{"id":8746,"date":"2016-02-13T00:45:36","date_gmt":"2016-02-12T23:45:36","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=8746"},"modified":"2022-04-24T20:27:56","modified_gmt":"2022-04-24T18:27:56","slug":"aspects-of-the-ornstein-uhlenbeck-process","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2016\/02\/13\/aspects-of-the-ornstein-uhlenbeck-process\/","title":{"rendered":"Aspects of the Ornstein-Uhlenbeck process"},"content":{"rendered":"
\"George<\/a>
George Uhlenbeck (1900-1988)<\/figcaption><\/figure>\n

The Ornstein-Uhlenbeck process \\( {X={(X_t)}_{t\\in[0,\\infty)}} \\) on \\( {\\mathbb{R}^n} \\) is the solution of the stochastic differential equation<\/p>\n

\\[ dX_t=\\sqrt{2}dB_t-X_tdt \\]<\/p>\n

where \\( {{(B_t)}_{t\\in[0,\\infty)}} \\) is a standard Brownian motion. Since the diffusion coefficient is constant and the drift is affine, it follows that \\( {X} \\) is a Gaussian process. The computation of the mean and of the variance of \\( {X_t} \\) conditional on \\( {\\{X_0=x\\}} \\) yields<\/p>\n

\\[ \\mathrm{Law}(X_t\\mid X_0=x)=\\mathcal{N}(xe^{-t},\\sqrt{1-e^{-2t}}I_n). \\]<\/p>\n

This shows that for any \\( {x} \\) and conditional on \\( {\\{X_0=x\\}} \\), \\( {X} \\) converges in distribution:<\/p>\n

\\[ X_t\\underset{t\\rightarrow\\infty}{\\overset{d}{\\longrightarrow}}\\gamma_n \\]<\/p>\n

where \\( {\\gamma_n=\\mathcal{N}(0,I_n)} \\) has density \\( {(2\\pi)^{-\\frac{n}{2}}e^{-\\frac{1}{2}|x|^2}} \\). This shows also that \\( {\\gamma_n} \\) is invariant<\/b>:<\/p>\n

\\[ X_0\\sim\\gamma_n\\quad\\Rightarrow\\quad\\forall t\\geq0,\\quad X_t\\sim\\gamma_n. \\]<\/p>\n

Actually a stronger property holds true: the law \\( {\\gamma_n} \\) is reversible<\/b> in the sense that<\/p>\n

\\[ X_0\\sim\\gamma_n\\quad\\Rightarrow\\quad\\forall t\\geq0,\\quad (X_0,X_t)\\overset{d}{=}(X_t,X_0). \\]<\/p>\n

The explicit law of the process allows computations, for instance for any \\( {s,t\\geq0} \\),<\/p>\n

\\[ \\mathrm{Cov}(X_s,X_t)=e^{-|t-s|}(1-e^{-2\\min(s,t)}). \\]<\/p>\n

For any bounded and measurable \\( {f:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\), any \\( {x\\in\\mathbb{R}} \\) and \\( {t\\in[0,+\\infty)} \\), we set<\/p>\n

\\[ P_t(f)(x)=\\mathbb{E}(f(X_t)\\mid X_0=x). \\]<\/p>\n

We have \\( {P_t(\\mathbf{1}_A)(x)=\\mathbb{P}(X_t\\in A\\mid X_0=x)} \\). The family \\( {{(P_t)}_{t\\in[0,\\infty)}} \\) is a semigroup<\/b> of linear operators acting on continuous and bounded functions, in the sense that<\/p>\n

\\[ P_0=id, \\quad \\forall s,t\\geq0, \\quad P_t\\circ P_s = P_{t+s}. \\]<\/p>\n

These operators are Markov operators, in the sense that for any \\( {t\\in[0,\\infty)} \\),<\/p>\n

\\[ \\forall c\\in\\mathbb{R},\u00a0P_t(c)=c, \\quad\\text{and}\\quad \\forall f,\u00a0f\\geq0\u00a0\\Rightarrow\u00a0P_t(f)\\geq0. \\]<\/p>\n

The explicit law of the process provides the Mehler formula<\/b> for the semigroup<\/p>\n

\\[ P_t(f)(x) =\\int\\!f(xe^{-t}+\\sqrt{1-e^{-2t}}y)\\gamma_n(dy)\\\\ =\\mathbb{E}(f(xe^{-t}+\\sqrt{1-e^{-2t}}Z)). \\]<\/p>\n

This gives the following commutation<\/b> with the gradient when \\( {f} \\) is smooth:<\/p>\n

\\[ (\\nabla P_t f)(x)=e^{-t}P_t(\\nabla f)(x), \\]<\/p>\n

where in the right hand side, \\( {P_t} \\) acts on each coordinates of the vector \\( {\\nabla f} \\).<\/p>\n

The infinitesimal generator<\/b> is the unbounded operator in \\( {L^2(\\gamma_n)} \\) given by<\/p>\n

\\[ Af =\\partial_{t=0^+}P_tf =\\lim_{t\\rightarrow0^+}\\frac{P_tf-f}{t} =\\Delta f-\\langle x,\\nabla f\\rangle. \\]<\/p>\n

The Chapman-Kolmogorov evolution equation<\/b> writes<\/p>\n

\\[ \\partial_tP_t=AP_t=P_tA. \\]<\/p>\n

If we fix \\( {f} \\) and write \\( {u_t(x)=P_t(f)(x)} \\) for any \\( {x} \\) and \\( {t} \\) then<\/p>\n

\\[ u_0=f\\quad\\text{and}\\quad \\partial_t u_t=Au_t=\\Delta u_t-\\langle x,\\nabla u_t\\rangle. \\]<\/p>\n

The operator \\( {A} \\) (and \\( {P_t} \\) for any \\( {t\\geq0} \\)) is symmetric in \\( {L^2(\\gamma_n)} \\), in other words an integration by parts<\/b> holds, meaning that for any \\( {f} \\) and \\( {g} \\),<\/p>\n

\\[ -\\int\\!fAg\\,d\\gamma_n =\\int\\!\\nabla f\\cdot\\nabla g\\,d\\gamma_n. \\]<\/p>\n

If \\( {X_0} \\) has density \\( {f_0} \\) with respect to \\( {\\gamma_n} \\) then \\( {X_t} \\) has also a density with respect to \\( {\\gamma_n} \\) given by \\( {f_t=P_tf_0} \\). If \\( {g} \\) is the Lebesgue density of \\( {\\gamma_n} \\), then \\( {g_t=f_tg} \\) is the Lebesgue density of \\( {X_t} \\). The evolution of \\( {g_t} \\) with respect to \\( {t} \\) is described by the Fokker-Planck equation<\/b>, dual of the Chapman-Kolmogorov equation,<\/p>\n

\\[ \\partial_tg_t=\\Delta g_t+\\mathrm{div}(xg_t). \\]<\/p>\n

If \\( {\\mu} \\) and \\( {\\nu} \\) are probability measures on \\( {\\mathbb{R}^n} \\) with \\( {\\nu\\ll\\mu} \\) then the Kullback-Leibler divergence<\/b> or relative entropy<\/b> of \\( {\\nu} \\) with respect to \\( {\\mu} \\) is defined by<\/p>\n

\\[ H(\\nu\\mid\\mu)=\\int\\!f\\log f\\,d\\mu=\\int\\!\\log f\\,d\\nu \\quad\\text{where}\\quad f=\\frac{d\\nu}{d\\mu}. \\]<\/p>\n

We take the convention \\( {H(\\nu\\mid\\mu)=+\\infty} \\) if \\( {f\\log f\\not\\in L^1(\\mu)} \\) or if \\( {\\nu\\not\\ll\\mu} \\). Note that Jensen's inequality shows that \\( {H(\\nu\\mid\\mu)\\geq0} \\) with equality iff \\( {\\mu=\\nu} \\).<\/p>\n

In the case where \\( {\\mu} \\) is a Boltzmann-Gibbs measure with Lebesgue density \\( {g(x)=e^{-V(x)}} \\), the quantity \\( {H(\\nu\\mid\\mu)} \\) becomes a Helmholtz free energy<\/b>, in the sense that<\/p>\n

\\[ H(\\nu\\mid\\mu)=\\int\\!V\\,d\\nu-S(\\nu) \\]<\/p>\n

where the first term in the right hand side is the mean energy<\/b> of \\( {\\nu} \\) while the second term in the right hand side is the Boltzmann-Shannon entropy<\/b><\/p>\n

\\[ S(\\nu)=\\displaystyle\\int\\!fg\\log(fg)\\,dx. \\]<\/p>\n

Suppose that the law \\( {\\mu_0} \\) of \\( {X_0} \\) has density \\( {f_0} \\) with respect to \\( {\\gamma_n} \\). Then the law \\( {\\mu_t} \\) of \\( {X_t} \\) has density \\( {f_t=P_tf_0} \\) with respect to \\( {\\gamma_n=\\mu_\\infty} \\). The free energy decays along the time. Namely, using the evolution equation and the integration by parts,<\/p>\n

\\[ \\begin{array}{rcl} \\frac{d}{dt}H(\\mu_t\\mid\\gamma_n) &=&\\displaystyle\\int\\!\\partial_t(f_t\\log f_t)\\,d\\gamma_n\\\\ &=&\\displaystyle\\int\\!(1+\\log f_t)Af_t\\,d\\gamma_n\\\\ &=&\\displaystyle-\\int\\!\\frac{|\\nabla f_t|^2}{f_t}\\,d\\gamma_n\\\\ &=&-J(\\mu_t\\mid\\gamma_n) \\leq0. \\end{array} \\]<\/p>\n

This is know as the de Bruijn<\/b> identity:<\/p>\n

\\[ \\frac{d}{dt}H(\\mu_t\\mid\\gamma_n)=-J(\\mu_t\\mid\\gamma_n)\\leq0. \\]<\/p>\n

The quantity<\/p>\n

\\[ J(\\nu\\mid\\mu) =\\int\\!\\frac{|\\nabla f|^2}{f}\\,d\\mu=\\int\\!|\\nabla\\log f|^2\\,d\\nu \\quad\\text{where}\\quad f=\\frac{d\\nu}{d\\mu} \\]<\/p>\n

is the Fisher information<\/b>. How it behaves along the O.-U. dynamics? Well, using commutation, two times Jensen's inequality, and the invariance of \\( {\\gamma_n} \\), we get<\/p>\n

\\[ \\begin{array}{rcl} J(\\mu_t\\mid\\gamma_n) &=&\\int\\!\\frac{|\\nabla f_t|^2}{f_t}\\,d\\gamma_n\\\\ &=&e^{-2t}\\int\\!\\frac{|(\\nabla f)_t|^2}{f_t}\\,d\\gamma_n\\\\ &\\leq& e^{-2t}\\int\\!\\frac{(|\\nabla f|)_t^2}{f_t}\\,d\\gamma_n\\\\ &\\leq& e^{-2t}\\int\\!\\left(\\frac{|\\nabla f_0|^2}{f_0}\\right)_t\\,d\\gamma_n\\\\ &=& e^{-2t}J(\\mu_0\\mid\\gamma_n), \\end{array} \\]<\/p>\n

in other words the Fisher information decays exponentially:<\/p>\n

\\[ \\forall \\mu_0\\ll\\gamma_n,\u00a0 \\forall t\\geq0,\\quad J(\\mu_t\\mid\\gamma_n)\\leq e^{-2t}J(\\mu_0\\mid\\gamma_n). \\]<\/p>\n

In particular we get \\begin{align*} H(\\mu_0\\mid\\gamma_n) =-\\int_0^\\infty\\!\\frac{d}{dt}H(\\mu_t\\mid\\gamma_n)\\,dt =\\int_0^\\infty\\!J(\\mu_t\\mid\\gamma_n)\\,dt \\leq \\frac{1}{2}J(\\mu_0\\mid\\gamma_n). \\end{align*} This inequality is known as a logarithmic Sobolev inequality<\/b>:<\/p>\n

\\[ \\forall \\nu\\ll\\gamma_n,\\quad H(\\nu\\mid\\gamma_n)\\leq\\frac{1}{2}J(\\nu\\mid\\gamma_n). \\]<\/p>\n

This inequality is optimal in the sense that equality is achieved when \\( {d\\nu(x)\/d\\gamma_n(x)=e^{ax}} \\) for some \\( {a\\in\\mathbb{R}} \\). Using this inequality for \\( {\\nu=\\mu_t} \\) yields<\/p>\n

\\[ \\frac{d}{dt}H(\\mu_t\\mid\\gamma_n) =-J(\\mu_t\\mid\\gamma_n) \\leq -\\frac{1}{2}H(\\mu_t\\mid\\gamma_n). \\]<\/p>\n

which gives, by Gronwall's lemma, an exponential decay<\/b> of the free energy, namely<\/p>\n

\\[ \\forall\\mu_0\\ll\\gamma_n,\u00a0 \\forall t\\geq0,\u00a0 H(\\mu_t\\mid\\gamma_n)\\leq e^{-2t}H(\\mu_0\\mid\\gamma_n). \\]<\/p>\n

Since both sides are equal for \\( {t=0} \\), taking the derivative at time \\( {t=0} \\) allows to recover from this exponential decay the logarithmic Sobolev inequality!<\/p>\n

Hypercontractivity.<\/b> For any \\( {t\\in[0,\\infty)} \\) and any \\( {p\\in[1,\\infty]} \\), Mehler's formula shows immediately that \\( {P_t} \\) can be extended into a linear operator on \\( {L^p(\\gamma_n)} \\). In fact \\( {P_t} \\) is always a contraction:<\/p>\n

\\[ \\forall p\\geq1,\u00a0\\forall t\\in[0,\\infty),\u00a0\\forall f\\in L^p(\\gamma_n),\\quad \\Vert P_tf\\Vert_p\\leq\\Vert f\\Vert_p. \\]<\/p>\n

Namely, using Jensen's inequality and the invariance of \\( {\\gamma_n} \\),<\/p>\n

\\[ \\begin{array}{rcl} \\Vert P_tf\\Vert_p^p &=&\\int\\!|\\mathbb{E}(f(X_t)\\mid X_0=x)|^p\\,d\\gamma_n(x)\\\\ &\\leq& \\int\\!\\mathbb{E}(|f(X_t)|^p\\mid X_0=x)\\,d\\gamma_n(x)\\\\ &=&\\int\\!P_t(|f|^p)\\,d\\gamma_n(x)\\\\ &=&\\int\\!|f|^p\\,d\\gamma_n(x)\\\\ &=&\\Vert f\\Vert_p^p. \\end{array} \\]<\/p>\n

Since equality is achieved for constant functions, it follows that \\( {\\Vert P_t\\Vert_{p\\rightarrow p}=1} \\). The semigroup \\( {{(P_t)}_{t\\in[0,\\infty)}} \\) is in fact hypercontractive<\/b>:<\/p>\n

\\[ \\forall p\\geq1,\u00a0\\forall t\\geq0,\u00a0\\forall f\\in L^p(\\gamma_n), \\quad \\Vert P_t f \\Vert_{p(t)} \\leq \\Vert f \\Vert_p, \\]<\/p>\n

where \\( {p(t) = 1 + (p-1)e^{2t}} \\), in other words \\( {\\Vert P_t\\Vert_{p\\rightarrow p(t)}=1} \\), and moreover this value \\( {p(t)} \\) is critical in the sense that if \\( {q > p(t)} \\) then \\( {\\Vert P_t\\Vert_{p\\rightarrow q}=+\\infty} \\).<\/p>\n

Let us give a proof. One can assume that \\( {f\\geq0} \\) since \\( {|P_t f|\\leq P_t|f|} \\) by Jensen's inequality. Note that \\( {p(0)=0} \\) and \\( {p(t)>p} \\) if \\( {t>0} \\). Set \\( {\\alpha(t)=\\log\\Vert P_t f\\Vert_{p(t)}} \\). To lighten the notation, let us set \\( {f_t=P_tf} \\). We have, for any \\( {t\\geq0} \\),<\/p>\n

\\[ \\begin{array}{rcl} \\alpha'(t) &=&\\left(\\frac{1}{p(t)}\\log\\int\\!f_t^{p(t)}\\,d\\gamma_n\\right)'\\\\ &=&-\\frac{p'(t)}{p(t)^2}\\log\\int\\!f_t^{p(t)}\\,d\\gamma_n +\\frac{1}{p(t)}\\frac{\\left(\\displaystyle\\int\\!f_t^{p(t)}\\,d\\gamma_n\\right)'}{\\displaystyle\\int\\!f_t^{p(t)}\\,d\\gamma_n}\\\\ &=&-\\frac{p'(t)}{p(t)^2}\\log\\int\\!(f_t)^{p(t)}\\,d\\gamma_n +\\frac{1}{p(t)}\\frac{\\displaystyle\\int\\!\\left(p'(t)\\log f_t+p(t)\\frac{Af_t}{f_t}\\right)f_t^{p(t)}\\,d\\gamma_n}{\\displaystyle\\int\\!f_t^{p(t)}\\,d\\gamma_n}\\\\ &=&-\\frac{p'(t)}{p(t)^2}\\log\\int\\!f_t^{p(t)}\\,d\\gamma_n +\\frac{p'(t)}{p(t)^2}\\frac{\\displaystyle\\int\\!f_t^{p(t)}\\log f_t^{p(t)}\\,d\\gamma_n}{\\displaystyle\\int\\!f_t^{p(t)}\\,d\\gamma_n} +\\frac{\\displaystyle\\int\\!(Af_t)f_t^{p(t)-1}\\,d\\gamma_n}{\\displaystyle\\int\\!f_t^{p(t)}\\,d\\gamma_n}\\\\ &=&\\frac{p'(t)}{p(t)^2}\\left(H(h_t^{p(t)}\\gamma_n\\mid\\gamma_n)+\\frac{p(t)^2}{p'(t)}\\int\\!(Ah_t)h_t^{p(t)-1}\\,d\\gamma_n\\right) \\end{array} \\]<\/p>\n

where \\( {h_t=f_t\/\\Vert f_t\\Vert_{p(t)}} \\). Now the logarithmic Sobolev inequality and the integration by parts give, for any \\( {h\\geq0} \\) such that \\( {h^p} \\) is a probability density with respect to \\( {\\gamma_n} \\),<\/p>\n

\\[ \\begin{array}{rcl} \\mathrm{H}(h^p\\gamma_n\\mid\\gamma_n) &\\leq&\\frac{1}{2}\\int\\!\\frac{|\\nabla h^p|^2}{h^p}\\,d\\gamma_n\\\\ &=&\\frac{p^2}{2}\\int\\!|\\nabla h|^2h^{p-2}\\,d\\gamma_n\\\\ &=&\\frac{p^2}{2(p-1)}\\int\\!\\left<\\nabla h,\\nabla h^{p-1}\\right>\\,d\\gamma_n\\\\ &=&-\\frac{p^2}{2(p-1)}\\int\\!(Ah)h^{p-1}\\,d\\gamma_n. \\end{array} \\]<\/p>\n

Using this inequality for \\( {h=h_t} \\) and \\( {p=p(t)} \\), and using \\( {2(p(t)-1)=p'(t)} \\), we obtain that \\( {\\alpha'(t)\\leq0} \\) for any \\( {t\\geq0} \\), and as a consequence<\/p>\n

\\[ \\log\\Vert P_tf\\Vert_{p(t)} = \\alpha(t) \\leq\\alpha(0)=\\log\\Vert f\\Vert_p. \\]<\/p>\n

Finally, if now \\( {q>p(t)} \\) then taking \\( {f_\\lambda(x)=e^{\\langle\\lambda,x\\rangle}} \\) for some \\( {\\lambda\\in\\mathbb{R}^n} \\) gives<\/p>\n

\\[ \\Vert f_\\lambda\\Vert_p=e^{p|\\lambda|^2\/2} \\quad\\text{and}\\quad P_t f_\\lambda=e^{|\\lambda|^2(1-e^{-2t})\/2}f_{\\lambda e^{-t}} \\]<\/p>\n

and therefore<\/p>\n

\\[ \\frac{\\Vert P_t f_\\lambda\\Vert_{q}}{\\Vert f_\\lambda\\Vert_p} =e^{|\\lambda|^2(e^{-2t}(q-1)+1-p)\/2}, \\]<\/p>\n

a quantity which tends to \\( {+\\infty} \\) as \\( {|\\lambda|\\rightarrow\\infty} \\) since \\( {q>p(t)=1+(p-1)e^{2t}} \\).<\/p>\n

The proof shows that conversely, from the hypercontractive statement, one can extract the logarithmic Sobolev inequality by taking the derivative at \\( {t=0} \\).<\/p>\n

Polynomials.<\/b> The set of polynomials \\( {\\mathbb{R}[X]} \\) is dense in \\( {L^2(\\gamma_1)} \\). To see it, let us take \\( {f\\in L^2(\\gamma_1)} \\), then the Laplace transform \\( {\\varphi_\\mu} \\) of the signed measure \\( {\\mu(dx)=f(x)\\gamma_1(dx)} \\) is finite on \\( {\\mathbb{R}} \\) since for any \\( {\\theta\\in\\mathbb{R}} \\), by the Cauchy-Schwarz inequality,<\/p>\n

\\[ (\\varphi_\\mu(\\theta))^2=\\left(\\int\\! \\exp(\\theta x)\\,\\mu(dx)\\right)^2 \\leq \\int\\!f^2\\,d\\gamma_1\\int\\!\\exp(2\\theta x)\\,\\gamma_1(dx)<+\\infty, \\]<\/p>\n

and in particular, \\( {\\varphi_\\mu} \\) is analytic on a neighborhood of \\( {0} \\). Now since for any \\( {k\\in\\mathbb{N}} \\),<\/p>\n

\\[ \\varphi_\\mu^{(k)}(0) =\\int\\!x^kf(x)\\,\\gamma_1(dx) =\\langle P_k,f\\rangle_{L^2(\\gamma_1)}\\quad\\text{where}\\quad P_{k}(x) =x^{k}, \\]<\/p>\n

and if \\( {f\\perp\\mathbb{R}[X_1,\\ldots,X_n]} \\) in \\( {L^2(\\mathbb{R})} \\), then the derivatives of any order of \\( {\\varphi_\\mu} \\) vanish at \\( {0} \\), and since \\( {\\varphi_\\mu} \\) is analytic, we get \\( {\\varphi_\\mu\\equiv0} \\) and then \\( {\\mu=0} \\) and then \\( {f=0} \\) in \\( {L^2(\\gamma_n)} \\).<\/p>\n

Hermite polynomials.<\/b> Hermite's polynomials \\( {{(H_k)}_{k\\in\\mathbb{N}}} \\) are the orthogonal polynomials obtained using the Gram-Schmidt algorithm in \\( {L^2(\\gamma_1)} \\) from the canonical basis of \\( {\\mathbb{R}[X]} \\). They are normalized in such a way that the coefficient of the term of highest degree in \\( {H_k} \\) is \\( {1} \\) for any \\( {k\\geq0} \\). We find<\/p>\n

\\[ H_0(x)=1,\\quad H_1(x)=x,\\quad H_2(x)=x^2-1,\\quad\\ldots \\]<\/p>\n

It can be checked that Hermite's polynomials \\( {{(H_k)}_{k\\geq0}} \\) satisfy<\/p>\n