Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker–Planck Equation with Confining Potential

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker–Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}H1-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2-distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal O\big ( (1+t)e^{-t\nu /2}\big )$$\end{document}O((1+t)e-tν/2), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}ν the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2-space to a weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}H1-space).


Introduction
This paper is devoted to the study of the long time behavior of the kinetic Fokker-Planck equation describing the time evolution of the phase space probability density f (t, x, v), e.g. in a plasma [31].
Applications range from plasma physics [29,13] to stellar dynamics [17,18].Here V = V (x) is a given smooth, bounded below confinement potential for the system, and ν > 0, σ > 0 denote the friction and diffusion parameters, respectively.This equation is associated with the Langevin stochastic differential equation where {B t } t≥0 is a Brownian motion in R n with covariance B t , B t ′ = δ t−t ′ .Since the equation conserves mass, i.e., we shall always assume (without restriction of generality) that R 2n f 0 (x, v)dxdv = 1.The unique normalized steady state of ( 1) is given by where c V is a positive constant such that R 2n f ∞ (x, v)dxdv = 1.The following equation is also considered as the kinetic Fokker-Planck equation: and to switch from (1) to (3) it suffices to set h := f /f ∞ .It was shown in [22] that, if V ∈ C ∞ (R n ), (3) generates a C ∞ regularizing contraction semigroup in L 2 (R d , f ∞ ) := {g : R d → R : g is measurable and R d g 2 f ∞ dxdv < ∞}, d = 2n.For well-posedness with non-smooth potentials, we refer to [32,Theorem 6,Theorem 7].
The long time behavior and exponential convergence of the solution to the steady state has been studied and there are various results: in [19], algebraic decay was proved for potentials that are asymptotically quadratic (as |x| → ∞) and for initial conditions that are bounded below and above by Gaussians.The authors used logarithmic Sobolev inequalities and entropy methods.In [24], exponential decay was obtained also for faster growing potentials and more general initial conditions.That proof is based on hypoellipticity techniques.By using hypoelliptic methods, Villani proved exponential convergence results in [32,Theorem 35] and in L 2 (R d , f ∞ ) [32,Theorem 37].The main conditions in Villani's theorems above, as well as in [20,9,10,33,14,15], are the validity of the Poincaré inequality (5) and the criterion where denotes the Frobenius norm of When ∂ 2 V ∂x 2 is bounded, Villani also proved that the solution converges to the steady state exponentially in the logarithmic entropy [32,Theorem 39].This result was extended in [16] to potentials V satisfying a weighted log-Sobolev inequality and the condition that V −2η ∂ 2 V ∂x 2 is bounded for some η ≥ 0.Even though Villani's result allows for a general class of potentials, the growth condition (4) is not satisfied by potentials with singularities.This type of potentials, such as Lennard-Jones type interactions with confinement, are considered in [10] and their method relies on an explicit construction of a Lypunov function and Gamma calculus.In [20], Dolbeault, Mouhot, and Schmeiser developed a method to get exponential decay in L 2 for a large class of linear kinetic equations, including (1).Their method was also used to study the long time behavior of (1) when the potential V is zero or grows slowly as |x| → ∞, see [11,12].Based on a probabilistic coupling method, Eberle, Guillin, and Zimmer [21] obtained an exponential decay result in Wasserstein distance.
The associated semigroup of the kinetic Fokker-Planck equation has instantaneous regularizing properties which is called hypoellipticity [26].This hypoelliptic regularization is obvious when the confining potential V is zero or quadratic as the fundamental solution can be explicitly computed (see [28], [26]).For potentials such that ∂ 2 V ∂x 2 is bounded, Hérau [23] obtained short time estimates for a L 2 (R d , f ∞ ) → H 1 (R d , f ∞ ) regularization by constructing a suitable Lyapunov functional.Based on interpolation inequalities and a system of differential inequalities, Villani [32,Appendix A.21] extended Hérau's result for potentials satisfying (4).
We provide a new method to establish exponential decay of the solution to the steady state in H 1 (R d , f ∞ ) for a wide class of potentials: Our method extends [32,1,3] by allowing for more general Lyapunov functionals.Generalizing the previous approaches, the weight matrix in the dissipation functional (a generalized Fisher information) may now depend on x and v.This leads to a new criterion on the potential V.For this entropy method we need the time derivative of the dissipation functional, but we also provide its (x, v)-pointwise analog, in the spirit of the Gamma calculus [9].We provide a formula to estimate easily the exponential decay rate depending on the parameters of the equation, the constants appearing in the Poincaré inequality (5) and the growth condition on the potential (see (6) below).As a test of the effectiveness of our method, we show that our estimate on the decay rate is sharp when the potential is a quadratic polynomial.Moreover, our method lets us obtain estimates on the hypoelliptic regularization for potentials that are more general than in [23].
The organization of this paper is as follows.In Section 2, we define the assumptions on the potential, state the main results, and present concrete examples of such potentials.In Section 3, we present the intuition and explain our method.Section 4 contains important lemmas about matrix inequalities which are important to construct suitable Lyapunov functionals.The final section presents the proof of the main results.

Main results
We make the following assumptions.
Assumption 2.1.There exists a constant C P I > 0 such that the Poincaré inequality Sufficient conditions on the potential appearing in f ∞ so that the Poincaré inequality holds, e.g. the Bakry-Emery criterion, are presented in [8,Chapter 4].Assumption 2.2.There are constants c ∈ R and τ ∈ [0, ν) such that the following R m×m matrix, is positive semi-definite for all x ∈ R n , where I ∈ R n×n denotes the identity matrix.
Roughly speaking, Assumption 2.2 essentially means that the second order derivatives of V control the third order ones.It implies that ∂x 2 + cI is positive semi-definite for all x ∈ R n , and hence the eigenvalues of are uniformly bounded from below.We note that, in contrast to the Bakry-Emery strategy [7], the eigenvalues here may take negative values.
Let α(x) ∈ R denote the smallest eigenvalue of Then the following condition implies Assumption 2.2.For its proof see Appendix 6.1.Assumption 2.2'.There are constants c ∈ R and τ for all x ∈ R n and i ∈ {1, ..., n}.
We denote and assume in the sequel that α 0 > −∞.Hence Assumption 2.2 can only hold for some c ≥ −α 0 .
In the following results, we require that We now state our first result, i.e. exponential decay of a functional that is a linear combination of the weighted L 2 −norm and a Fisher information-type functional: Theorem 2.3.Let V be a C ∞ potential in R n satisfying Assumptions 2.1 and 2.2.Let C P I , c, τ, and α 0 be the constants in (5), (6), and (8).Suppose the initial data f 0 satisfies and Then there are explicitly computable constants C > 0 and λ > 0 (independent of f 0 ) such that the solution f (t) of (1) satisfies for all t ≥ 0.Moreover, we have: , where ∂x 2 is positive definite, then Assumptions 2.1 and 2.2 are satisfied with τ = 0, c = −α 0 (this rules out the conditions in the case of (c)).Moreover, the decay rates λ in (a) and (d) are sharp and, in the case of (d), ν ≥ A −1 2 + √ ν 2 − 4α 0 holds and so 2λ = ν − √ ν 2 − 4α 0 .In the case of (b), the decay rate 2λ = ν − ε is sharp in the sense that (9) holds with the rate 2λ = ν − ε for any small fixed ε ∈ (0, ν), but it does not hold with the rate 2λ = ν.Remark 2.4.
1.It is possible to make weaker regularity hypothesis on the potential V, but we maintain the assumption that V ∈ C ∞ to keep the presentation simple.(9) implies that the solution converges exponentially to the steady state in

Since
∂x 2 are uniformly bounded, then (9) is equivalent to the exponential decay of the solution to the steady state in H 1 (R 2n , f ∞ ).Due to the Poincaré inequality (5), the L 2 −term on the right hand side of (9) could be omitted.
3. If V satisfies Assumption 2.2 with some constants c ∈ R and τ ∈ [0, ν), then V also satisfies Assumption 2.2 with any c ≥ c and τ ∈ [τ, ν).Therefore, these constants are not unique.But the exponential decay rate λ obtained in Theorem 2.3 depends on the choice of c and τ.To obtain a better rate, one has to optimize λ = λ(c, τ ) with respect to all c and τ satisfying Assumption 2.2.
5. The highest exponential rate is ν 2 which can be attained by the quadratic potentials V with When V is a quadratic polynomial as in Theorem 2.3 (e), we prove the following sharp estimates.Proposition 2.5.Let V be a quadratic polynomial and ∂2 V ∂x 2 be positive definite.Let α 0 > 0 be the smallest eigenvalue of We shall use this proposition to prove the sharpness of the decay rates in Theorem 2.3 (e).When V is a quadratic polynomial and −α 0 = − ν 2 4 =: c, Theorem 2.3 (e) shows that the decay in ( 9) can be e −(ν−ε)t for any small fixed ε ∈ (0, ν), but it can not be e −νt .In this case, it is natural to expect a decay between e −νt and e −(ν−ε)t : Proposition 2.5 shows that this is indeed the case for the square of the L 2 −norm, with the decay (1 + t) 2 e −νt .But an analogous extension of this result for the functional on the left hand side of (9) (i.e., to replace the term Ce −(ν−ε)t with C(1 + t) 2 e −νt ) has not been obtained so far.
Our next result is about the estimates on the hypoelliptic regularization.
Theorem 2.7.Assume V is a C ∞ potential on R n and there are constants c ∈ R and τ ≥ 0 such that the matrix (6) is positive semi-definite for all x ∈ R n .Suppose the initial data f 0 satisfies Then, for any t 0 > 0, there are explicitly computable If we use the estimates in Theorem 2.7, this condition can be relaxed: Corollary 2.8.Let V be a C ∞ potential in R n satisfying Assumptions 2.1 and 2.2.Suppose the initial data f 0 satisfies Then, for any t 0 > 0, there is holds for all t ≥ t 0 with λ defined in Theorem 2.3.
1.In contrast to Theorem 2.3, Theorem 2.7 holds even if the Poincaré inequality (5) is not satisfied by f ∞ .Also, τ can be larger than ν.
2. The exponents of t in (11) and (12) are optimal when V is a quadratic polynomial (see [33, Appendix A]).
To illustrate our result, we present concrete examples of potentials V satisfying our Assumption 2.1 and Assumption 2.2: x ∈ R n with a positive definite covariance matrix M −1 ∈ R n×n , a constant vector p ∈ R n and a constant q ∈ R, the convergence rate is (case (a)) , and it is sharp for α 0 = ν 2 4 , where α 0 is the smallest eigenvalue of M −1 (see Theorem 2.3 (e)).
b) More generally, we consider potentials of the form where r > 0, k ∈ N and V 0 : R n → R is a polynomial of degree j < 2k.Since we have already considered quadratic potentials, we assume k ≥ 2. V satisfies the Poincaré inequality (5); this can be proven, for example, by showing that V satisfies one of the sufficient conditions given in [6, Corollary 1.6].Concerning Assumption 2.2' we have Since V 0 has degree j < 2k, there is a constant A > 0 such that Therefore, we can estimate We also observe that there exists a positive constant B such that for all i ∈ {1, ..., n}.(14) shows that the smallest eigenvalue of there are constants c and τ ∈ [0, ν) such that ( 7) is satisfied.Thus, Theorem 2.3 applies to this type of potentials.In particular, it applies to double-well potentials of the form Remark 2.11.
1. Our decay and regularization results above extend those of [23], where a stronger assumption, i.e.
) for all i, j ∈ {1, ..., n}, was made.By contrast, we did not require the boundedness of the second and higher derivatives of V. [32,20,9,10,33,14,15]) used the growth condition (4) to get some weighted Poincaré type inequalities (see [32,Lemma A.24]), which are crucial in these works -and additional to the Poincaré inequality (5).Our technique is rather different, based on construction of appropriate state dependent matrices and state dependent matrix inequalities so that the (modified) dissipation functional (see (20) below) decays exponentially.

Most of the previous works on the exponential convergence
3. Most of the previous methods for proving the exponential convergence do not give an accurate decay rate, λ is typically much too small there (see [32,Section 7.2], [20, Section 1.4]).For example, in [32, Section 7.2], the exponential decay rate λ = 1 40 was obtained for V (x) = |x| 2 2 and ν = σ = 1.Since our decay rates are sharp for quadratic potentials, in this setting, the true rate λ = 1 2 is given by Theorem 2.3 (a) and (e).

Modified entropy methods for degenerate Fokker-Planck equations
We first consider the following degenerate and non-symmetric Fokker-Planck equation [2,1]: where The weak maximum principle for degenerate parabolic equations [25] can be applied to (15) and we can prove that f (t, ξ) ≥ 0 for all t > 0, ξ ∈ R d .The divergence structure implies that the initial mass is conserved and f (t, •) describes the evolution of a probability density We are interested in the large-time behavior of the solution, in particular, when rank(D) is less than the dimension d.When D is positive definite (rank(D) = d), the large time behavior and exponential convergence have been studied comprehensively (see [7], [4], [2]).One of the well-know conditions which provides the exponential decay of the solution to the steady state is called the Bakry-Emery condition (see ( 16) below) leading to: Then To prove the theorem above, one considers the time derivative of the L 2 −norm and we see that it decreases It can be proven that, under the Bakry-Emery condition, Integrating this inequality from (t, ∞) and using the convergences and, by Grönwall's lemma, we get the desired result.When D is only positive semi-definite, i.e. rank(D) < d, one observes that I(f (t)|f ∞ ) may vanish for certain probability densities f = f ∞ .Hence the inequalities (18) and (19) will not hold in general.Since the above problems stem from the singularity of D, one can modify the dissipation function and define a modified dissipation functional (see also [1,3]) where P : R d → R d×d is a symmetric positive definite matrix which will be chosen later.Extending the approach of [1,3], we allow the matrix P here to depend on ξ ∈ R d .Our goal is to derive a differential inequality similar to (18) (like the dissipation functional satisfied for non-degenerate equations), i.e.
for some λ > 0 and a "good" choice of the matrix P. If this holds true, we would obtain If we can choose such P = P (ξ) ≥ ηI for some η > 0 and all ξ ∈ R d , under the validity of the Poincaré which implies the exponential decay of the L 2 −norm More generally, since the quadratic entropy is also a decreasing function of time t, instead of proving (21), we can consider the functional and choose a suitable parameter γ ≥ 0 and a matrix P such that for some λ > 0. This idea and method were successfully applied in [3] to (15) when the potential E is quadratic.
We shall apply this method to the kinetic Fokker-Planck equation with non-quadratic V (x).
First, we denote Then the kinetic Fokker-Planck equation ( 1) can be written in the form of (15), with The rank of the diffusion matrix D is n < d = 2n.Thus, (1) is both non-symmetric and degenerate and the arguments above apply to the equation.We will develop a modified entropy method.We will choose ξ−dependent matrix P in the modified dissipation functional (20) so that (23) holds and λ > 0 is as large as possible.
We also mention that when the potential E is quadratic in (15), the question about the long time behavior can be reduced to an ODE problem: for some positive definite matrix K. Assume (D + R)K −1 is positive stable and there is no non-trivial subspace of KerD which is invariant under Proof.See [5,Theorem 3.4].
One consequence of Theorem 3.2 is that the decay estimate of the ODE-solution carries over to the corresponding Fokker-Planck equation.4 The choice of the matrix P For future reference (in the proof of Theorem 2.7) we shall now also allow the matrix P to be time dependent.Hence we shall next consider the generalized functional The following lemmas will play a crucial role in our arguments.
Lemma 4.1.Let P : [0, ∞) × R 2n → R 2n×2n be smooth and f be the solution of (1), then where , and denotes a scalar differential operator that is applied to each element of the matrix P = P (t, x, v).
Proof.We denote These equations can be written with respect to u = u 1 u 2 : It allows us to compute the time derivative of the modified dissipation functional First, we consider the term in the second line of (28) and use By integrating by parts the last term of ( 29) we obtain and we find If we use this equality in (29), we get 4σ Next, we integrate by parts in the terms in the third line of (28): (31) and (32) show that the third line of (28) equals Combining ( 28), (30), and (33) we obtain the statement (27).
Remark 4.2.We give now a (formal) generalization of the above result (27) to Markovian evolution equations using the Gamma calculus, see, e.g., [8,9,10]: First, let L be the generator of some Markovian evolution on R d with corresponding invariant measure f ∞ dξ.Let P = P (ξ) be a smooth matrix function (but it does not have to be symmetric or positive definite).We define the first order bilinear form For a solution h(t) of ∂ t h = Lh, these definitions give We use Γ P to define the modified dissipation functional We obtain by integrating (34): where we used that In particular, let L be the generator of the kinetic Fokker-Planck equation (3), and we recall that Then, a straightforward (but lengthy) computation shows that One can check (by integrating by parts the term 4σ R d n i=1 u T (∂ vi P )∂ vi uf ∞ dξ in the right hand side of (35)) that (35) coincides with (27).Hence, (35) reproduces (27).But in contrast to (27), the preceding statement (34) is local in ξ and therefore stronger.
The key question for using the modified entropy dissipation functional S(f ) is how to choose the matrix P. To determine P we shall need the following algebraic result: We consider the matrix function which appears in (27).We want to construct a symmetric positive definite matrix P (x) such that Q(x)P (x) + P (x)Q T (x) is positive definite and for some µ > 0 and for all x ∈ R n .We recall α(x) := min i∈{1,..,n} be positive definite for some x ∈ R n .Then: 4 , then µ = ν 2 and there exists a symmetric positive definite matrix P (x) such that and there exists a symmetric positive definite matrix P (x) such that Q(x)P (x) + P (x)Q T (x) ≥ 2µP (x).
(c) If α 0 = ν 2 4 , then µ = ν 2 and, for any ε ∈ (0, ν), there exists a symmetric positive definite matrix Proof.Part 1) Let x be any point of R n , we compute the eigenvalues β(x) of Q(x).If β(x) = 0 we have the condition Hence the non-zero eigenvalues of Q(x) are where i = √ −1.Moreover, β(x) = 0 can be an eigenvalue of Q(x) iff one of the eigenvalues of is zero.This shows that Q(x) is positive stable (i.e., the eigenvalues β i (x) have positive real part) iff For Part 2) we shall construct matrices P (x), which relies on the proof of Lemma 4.3 (Lemma 4.3 in [3]).
(a) Let α 0 > ν 2 4 .In this case, because of (37) the matrix Q(x) is positive stable and µ = ν 2 > 0. We define the matrix , and for this choice, it is easy to check that To make sure that P (x) is positive definite, we compute the eigenvalues η(x) of P (x) at each x ∈ R n : For η(x) = 2 we have the condition η(x) = 2 is not an eigenvalue of P (x) and so the eigenvalues of P (x) satisfy We conclude that the eigenvalues are Since we assumed α i (x) ≥ α(x) ≥ α 0 > ν 2 4 for all i ∈ {1, ..., n}, the eigenvalues are positive and satisfy η := inf Thus, P (x) is positive definite and . Let ε > 0 be a fixed small number.We define and consider the matrix We compute its eigenvalues η(x) by a similar computation as above: where We also have Thus, P (x) is positive definite and P (x) ≥ ηI for all x ∈ R n .Then we compute Since ∂ 2 V ∂x 2 ≥ ωI, the second matrix in the last line of (39) is bounded below by Consequently, we get is not positive definite at some x ∈ R n (and hence α 0 ≤ 0), then Q(x) is not positive stable.In this case, it is not possible to find a positive constant µ and a positive definite matrix P (x) such that Q(x)P (x) + P (x)Q T (x) ≥ µP (x).If α 0 is just finite and not necessarily positive, we have the following modified inequality.Lemma 4.5.Let α 0 > −∞.Then there exist γ ≥ 0, δ ∈ [0, ν), and a symmetric positive definite matrix function P (x) such that where D = 0 0 0 σI ∈ R 2n×2n is the matrix defined in (24).

Proof of Theorem 2.3
Proof.We denote We consider the modified dissipation functional for some symmetric positive definite matrix P = P (x, v) ∈ R 2n×2n .By Lemma 4.1 (for a t-independent matrix P ) we have + cI is also positive semi-definite and so ≥ −cI for all x ∈ R n .We define the matrix P depending on the constant c.
Case (a): . By Lemma 4.4 (2a) and by its proof, the matrix For this choice of the matrix P, Then (47) can be written as We shall now consider each term of this equation.First we compute Now we consider the last term in (49) 4 where we integrated by parts and used 49), ( 51), (52), and (50) we obtain The right hand side of this inequality is a quadratic polynomial with respect to ∂ vi u 2 , i ∈ {1, ..., n}, and u 2 .The corresponding matrix of this quadratic polynomial is The assumption ∂x 2 +cI and the Assumption 2.2 imply that (53) is positive semi-definite.
Thus we have obtained and by Grönwall's lemma The estimate P (x) ≥ ηI and the Poincaré inequality (5) imply The matrix inequalities (see Lemma 6.1 in Appendix 6.2) show that S(f (t)) is equivalent to the functional This equivalence, and (55) let us obtain (9).
Case (b): Then by Lemma 4.4 (2c), for any ε ∈ (0, ν − τ ), the matrix With this matrix we have and by using (48 (47), ( 57), ( 58), ( 59), (60), and similar estimates as for Case a) show that The right hand side of this inequality is a quadratic polynomial with respect to ∂ vi u 2 , i ∈ {1, ..., n}, and u 2 .The corresponding matrix of this quadratic polynomial is 61) is positive definite and we get and by Grönwall's lemma Similar to (55), we have The functional and S(f (t)) are equivalent because of (see Lemma 6.1 in Appendix 6.2) (64) This equivalence, and (63) imply (9).
The choice of the matrix P in (67), (66), and (68) lets us estimate Similar computations as for Case (a) as well as (58) (but with ε 2 = 2a) lead to The two integrands of the right hand side are together a quadratic polynomial of ∂ vi u 2 , i ∈ {1, ..., n}, and u 2 , and its corresponding matrix is Because of a − ν 2 4 ≥ c and Assumption 2.2, the matrix (70) is positive semi-definite, thus, we have The estimate P (x) ≥ ηI (η > 0 defined in (41)) and the Poincaré inequality (5) imply and so This estimate and (71) let us conclude One can check that (see Lemma 6.1 in Appendix 6.2) (75) Hence, S(f (t)) is equivalent to the functional Subsequently, Φ(f (t)) and the functional on the left hand side of ( 9) are equivalent.This equivalence and (74) let us obtain (9).
Case (c) and (d), estimated decay rate: Next, we shall estimate λ from (73) explicitly, and we shall choose the parameters a and γ such that λ is (rather) large.By (41) and ( 46), η = η(a) and δ = δ(a, γ) are functions of a Since δ > 0, and η is monotonically increasing up to 2, we have the following uniform estimate and choice of the decay rate: Next, we shall estimate this supremum (in fact it is a maximum).First we introduce a new variable With the notations A(a) := 1 + a + α 0 + (a + α 0 − 1) 2 + ν 2 2σC P I > 0 and B(a Step 1, reformulation as an ODE-problem: To this end we use Theorem 3.2.We check the conditions of this theorem for the kinetic Fokker-Planck equation.With the notation ξ = x v , we write where α i , i ∈ {1, ..., n} denote the eigenvalues of M −1 .By solving the latter equation, we find that the eigenvalues of Hence µ is positive, so K −1/2 (D + R)K −1/2 and (D + R)K −1 are positive stable.Therefore, Theorem 3.2 applies to the kinetic Fokker-Planck equation.
Step 2, decay rates of the ODE-solution: We consider the ODE ξ with the initial data ξ(0) = ξ 0 .Since K −1/2 (D + R)K −1/2 is positive stable, the solution ξ(t) is stable.To quantify the decay rate, we continue to analyze the eigenvalues of Let m i be the multiplicity of α i > 0 as an eigenvalue of M −1 (now the α i with i ∈ {1, ..., ñ} are labeled without multiplicity).Since M −1 is symmetric, there are linearly independent eigenvectors Then we can check that the vectors are linearly independent eigenvectors of K −1/2 (D + R)K −1/2 corresponding to β − i , i ∈ {1, ..., ñ}.Moreover, these vectors form a basis of the space of eigenvectors corresponding to β − i .Similarly, the vectors − Proof of Theorem 2.7.
Step 1, an auxiliary inequality: As we assume the matrix ( 6) is positive semi-definite, then the following submatrices of ( 6) are positive semi-definite: Letting δ > 0, we consider X δ is positive semi-definite as it is the Kronecker product [27, Corollary 4.2.13] of two positive semi-definite matrices.Hence, we have for all k ∈ {1, ..., n} : Tr(X and by minimizing the constant on the left hand side of (89) with respect to δ (i.e., by choosing Step 2, growth estimate for the r.h.s. of (11), ( 12): We denote and by integrating by parts, we obtain Next, we compute (with || • || denoting the Frobenius norm) Integrating by parts with respect to v, we obtain 2 Next, we work on the term in the second line of (92): 2 where we integrated by parts twice, and used − ν σ vf ∞ = ∇ v f ∞ and the notations Using the identity n i,j=1 the estimate (90), and the discrete Hölder inequality, (94) can be estimated as Combining the equations from (92) to (95) and the identity we get (96) can be reformulated as Step 3, t−dependent functional Ψ: In order to prove the short-time regularization of ( 11) and ( 12) we introduce now an auxiliary functional that depends explicitly on time.Our strategy is the generalization of the approach in [32, Theorem A.12], [23, Theorem 1.1], [3, Theorem 4.8].
For t ∈ (0, t 0 ], we consider the following functional with the t− and x−dependent matrix in R 2n×2n , ε, γ 1 , and γ 2 are positive constants which we shall fix later.We note that, for all t ∈ (0, t 0 ], as ∂ 2 V ∂x 2 + cI is positive semi-definite.Thus, Ψ(t, f (t)) is non-negative and satisfies Our goal is to show that Ψ(t, f (t)) decreases.To this end we estimate the time derivative of the second term in (98).First, (27) yields . We consider each terms of (102).Because of (100), the first term can be estimated as ) For the third term of (102) we have For the second term of (102) we compute Using the estimates We fix ε = ε(t 0 ) > 0 so that the element in the upper left corner of the matrix in ( 106) is positive for t > 0; more precisely we require Then, the matrix in the last line of (106) can be estimated as Using this matrix inequality, we obtain from (106): (102), ( 103), (104), and (108) show that As the matrix ( 6) is positive semi-definite, we have Step 4, decay of the functional Ψ: We estimate the time derivative of (98): Combining (91), (97), and (109) yield We fix γ 1 > 0 and γ 2 > 0 such that for all x ∈ R n and t ∈ [0, t 0 ].We recall that we have fixed ε = ε(t 0 ) so that (107) holds, which makes the above denominator positive.The existence of such γ 1 > 0 and γ 2 > 0 can be proven by the following arguments: We can consider the left hand side of (111) as a quadratic polynomial of ∂ 2 V ∂x 2 + cI ∈ [0, ∞).As time t varies in a bounded interval [0, t 0 ], the terms containing t are bounded.Therefore, we can choose large values for γ 1 = γ 1 (t 0 ) and γ 2 = γ 2 (t 0 ) so that this quadratic polynomial is non-negative for all t ∈ [0, t 0 ].
To show that ( 6) is positive semi-definite, it is enough to show the quadratic form above is non-negative.Assumption 2. Therefore, we get the desired result Proof.We consider, for some k ∈ R to be chosen later as We set k := 1 2c 2 > 0.
These two cases show that inequality (80) holds.