta'(\zeta)}{\eta(\zeta)} +\frac{\sqrt{\kappa}}{N}\frac{d-2}{2}\frac{1}{\zeta} \right), \\ & \; (\text{use \eqref{lem295ineq}}), \\ & \le 0. \end{align}\] Therefore, since \(G''\) is bounded on \([1,2]\), we get \[-2\Delta_z G(\zeta) \le -2G''(\zeta)|\nabla_z \zeta|^2 \le \frac{C}{s}\] Thus we conclude 60 .

This ends the proof of 58 .

Step 2. As explained in the beginning of Step 1, armed with 58 , it remains to estimate \(-\Delta\log\psi=-\sum_{i=1}^N \nabla_i (\frac{\nabla \psi}{\psi})_i\) to conclude 46 . In the case \(d \geq 4\), we have \[-(\frac{\nabla \psi}{\psi})_i=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N\frac{x^i-x^j}{|x^i-x^j|^2_\varepsilon},\] so \[\begin{align} \nabla_i (\frac{\nabla \psi}{\psi})_i=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N\nabla _i \biggl(\frac{x^i-x^j}{|x^i-x^j|^2_\varepsilon}\biggr)=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N\biggl( d\frac{1}{|x^i-x^j|_\varepsilon^2} - 2 \frac{|x^i-x^j|^2}{|x^i-x^j|_\varepsilon^4}\biggr). \end{align}\] It remains to apply inequality \(d\frac{1}{|x|_\varepsilon^2} - 2 \frac{|x|^2}{|x|_\varepsilon^4} \leq \frac{d-2}{|x|^2}\) (using \(d \geq 4\)) to obtain 46 . (Note that in 46 we sum only above the diagonal, hence the extra factor \(2\).)

In the case \(d=3\), we use a different regularization of the Euclidean norm in the definition of \(\psi=\psi_{s,\varepsilon}\). The next argument also works in dimensions greater than \(3\), so we will continue writing \(d\) since this makes the argument easier to follow. We have (taking into account that \(s\) gets cancelled out in \(\frac{\nabla \psi}{\psi}\)) \[\begin{align} -\nabla_i (\frac{\nabla \psi}{\psi})_i=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N \Delta_{i}\log |x^i-x^j|_{[\varepsilon]}. \end{align}\] The Laplacian \(\Delta_i\) of a radial function \(f(r)\) (\(r=|x^i-x^j|\), we can make a change of variables to put \(x^j\) at the origin) is \(f''+\frac{d-1}{r}f'\), so, since for \(f(r)=\log(r+\varepsilon)\) we have \(f'(r)=\frac{1}{r+\varepsilon}\), \(f''(r)=-\frac{1}{(r+\varepsilon)^2}\), \[\begin{align} \Delta_{i}\log |x^i-x^j|_{[\varepsilon]} & = - \frac{1}{\big(|x^i-x^j|+\varepsilon\big)^2} + \frac{d-1}{|x^i-x^j|\big(|x^i-x^j|+\varepsilon\big)} \\ & (\text{use -\frac{1}{(r+\varepsilon)^2} + \frac{d-1}{r(r+\varepsilon)} \leq \frac{(d-2)}{r^2}}) \\ & \leq (d-2)\frac{1}{|x^i-x^j|^2}, \end{align}\] hence \[-\nabla_i (\frac{\nabla \psi}{\psi})_i \leq \sqrt{\kappa}\frac{(d-2)^2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N \frac{1}{|x^i-x^j|^2},\] i.e. we have proved 46 . 0◻

7 Alternative approach to estimating heat kernel of operator \(L\)↩︎

One limitation of Theorem 1 is that it deals with operator \(\Lambda_\varepsilon\) and not with operator \[L_\varepsilon=-\Delta + \frac{d-2}{2}\frac{\sqrt{\kappa}}{N} \sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|_\varepsilon^2} \cdot \nabla_{x_i},\] where \(|x^i-x^j|_\varepsilon:= \sqrt{|x^i-x^j|^2+\varepsilon}\). To handle operator \(L_\varepsilon\), we can use Moser’s iterations (Theorem 2) or, alternatively, we can use the desingularization procedure of Theorem A. This is what we do below. Now, verifying condition ?? of Theorem A will be more interesting. In what follows, \(d \geq 3\).

We will simplify the problem somewhat and work in the domain in \(\mathbb{R}^{dN}\) \[D_R:=\bigcap_{1 \leq i<j\leq N}\{x \in \mathbb{R}^{dN} \mid |x^i-x^j|<R\}\] for a fixed large \(R>0\). When the trajectory \(X_t=X_t^\varepsilon=(X_t^{i,\varepsilon})_{i=1}^N\) of the smoothed out “higher-dimensional Keller-Segel system” \[\label{ks5} dX_t^{i} = - \frac{\nu}{N}\sum_{j=1, j \neq i}^N \frac{X_t^i-X_t^j}{|X_t^i-X_t^j|_\varepsilon^2}dt + \sqrt{2}dB_t^i, \quad i=1,\dots,N,\tag{62}\] hits the boundary of \(D_R\), i.e. the distance between at least one pair of particles becomes equal to \(R\), we stop. Let \(X_{t}^{0,\varepsilon}\) denote the corresponding stopped process. This corresponds to considering the initial-boundary value problem in \(]0,T] \times D_R\) \[\big(\partial_t -\Delta + \frac{\nu}{N}\sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|_\varepsilon^2} \cdot \nabla_{x_i}\big)u=0, \quad u|_{\partial D_R}=0, \quad u|_{t=0}=f,\] where \(f\) has support in \(D_R\). That is, \[\mathbb{E}_{X_{t=0}^{0,\varepsilon}=x}[f(X_{t}^{0,\varepsilon})]=\int_{D_R}k_\varepsilon(t,x,y)f(y)dy,\] where \(k_\varepsilon\) denotes the heat kernel that corresponds to \(L^0_\varepsilon\), i.e. the operator \[-\Delta + \frac{\nu}{N}\sum_{i=1}^N \sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|_\varepsilon^2} \cdot \nabla_{x_i},\] with the Dirichlet boundary conditions on \(\partial D_R\).

Returning to the earlier notations, in Theorem 3 we desingularize operator \(-\Delta - \frac{\nabla \psi_\varepsilon}{\psi_\varepsilon}\cdot \nabla\) using the same weight \(\varphi_\varepsilon=\psi_\varepsilon+1\). The crucial step in the proof, which uses Theorem A, is the verification of the “desingularizing bound” ?? , i.e. the verification that \[\begin{align} (\psi_\varepsilon+1)(-\Delta - \frac{\nabla \psi_\varepsilon}{\psi_\varepsilon} \cdot \nabla)(\psi_\varepsilon+1)^{-1}v = -\Delta v + \nabla \cdot \big(\frac{\psi_\varepsilon-1}{\psi_\varepsilon(\psi_\varepsilon+1)}(\nabla \psi_\varepsilon) v\big) + \frac{1}{\psi_\varepsilon+1}\bigg({\rm div\,}\frac{\nabla \psi_\varepsilon}{\psi_\varepsilon}\bigg)v \end{align}\] is the generator of an \(L^1\) quasi contraction on \(\bar{D}_R\) unformly in \(\varepsilon>0\). The difficulty is in dealing with the last term, i.e. the potential, which is singular (as \(\varepsilon \downarrow 0\)), but not very singular. At this step, we appeal to the regularity results for the elliptic equations in [21] obtained using De Giorgi’s method. De Giorgi’s method is a powerful technique, but this approach seems reasonable if one keeps in mind that we use little specifics about the weight \(\psi\); the proof is rather abstract and should work for other particle systems.

8 Proof of Theorem A (abstract desingularization)↩︎

In the proof of Theorem A we use the following weighted variant of the Coulhon-Raynaud extrapolation theorem [41].

9 Trotter’s approximation theorem↩︎

Consider a sequence \(\{e^{-tA_k}\}_{k=1}^\infty\) of strongly continuous semigroups on a Banach space \(Y\).

The first condition of the theorem is satisfied if e.g. \(\sup_k\|(z+A_k)^{-1}\|_{Y \rightarrow Y} \leq C|z-\omega|^{-1}\), \({\rm Re\,}z>\omega\).

10 Immersion of particle system in a turbulent flow↩︎

We mentioned in the comments after Theorem 2 that its proof extends to operator 22 . Here are the details. Recall notation \(\phi_\alpha(x)=e^{\alpha \cdot x}\). Put \[A_\alpha^c:=\phi_\alpha A^c \phi_\alpha^{-1},\] where \[A^c =- \nabla \cdot (I+C) \cdot \nabla - \frac{\nabla \hat{\varphi}}{\hat{\varphi}}\cdot (I+C)\cdot \nabla,\] and set \[u(t)=e^{-t(\omega + A_\alpha^c)}f, \quad \text{for}\, f \in C_c^\infty.\] We assume without loss of generality that \(f \geq 0\), so \(u(t) \geq 0\) for all \(t>0\). The counterpart of 52 in the proof of Theorem 2 is \[\omega \langle u^{\frac{p}{2}},u^{\frac{p}{2}}\rangle_{\hat{\varphi}} + \frac{1}{p}\partial_t \langle u^{\frac{p}{2}},u^{\frac{p}{2}}\rangle_{\hat{\varphi}} + \langle A^c_\alpha u,u^{p-1}\rangle_{\hat{\varphi}}=0,\] where we evaluate, for \(p \geq 2\), \[\begin{align} \langle A_\alpha^c u,u^{p-1} \rangle_{\hat{\varphi}} & =\langle A^c \phi_{-\alpha}u,\phi_\alpha u^{p-1} \hat{\varphi}\rangle \\ & = \langle - \nabla \cdot(I +C) \cdot \nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \hat{\varphi}\rangle - \langle \frac{\nabla \hat{\varphi}}{\hat{\varphi}}(I+C)\nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \hat{\varphi}\rangle\\ & = \langle A_\alpha u,u^{p-1}\rangle_{\hat{\varphi}} + \langle - \nabla \cdot C \cdot \nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \hat{\varphi}\rangle - \langle \frac{\nabla \hat{\varphi}}{\hat{\varphi}}C\cdot \nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \hat{\varphi}\rangle\\ & =: \langle A_\alpha u,u^{p-1}\rangle_{\hat{\varphi}} + I - J, \end{align}\] where \(\langle A_\alpha u,u^{p-1}\rangle_{\hat{\varphi}}\) was already evaluated in 53 . Next, integrating by parts, we evaluate \[\begin{align} I=\langle - \nabla \cdot C \cdot \nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \hat{\varphi}\rangle & = \langle C \cdot \nabla (\phi_{-\alpha}u),\nabla (\phi_\alpha u^{p-1} \hat{\varphi})\rangle \\ & =\langle C \cdot \nabla (\phi_{-\alpha}u),\nabla (\phi_\alpha u^{p-1}) \hat{\varphi} \rangle + \langle C \cdot \nabla (\phi_{-\alpha}u),\phi_\alpha u^{p-1} \nabla \hat{\varphi}\rangle, \end{align}\] where the last term coincides with \(J\). Therefore, \[I-J=\langle C \cdot \nabla (\phi_{-\alpha}u),\nabla (\phi_\alpha u^{p-1}) \hat{\varphi} \rangle.\] In turn, taking into account skew-symmetry of \(C\), \[\begin{align} \langle C \cdot \nabla (\phi_{-\alpha}u),\nabla (\phi_\alpha u^{p-1} )\hat{\varphi} \rangle & = \langle C \cdot (-\alpha u + \nabla u),(\alpha u^{p-1} + (p-1)u^{p-2}\nabla u) \hat{\varphi}\rangle \\ & = \langle C \cdot (-\alpha) u,(p-1)u^{p-2}(\nabla u)\hat{\varphi}\rangle + \langle C \cdot \nabla u,\alpha u^{p-1}\hat{\varphi} \rangle \\ & = \langle \alpha u,(p-1)u^{p-2}(C \cdot \nabla u)\hat{\varphi}\rangle + \langle C \cdot \nabla u,\alpha u^{p-1}\hat{\varphi} \rangle \\ & =p\langle C \cdot \nabla u, \alpha u^{p-1}\hat{\varphi}\rangle = 2\langle C \cdot \nabla v, \alpha v\hat{\varphi}\rangle, \end{align}\] where we have put \(v=u^{\frac{p}{2}}\) as in the proof of Moser’s Lemma 4. Therefore, \[|I-J| \leq \|C\|_\infty \biggl[ \varepsilon_0 \langle |\nabla v|\rangle_{\hat{\varphi}} + \frac{|\alpha|^2}{\varepsilon_0}\langle v,v\rangle_{\hat{\varphi}} \biggr]\] for any \(\varepsilon_0>0\), where, recall, \(\langle |\nabla v|\rangle_{\hat{\varphi}} \equiv a[v,v]\). Take \(\varepsilon_0=\frac{1}{2}\frac{4(p-1)}{\|C\|_\infty p^2}\). Then \(\varepsilon_0^{-1} \leq \frac{p}{2}\|C\|_\infty\). With this choice of \(\varepsilon_0\), we arrive at \[\begin{align} \langle A_\alpha^c u, & u^{p-1} \rangle_{\hat{\varphi}} = \langle A_\alpha u,u^{p-1}\rangle_{\hat{\varphi}} \\ & + \text{ term whose absolute value is smaller than }\frac{1}{2}\frac{4(p-1)}{p^2} a[v,v] + |\alpha|^2 \frac{p\|C\|_\infty^2}{2}\langle v,v\rangle_{\hat{\varphi}}. \end{align}\] Now we can plug this estimate in the proof of Moser’s Lemma 4 and argue as before.

11 Construction of a posteriori heat kernel for operator \(L\)↩︎

Let \(d \geq 3\). Let \(b_\varepsilon(x)=(b_\varepsilon^i(x))_{i=1}^N:\mathbb{R}^{dN} \rightarrow \mathbb{R}^{dN}\) be the drift in operator \(L_\varepsilon=-\Delta + b_\varepsilon \cdot \nabla\) in Theorem 2, that is, \[b_\varepsilon^i(x)=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N \frac{x^i-x^j}{|x^i-x^j|^2 + \varepsilon}\] if \(d \geq 4\), and \[b_\varepsilon^i(x)=\sqrt{\kappa}\frac{d-2}{2}\frac{1}{N}\sum_{j=1, j \neq i}^N \frac{1}{|x^i-x^j|+\varepsilon}\frac{x^i-x^j}{|x^i-x^j|}\] if \(d=3\). The following estimates are consequences of the many-particle Hardy inequality 19 . They will be needed in Proposition 1 where we construct the limiting semigroup \(\lim_{\varepsilon \downarrow 0}e^{-t L_\varepsilon}\).

We will also need a weighted variant of the previous estimate. Set \[\label{rho95def} \rho(x):=(1+\sigma^2|x|^{2})^{-\gamma}, \quad \sigma>0,\tag{69}\] where \(\gamma>dN+2\,(>\frac{dN}{2})\) is fixed, so that \(\langle \rho\rangle<\infty\) and also \(\langle |\nabla \sqrt{\rho}|^2 \rangle<\infty\). This weight has property \[\label{rho95est} |\nabla \rho| \leq C\sigma\rho.\tag{70}\] Let us emphasize that the choice of \(\sigma\) is up to us, so using the previous estimate we can replace all occurrences of \(\nabla \rho\) by \(\rho\) with a small coefficient.

The following proposition is a part of Theorem 2(ii).

12 Two-particle case↩︎

In this appendix we specify Theorems 1 and 2 to the heat kernel of the operators \[\Lambda:=-\Delta - \frac{\nabla \varphi}{\varphi} \cdot \nabla \text{ on } \mathbb{R}^2, \quad \varphi(x)=|x|^{-\nu}+1\] and \[L:=-\Delta + \nu \frac{x}{|x|^2} \cdot \nabla \text{ on } \mathbb{R}^d, \quad d \geq 3.\] The upper bounds on their heat kernels allow us to estimate from above the density of the law of the relative displacement \(Z_t:=X_t^1-X_t^2\) of the two-particle system, i.e., say, in dimension two, \[\left\{ \begin{array}{l} dX_t^1=\frac{1}{2}\frac{\nabla \varphi}{\varphi} (X_t^1-X_t^2)dt + \frac{1}{\sqrt{2}}dB^1_t, \\ dX_t^2=\frac{1}{2}\frac{\nabla \varphi}{\varphi} (X_t^2-X_t^1)dt + \frac{1}{\sqrt{2}}dB^2_t, \end{array} \right.\] where \(\{B_t^1\}\), \(\{B_t^2\}\) are independent two-dimensional Brownian motions; this can be easily seen by subtracting the corresponding SDEs and using the fact that \(\varphi\) is an even function. Similarly in dimensions \(d \geq 3\).

We define the \(\varepsilon\)-regularized drifts and operators \(\Lambda_\varepsilon\), \(L_\varepsilon\) in the same way as we did in Theorems 1 and 2, and denote by \(p_\varepsilon\), \(k_\varepsilon\) the corresponding heat kernels \(:=\) integral kernels of semigroups \(e^{-t\Lambda_\varepsilon}\), \(e^{-tL_\varepsilon}\). Define in dimension \(d=2\) \[\varphi_\varepsilon(y):=|y|_\varepsilon^{-\nu}+\varepsilon, \quad |y|_\varepsilon:=\sqrt{|y|^2+\varepsilon}.\] In dimensions \(d \geq 4\), we define \(\hat{\varphi} \in W^{2,\infty} (\mathbb{R}^d \setminus \{0\})\) via \[\hat{\varphi}(y)=\left\{ \begin{array}{ll} |y|_\varepsilon^{-\nu}, & |y| \leq \sqrt{t}, \\ \frac{1}{2}, & |y| \geq 2\sqrt{t} \end{array} \right.\] and set \(\hat{\varphi}_t(y):=\hat{\varphi}(y/\sqrt{t})\). In dimension \(d=3\) replace \(|y|_\varepsilon\) with \(|y|_{[\varepsilon]}:=|y|+\varepsilon\).

Regarding the case \(d \geq 3\), see also comments after Theorems 1, 2.

References↩︎