Hamilton Type Gradient Estimate for the Sub-Elliptic Operators

Qian, Bin

doi:10.1007/s11118-014-9448-z

Cited by 2 publications

(2 citation statements)

References 20 publications

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“…Then Baudoin and Garofalo introduced in [9] a property, called "a generalized curvaturedimension inequality", which has to be thought of as a lower bound on a sub-Riemannian generalization of the Ricci tensor. Under such a condition, the heat kernel generated by L satisfies (UE) as well as (Lip); see [54]. We refer the reader to [9] for some examples of such sub-elliptic settings and the fact that the heat kernel also satisfies in that case some Gaussian lower bound.…”

Section: Functional Calculus Adapted To the Heat Semigroupmentioning

confidence: 99%

Heat semigroup and singular PDEs

Bailleul

Bernicot

2016

Journal of Functional Analysis

222

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We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character in Hölders spaces, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a weighted noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev scales of spaces rather than Hölder spaces. Contents 1 2 B.4 Resolution of PAM in such a 2-dimensional setting 79 of paracontrolled calculus described in a second. Recall that if M and N are two continuous martingales one has d(M N ) = M dN + N dM + d M, N .The above space of functions u = Π v (X) + u ♯ , can be turned into a Banach space. Once this ansatz for the solution space has been chosen, remark that the product u ξ can formally be written asSince u ♯ has Hölder regularity strictly bigger than 1 and ξ is −1 − -regular, the sum of their regularity indices is positive, and the term Π(u ♯ , ξ) is perfectly well-defined. This lives us with Π Π v (X), ξ as the only undefined term. The following fact is the workhorse of paracontrolled calculus. The trilinear maphappens to depend continuously on a, b and c provided they are Hölder distributions, with the sum of their Hölder exponents positive. Note the paralell between the continuity of this 'commutator' and the rule for stochastic differentials, for which, given another continuous martingale P , we haveThe formal product u ξ can thus be written as a sum of well-defined terms plus the formal product v Π(X, ξ), with a diagonal term Π(X, ξ) still undefined on a purely analytic basis. This is where probability comes into play. If one regularizes ξ into ξ ε , with X ε defined accordingly, one can prove that there exists a function/constant C ε such that the renormalized quantity ξ (2),ε := Π X ε , ξ ε − C ε converges in probability to some limit distribution ξ (2) of Hölder regularity 0 − = 1 − + (−1 − ); this is enough to make sense of the product v ξ (2) on an analytical basis; but replacing Π X ε , ξ ε by Π X ε , ξ ε − C ε in the decomposition of u ξ ε amounts to looking at the product u ξ ε − C ε . The enhancement ξ := ξ, ξ (2) of ξ is called a rough, or enhanced, distribution, and one can use it to define the product u ξ from the above formulas. At that point, it does not come as a surprise that one can then set (PAM) equation as a fixed point problem in the ansatz...

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Section: Functional Calculus Adapted To the Heat Semigroupmentioning

confidence: 99%

Heat semigroup and singular PDEs

Bailleul

Bernicot

2016

Journal of Functional Analysis

222

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“…Functional inequalities such as Poincaré inequality, Log-Sobolev inequality; · · · · · · , which generalizes the works of S. T. Yau, D. Bakry, M. Ledoux etc., see [17,1,15] and references therein. See also [11,23,24,26] for recent works on gradient estimate and curvature property for subelliptic operators.…”

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confidence: 99%

Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

Qian

2017

Nonlinear Analysis

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In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities (Theorem 2.1) for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality (Theorem 4.1). Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity (Theorem 5.3).

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Hamilton Type Gradient Estimate for the Sub-Elliptic Operators

Cited by 2 publications

References 20 publications

Heat semigroup and singular PDEs

Heat semigroup and singular PDEs

Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

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