{"id":11471,"date":"2019-05-21T14:40:19","date_gmt":"2019-05-21T12:40:19","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=11471"},"modified":"2024-05-09T20:57:28","modified_gmt":"2024-05-09T18:57:28","slug":"kernels","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2019\/05\/21\/kernels\/","title":{"rendered":"Kernels"},"content":{"rendered":"
\"Charles-Augustin<\/a>
Charles-Augustin de Coulomb (1736-1806)<\/a><\/figcaption><\/figure>\n

The Coulomb or Newton kernel<\/b> in \\( {\\mathbb{R}^d} \\), \\( {d\\geq1} \\), is often defined for all \\( {x\\neq0} \\) as<\/p>\n

\\[ g(x):=\\begin{cases} -|x| & \\mbox{if }d=1,\\\\ \\log\\frac{1}{|x|} & \\mbox{if }d=2,\\\\ \\frac{1}{|x|^{d-2}} & \\mbox{if }d\\geq3, \\end{cases} \\]<\/p>\n

where \\( {|x|:=\\sqrt{x_1^2+\\cdots+x_d^2}} \\) is the Euclidean norm. It is the fundamental solution of the Laplace or Poisson equation, in the sense that<\/p>\n

\\[ -\\Delta g=c_d\\delta_0 \\quad\\mbox{where}\\quad c_d:= \\begin{cases} 2 & \\mbox{if }d=1,\\\\ 2\\pi & \\mbox{if }d=2,\\\\ (d-2)|\\mathbb{S}^{d-1}| & \\mbox{if }d\\geq 3, \\end{cases} \\]<\/p>\n

where \\( {\\mathbb{S}^{d-1}:=\\{x\\in\\mathbb{R}^d:|x|=1\\}} \\), and, denoting \\( {\\Gamma} \\) the Euler Gamma function,<\/p>\n

\\[ |\\mathbb{S}^{d-1}| =2\\frac{\\pi^{d\/2}}{\\Gamma(d\/2)}. \\]<\/p>\n

This partial differential equation is in the sense of Schwartz distributions \\( {\\mathcal{D}'(\\mathbb{R}^d)} \\). Note that on \\( {\\mathbb{R}^d\\setminus\\{0\\}} \\), the function \\( {g} \\) is \\( {\\mathcal{C}^\\infty} \\) and harmonic<\/b> in the sense that \\( {\\Delta g(x)=\\partial_1^2g(0)+\\cdots+\\partial_d^2g(0)=0} \\) as a function for all \\( {x\\neq0} \\). The behavior at the origin makes \\( {g} \\) super-harmonic<\/b> (its Laplacian is \\( {\\leq0} \\)), which is a trace analogue of concavity<\/b>.<\/p>\n

The case \\( {d=1} \\) is intuitive: the derivative of \\( {|x|} \\) in the sense of distributions is the Heaviside function \\( {-\\mathbf{1}_{x<0}+\\mathbf{1}_{x>0}} \\) (essentially a jump at zero of height \\( {2} \\)), and the second derivative twice the Dirac mass, \\( {2\\delta_0} \\). The case \\( {d=2} \\) appears also as special: it blows up at infinity.<\/p>\n

The physical interpretation of \\( {g(x)} \\), up to physical constants, is the potential<\/b> generated at point \\( {x} \\) by a charge at the origin, the field<\/b> being \\( {-\\nabla g(x)} \\). The electric field if we model electrostatics (Coulomb), and the gravitational field if we model gravity (Newton). Note that \\( {-g(x)\\nabla g(x)} \\) vanishes at infinity if \\( {d\\geq2} \\), but not if \\( {d=1} \\), which makes a difference for integration by parts.<\/p>\n

For a probability measure \\( {\\mu} \\) on \\( {\\mathbb{R}^d} \\), the potential generated by \\( {\\mu} \\) at point \\( {x} \\) is<\/p>\n

\\[ U_\\mu(x)=(g*\\mu)(x)=\\int g(x-y)\\mu(\\mathrm{d}y). \\]<\/p>\n

In dimension \\( {d=1} \\) or \\( {d=2} \\), this is well defined as soon as \\( {\\mu} \\) integrates \\( {g} \\) at infinity. Note that \\( {g} \\) is Lebesgue locally integrable. The convolution operator<\/b> \\( {\\mu\\mapsto U_\\mu} \\) is the inverse of the Laplacian, and we have the inversion formula<\/b><\/p>\n

\\[ -\\Delta U_\\mu=(-\\Delta g)*\\mu=c_d(\\delta_0*\\mu)=c_d\\mu, \\]<\/p>\n

in \\( {\\mathcal{D}'(\\mathbb{R}^d)} \\). The potential \\( {U_\\mu} \\) is harmonic outside the support of \\( {\\mu} \\).<\/p>\n

Alternative formulation.<\/b> The following alternative definition is simpler:<\/p>\n

\\[ g(x):=\\begin{cases} \\frac{1}{(d-2)|x|^{d-2}} & \\mbox{if }d=1\\mbox{ or } d\\geq3,\\\\ \\log\\frac{1}{|x|} & \\mbox{if }d=2, \\end{cases} \\]<\/p>\n

which satisfies<\/p>\n

\\[ -\\Delta g=c_d\\delta_0 \\quad\\mbox{where}\\quad c_d:=|\\mathbb{S}^{d-1}| \\]<\/p>\n

with the convention \\( {|\\mathbb{S}^0|:=|\\{-1,1\\}|=2} \\). Indeed if \\( {d=1} \\) then \\( {\\frac{1}{(d-2)|x|^{d-2}}=-|x|} \\).<\/p>\n

This alternative formulation makes the equilibrium measure nicer in the quadratic confinement case. Namely, the electrostatic energy of a distribution of charges modeled by a probability measure \\( {\\mu} \\) on \\( {\\mathbb{R}^d} \\) with external field generated by a potential \\( {V:\\mathbb{R}^d\\rightarrow\\mathbb{R}} \\) is<\/p>\n

\\[ \\mathcal{E}_V(\\mu) :=\\iint g(x-y)\\mu(\\mathrm{d}x)\\mu(\\mathrm{d}y)+\\int V(x)\\mathrm{d}\\mu(x). \\]<\/p>\n

This functional is strictly convex and lower semi-continuous with respect to the narrow convergence of probability measures. Its minimizer, \\( {\\mu_*=\\arg\\inf\\mathcal{E}_V} \\), is called the equilibrium measure<\/b>. The Euler-Lagrange equation gives that \\( {\\mu_*} \\) has density<\/p>\n

\\[ \\frac{\\Delta V}{2c_d} \\]<\/p>\n

on its support. When \\( {V} \\) is strong enough at infinity then this support is compact. For instance when \\( {V(x)=|x|^2} \\) then we find that the equilibrium measure \\( {\\mu_*} \\) is the uniform distribution on the unit ball<\/b> of \\( {\\mathbb{R}^d} \\). Indeed, in this case<\/p>\n

\\[ \\frac{\\Delta V}{2c_d} =\\frac{2d}{2|\\mathbb{S}^{d-1}|} =\\frac{2d}{2d|\\mathbb{B}^d|} =\\frac{1}{|\\mathbb{B}^d|} \\]<\/p>\n

where \\( {\\mathbb{B}^d=\\{x\\in\\mathbb{R}^d:|x|=1\\}} \\) is the unit ball of \\( {\\mathbb{R}^d} \\). Recall that if \\( {s(r)} \\) and \\( {v(r)} \\) are respectively the surface of the sphere of radius \\( {r} \\) and the volume of the ball of radius \\( {r} \\) in \\( {\\mathbb{R}^d} \\) then \\( {v(r)=r^dv(1)} \\) and \\( {s(r)=v'(r)=dr^{d-1}v(1)} \\) hence \\( {s(1)=dv(1)} \\).<\/p>\n

An even more compact definition.<\/b> If we think the dimension \\( {d} \\) as being a real positive number, we may observe that for all \\( {x\\neq0} \\),<\/p>\n

\\[ \\lim_{d\\rightarrow2}\\frac{\\frac{1}{|x|^{d-2}}-1}{d-2} =\\partial_{s=0}\\frac{1}{|x|^{s}} =\\partial_{s=0}\\mathrm{e}^{-s\\log|x|} =-\\log|x|. \\]<\/p>\n

This means that the formula \\( {\\frac{1}{(d-2)|x|^{d-2}}} \\), already valid for \\( {d=1} \\) and \\( {d\\geq3} \\), is actually also valid for \\( {d=2} \\) provided that we remove a singularity<\/b>. This suggests to define the kernel, for all integers \\( {d\\geq1} \\) and even all real numbers \\( {d\\geq1} \\), and all \\( {x\\neq0} \\), as<\/p>\n

\\[ g(x):=\\frac{1}{(d-2)|x|^{d-2}}-\\frac{1}{d-2}, \\]<\/p>\n

which satisfies<\/p>\n

\\[ -\\Delta g=2\\frac{\\pi^{d\/2}}{\\Gamma(d\/2)}\\delta_0. \\]<\/p>\n

Note that physically, the potential is important only for defining the field, and as a consequence, the potential is defined up to an additive constant. In other words, what matters is the differences of potential values rather than the values themselves.<\/p>\n

Riesz kernels.<\/b> The Riesz kernel in dimension \\( {d\\geq2} \\) and parameter \\( {s\\geq0} \\) with \\( {0<s<d} \\) is defined for all \\( {x\\neq0} \\) by the formula<\/p>\n

\\[ g(x):=\\frac{1}{|x|^{d-s}}. \\]<\/p>\n

We recover the Coulomb\/Newton kernel when \\( {s=2} \\). The Riesz kernel is the fundamental solution of the fractional Laplacian<\/b>, which is a Fourier multiplier<\/b>, and a non-local operator<\/b> when \\( {d\\neq2} \\). Its inverse is a convolution operator<\/b>.<\/p>\n

Conclusion.<\/b> All in all, if we would like to incorporate all cases in a compact formula, we could consider in \\( {\\mathbb{R}^d} \\), \\( {d\\geq1} \\), for all \\( {s\\in\\mathbb{R}} \\) and \\( {x\\neq0} \\) the kernel<\/p>\n

\\[ g(x):=\\frac{1}{s|x|^s}, \\]<\/p>\n

with the convention \\( {g(x)=-\\log|x|} \\) if \\( {s=0} \\). The Coulomb\/Newton case corresponds to taking \\( {s=d-2} \\) as we have seen. Indeed, this is more or less the choice often made by my colleague Mathieu Lewin<\/a> for instance, see for example arXiv:1905.09138<\/a>.<\/p>\n

Entropy.<\/b> The logarithm appears as a derivative of power functions also in relation with entropy and hypercontractivity, as explained in a previous post<\/a>. The same for the derivation of the logarithmic Sobolev inequality from the Beckner inequalities.<\/p>\n

Further reading.<\/b><\/p>\n