Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|Δ\_{x}^{\frac α2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$for $v \in C\_0^{\infty}({\mathbb R}^N)$,where $L\_x$ is a non-local operator comparable with the ${\mathbb R}^d $-fractional Laplacian $Δ\_{x}^{\frac α2}$ in terms of symbols, $α\in (0,2)$. We require that when $L\_x$ is replaced by the classical ${\mathbb R}^d$-Laplacian $Δ\_{x}$, i.e., in the limit local case $α=2$, the operator$ Δ\_{x} + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} $ satisfy a weak type Hörmander condition with invariance by suitable dilations. {Such} estimates were only known for $α=2$. This is one of the first results on $L^p $ estimates for degenerate non-local operators under Hörmander type conditions. We complete our result on $L^p$-regularity for $ L\_{x } + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} $ by proving estimates like\begin{equation*} \|Δ\_{y\_i}^{\frac {α\_i} {2}} v \|\_{L^p({\mathbb R}^N)} \lec\_p \Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)},\end{equation*}involving fractional Laplacians in the degenerate directions $y\_i$ (here $α\_i \in (0, { {1\wedge α}})$ depends on $α$ and on the numbers of commutators needed to obtain the $y\_i$-direction). The last estimates are new even in the local limit case $α=2$ which is also considered.
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Even…
We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.
We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins (From Probability to Geometry, vol. in honor of J.M Bismut (2009)) to prove uniqueness of the martingale problem in the framework of non degenerated elliptic operators and the Mc Kean and Singer (Journal of Diff. Geometry, 1967) parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi, Journal of Funct. Analysis,…
We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD 2) + Az, D , where A, B are N x N matrices, z $\in$ R N , N $\ge$ 1, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of L, namely for L + Tr(S(t)D 2) where S(t) is a non-negative N x N matrix depending continuously…
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
We prove the local boundedness of the solutions to degenerate second order partial differential equations of Kolmogorov type with measurable coefficients in divergence form, under minimal integrability assumption on the lower order coefficients.
The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation L u = f with measurable coefficients in the dilation invariant case. Moreover, we prove Gaussian upper and lower bounds for it, and other related properties.
We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.
We consider a class of possibly degenerate second order elliptic operators $\cal A$ on $\R^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving ${\cal A}$. The Hölder function spaces are defined with respect to a non-euclidean metric related to the operator $\cal A$.
We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5 of [9] for the optimal spatial regularity.
In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previous…
The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathscr{L}u = f$, that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combi…
Let $P(D)$ be the differential operator induced by a polynomial $P$, and let ${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that $\|P(D)(f)\|_2\leq 1$. The problem of computing the asymptotic order of the Kolmogorov $n$-width $d_n({U^{[P]}_2},L_2)$ in the general case when ${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when $P(D)$ is non-degenerate.
We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.
We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.
We obtain the basic results concerning the problem of constructing operator realizations of the formal differential expression $\nabla \cdot a \cdot \nabla - b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ having critical-order singularities as the generators of Markov semigroups in $L^p$ and $C_\infty$.
The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second order partial differential equations with a Kolmogorov-type operator of the form $$\mathscr{L} :=\sum_{i,j=1}^m \partial^2_{x_i x_j } +\sum_{i,j=1}^N b_{ij}x_j\partial_{x_i}-\partial_t, $$ where $(x,t) \in \mathbb{R}^{N+1}$, $1 \leq m \le N$ and the matrix $B:=(b_{ij})_{i,j=1,\ldots,N}$ has real constant entries. In particular, we show that if the modulus of $L^p$-mean oscillation of $\mathscr{L} u$ at the origin is Dini, then the origin is a Lebesgue point of continuity in $L^p$ average f…
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle.
We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] in unbounded domains of the form \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,y_m,t)\}, \] where $ψ$ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we assume that $A$ is bounded, measurable, symme…
We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,t)\}. \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $ψ$ and $Ω$ we assume that $Ω$ is what we call an (unbounded) Lipschitz domain: $ψ$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euc…
We study the regularity properties of the second order linear operator in $\mathbb{R}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} where $A = \left( a_{jk} \right)_{j,k= 1, \dots, m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies Hörmander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second ord…
We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at "$t=- \infty$" for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by …
We analyze a class of partial differential equations that arise as "backwards Kolmogorov operators" in infinite population limits of the Wright-Fisher models in population genetics and in mathematical finance. These are degenerate elliptic operators defined on manifolds with corners. The classical example is the Kimura diffusion operator, which acts on functions defined on the simplex in R^n. We introduce anisotropic Holder spaces, and prove existence, uniqueness and regularity results for the heat and resolvent equations defined by this class of operators. This suffices to prove that the C^0-…
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can't be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type te…
We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations
Given a real and separable Hilbert space H we consider the measure-valued equation \begin{equation*} \int_Hϕ(x)μ_t(dx)- \int_Hϕ(x)μ(dx)= \int_0^t(\int_HK_0ϕ(x)μ_s(dx))ds, \end{equation*} where K_0 is the Kolmogorov differential operator \[ K_0ϕ(x)=\frac12\textrm{Trace}\big[BB^*D^2ϕ(x)\big]+< x,A^*Dϕ(x)>+< Dϕ(x),F(x)>, \] $x\in H$, $ϕ:H\to \Rset$ is a suitable smooth function, $A:D(A)\subset H\to H $ is linear, $F:H\to H$ is a globally Lipschitz function and $B:H\to H$ is linear and continuous. In order prove existence and uniqueness of a solution for the above equation, we show that $K_0$ is a…
In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.
We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a class of operators which satisfy our conditions.
In this paper we show that Markov uniqueness for symmetric pre-Dirichlet operators $L$ follows from the uniqueness of the corresponding Fokker-Planck-Kolmogorov equation (FPKE). Since in recent years a considerable number of uniqueness results for FPKE's have been achieved, we obtain new Markov uniqueness results in concrete cases. A selection of such will be presented in this paper. They include cases with killing and with degenerate diffusion coefficients.
The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.
We consider a semigroup of operators in the Banach space $C_b(H)$ of uniformly continuous and bounded functions on a separable Hilbert space $H$. In particular, we deal with semigroups that are related to solution of stochastic PDEs in $H$ and which are not, in general, strongly continuous. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to a large class of second order differential operators in $C_b(H)$.
We prove smoothing properties and optimal Schauder type estimates for a class of nonautonomous evolution equations driven by time dependent Ornstein-Uhlenbeck operators in a separable Hilbert space. They arise as Kolmogorov equations of linear nonautonomous stochastic differential equations with Gaussian noise.
We introduce a new class of strongly degenerate nonlinear parabolic PDEs $$((p-2)Δ_{\infty,X}^N+Δ_X)u(X,Y,t)+(m+p)(X\cdot\nabla_Yu(X,Y,t)-\partial_tu(X,Y,t))=0,$$ $(X,Y,t)\in\mathbb R^m\times \mathbb R^m\times \mathbb R$, $p\in (1,\infty)$, combining the classical PDE of Kolmogorov and the normalized $p$-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games introduced approximate solutions to the stated PDE when the parameter that controls …
We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
We consider a Kolmogorov-Fokker-Planck operator of the kind studied by Lanconelli-Polidoro in [Rend. Sem. Mat. Univ. Politec. Torino 52 (1994)], where the leading coefficients $a_{ij}$, instead of being constant, are bounded measurable functions of t. We construct an explicit fundamental solution for this operator, study its property, show a comparison result between this function and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem under various assumptions on the initial datum.
In this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case.
In this paper we survey some recent results concerned with nonautonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator
In this paper we consider nonautonomous elliptic operators ${\mathcal A}$ with nontrivial potential term defined in $I\times\mathbb R^d$, where $I$ is a right-halfline (possibly $I=\mathbb R$). We prove that we can associate an evolution operator $(G(t,s))$ with ${\mathcal A}$ in the space of all bounded and continuous functions on $\mathbb R^d$. We also study the compactness properties of the operator $G(t,s)$. Finally, we provide sufficient conditions guaranteeing that each operator $G(t,s)$ preserves the usual $L^p$-spaces and $C_0(\mathbb R^d)$.
This work is devoted to the study of the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problem…
In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then…
We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $ν$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(ν^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(ν^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide th…
We consider a class of ultra-parabolic Kolmogorov-type operators satisfying the Hormander's condition. We prove an intrinsic Taylor formula with global and local bounds for the remainder given in terms of the norm in the homogeneous Lie group naturally associated to the differential operator.
Consider the following Kolmogorov type hypoelliptic operator $$ \mathscr L_t:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+{\rm Tr} (a_t \cdot\nabla^2_{x_n}), $$ where $n\geq 2$, $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{\mathcal T_{s,t}; t\geq s\}$ be the time-dependent semigroup associated with $\mathscr L_t$; that is, $\partial_s {\mathcal T}_{s, t} f = - {\mathscr L}_s {\mathcal T}_{s, t}f$. For any $p\in(1,\infty)$, we show that there is a constant $C=C(p,n,d)>…
In this paper we consider the symmetric Kolmogorov operator $L=Δ+\frac{\nabla μ}μ\cdot \nabla$ on $L^2(\mathbb R^N,dμ)$, where $μ$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $μ$ we prove first weighted Rellich's inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,dμ)$ and $H^4(\mathbb R^N,dμ)$ respectively, generate analytic semigroups of contractions on $L^2(\mathbb R^N,dμ)$. We observe that $dμ$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we desc…
We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space $E$ of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identité du carré di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space $W^{1,2}(E;μ)$, where $μ$ is an invariant measure for the system, and we prove the validity of the Poinca…
We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $Γ$-calculus techniques. The advantages and drawbacks of each of these methods are discussed.
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
