Estimate for $P_{t}D$ for the stochastic Burgers equation

We prove gradient estimates for transition Markov semigroups (P t ) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C 1 -coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for DTo prove the result we use two main tools. First, we consider a Feynman-Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut-Elworthy-Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for D x P t ϕ without weights do not hold.2010 Mathematics Subject Classification AMS: 60H10, 60H30, 47D07.

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“…We prove weighted gradient estimates inspired by [DaDe03], [DaDe07] and [DaDe14], introducing a suitable potential V related to Hypothesis 1.1:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gradient estimates for SDEs without monotonicity type conditions

Prato

Priola

2018

Journal of Differential Equations

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“…In the recent paper [DaDe14] the following inequality involving the invariant measure ν of the Burgers equation was proved, H (−A) −α/2 Dϕ, z dν ≤ C p ϕ L p (H,ν) |z|, α > 1, (1.1) for all ϕ ∈ C 1 b (H), z ∈ H and all p > 1. Here A is the Laplace operator equipped with Dirichlet boundary conditions, α > 1 and D represents the gradient.…”

Section: Introductionmentioning

confidence: 99%

An integral inequality for the invariant measure of a stochastic reaction–diffusion equation

Prato

Debussche

2016

J. Evol. Equ.

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We consider a reaction-diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure ν of a stochastic reactiondiffusion equation. Then we discuss some consequences as an integration by parts formula which extends to ν a basic identity of the Malliavin Calculus. Finally, we prove the existence of a surface measure for a ball and a half-space of H.

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“…In the recent paper [DaDe14] the following inequality involving the invariant measure ν of the Burgers equation was proved H RDϕ, z dν ≤ C p ϕ L p (H,ν) |z|,…”

Section: Introductionmentioning

confidence: 99%

“…These estimates require some work because, due to the polynomial growth of the derivative of b, see (1.5), we cannot exploit the classical Bismut-Elworthy-Li formula, see [El92]. To overcome this problem we shall argue as in [DaDe03], [DaDe07] and [DaDe14], introducing a suitable potential (in the present case V (x) = K(1 + |x| 2N )) and the Feynman-Kac semigroup…”

Section: Introductionmentioning

confidence: 99%