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Multiple Sets Exponential Concentration and Higher Order Eigenvalues ; Herry, Ronan in Potential Analysis (in press) On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the ... [more ▼] On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung, Grigor’yan & Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117(2), 165–178 (1996). [less ▲] Detailed reference viewed: 49 (4 UL)Stochastic completeness and gradient representations for sub-Riemannian manifolds ; Thalmaier, Anton in Potential Analysis (2019), 51(2), 219-254 Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a ... [more ▼] Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of L/2, while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry. [less ▲] Detailed reference viewed: 210 (50 UL)Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow Li, Yi ; in Potential Analysis (2018) Detailed reference viewed: 120 (12 UL)Brownian bridges to submanifolds Thompson, James in Potential Analysis (2017) We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We ... [more ▼] We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold. [less ▲] Detailed reference viewed: 136 (37 UL)Algebraic Convergence Rate for Reflecting Diffusion Processes on Manifolds with Boundary Cheng, Li Juan ; in Potential Analysis (2016), 44(1), 91-107 A criteria for the algebraic convergence rate of diffusion semigroups on manifolds with respect to some Lipschitz norms in L2-sense is presented by using a Lyapunov condition. As application, we apply it ... [more ▼] A criteria for the algebraic convergence rate of diffusion semigroups on manifolds with respect to some Lipschitz norms in L2-sense is presented by using a Lyapunov condition. As application, we apply it to some diffusion processes with heavy tailed invariant distributions. This result is further extended to the reflecting diffusion processes on manifolds with non-convex boundary by using a conformal change of the metric. [less ▲] Detailed reference viewed: 80 (8 UL)Brownian motion and the distance to a submanifold Thompson, James in Potential Analysis (2016) We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance ... [more ▼] We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. We derive the concentration inequality using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficiently sharp to imply a comparison theorem. We provide numerous examples throughout. Further applications will feature in a subsequent article, where we see how the main results and methods presented here can be applied to certain study objects which appear naturally in the theory of submanifold bridge processes. [less ▲] Detailed reference viewed: 146 (26 UL)An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Potential Analysis (2015), 42(2), 483-497 We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is ... [more ▼] We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained. [less ▲] Detailed reference viewed: 353 (38 UL)Contraction of Measures on Graphs Hillion, Erwan in Potential Analysis (2014), 41 Given a finitely supported probability measure μ on a connected graph G, we construct a family of probability measures interpolating the Dirac measure at some given point o ∈ G and μ. Inspired by Sturm ... [more ▼] Given a finitely supported probability measure μ on a connected graph G, we construct a family of probability measures interpolating the Dirac measure at some given point o ∈ G and μ. Inspired by Sturm-Lott-Villani theory of Ricci curvature bounds on measured length spaces, we then study the convexity of the entropy functional along such interpolations. Explicit results are given in three canonical cases, when the graph G is either Z^n , a cube or a tree. [less ▲] Detailed reference viewed: 64 (3 UL)Density Estimates for Solutions to One Dimensional Backward SDE’s Bourguin, Solesne ; in Potential Analysis (2013), 38(2), 573-587 In this paper, we derive sufficient conditions for each component of the solution to a general backward stochastic differential equation to have a density for which upper and lower Gaussian estimates can ... [more ▼] In this paper, we derive sufficient conditions for each component of the solution to a general backward stochastic differential equation to have a density for which upper and lower Gaussian estimates can be obtained. [less ▲] Detailed reference viewed: 25 (0 UL)A Poincaré cone condition in the Poincaré group Tardif, Camille in Potential Analysis (2013), 38(3), 1001-1030 Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more ... [more ▼] Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more general context of a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincaré cone condition for a relativistic diffusion with values in the Poincaré group (i.e the group of affine direct isometries of the Minkowski space-time). [less ▲] Detailed reference viewed: 46 (1 UL)Stochastic particle approximations for the Ricci flow on surfaces and the Yamabe flow Philipowski, Robert in Potential Analysis (2011), 35 We present stochastic particle approximations for the normalized Ricci flow on surfaces and for the non-normalized Yamabe flow on manifolds of arbitrary dimension. Detailed reference viewed: 66 (15 UL)Derivative estimates of semigroups and Riesz transforms on vector bundles Thalmaier, Anton ; in Potential Analysis (2004), 20(2), 105-123 Detailed reference viewed: 139 (19 UL) |
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