Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). We derive the upper and lower bounds of the heat kernel on (X, d, µ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.
New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry-Émery Ricci curvature, the Escober-Lichnerowicz-Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry-Émery Ricci curvature and the m-Bakry-Émery Ricci curvature bounded from below by a non-positive constant, the Li-Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results. Mathematics Subject Classification (2010). Primary 58J05, 58J50; Secondary 35J92.
The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers [7,21]. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved. Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-2 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-4 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-12 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only.
The dimension-free Harnack inequality for the heat semigroup is established on the RCD(K , ∞) space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott-Sturm-Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strongenough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.
We generalize the results of Ambrosio [1] on the existence, uniqueness, and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. We then construct an explicit solution to the corresponding stochastic transport equation in terms of the stochastic flow. The approximate differentiability of the flow is also studied when the drift is a Sobolev vector field.
In this paper we present a unified treatment for the ordinary differential equations under the Osgood and Sobolev type conditions, following Crippa and de Lellis's direct method. More precisely, we prove the existence, uniqueness and regularity of the DiPerna-Lions flow generated by a vector field which is "almost everywhere Osgood continuous".
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