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Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces Thompson, James ; Thalmaier, Anton in Bernoulli (2020) In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian ... [more ▼] In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of RCD*(K,N) spaces. In each case the necessary ingredients are an Ito formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrodinger operators. [less ▲] Detailed reference viewed: 132 (37 UL)Derivative and divergence formulae for diffusion semigroups Thalmaier, Anton ; Thompson, James in Annals of Probability (2019), 47(2), 743-773 Detailed reference viewed: 425 (103 UL)Functional inequalities for Feynman-Kac semigroups Thompson, James in Journal of Theoretical Probability (2019) Using stochastic analysis, we prove various gradient estimates and Harnack inequalities for Feynman-Kac semigroups with possibly unbounded potentials. One of the main results is a derivative formula which ... [more ▼] Using stochastic analysis, we prove various gradient estimates and Harnack inequalities for Feynman-Kac semigroups with possibly unbounded potentials. One of the main results is a derivative formula which can be used to characterize a lower bound on Ricci curvature using a potential. [less ▲] Detailed reference viewed: 121 (29 UL)Uniform gradient estimates on manifolds with a boundary and applications Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Analysis and Mathematical Physics (2018), 8(4), 571-588 We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using ... [more ▼] We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform quantitative gradient estimates, firstly for bounded C^2 functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians. [less ▲] Detailed reference viewed: 215 (64 UL)Functional inequalities on manifolds with non-convex boundary Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Science China Mathematics (2018), 61(8), 1421-1436 In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non ... [more ▼] In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary. [less ▲] Detailed reference viewed: 221 (40 UL)Quantitative C1-estimates by Bismut formulae Cheng, Li Juan ; Thalmaier, Anton ; Thompson, James in Journal of Mathematical Analysis and Applications (2018), 465(2), 803-813 For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a ... [more ▼] For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δu. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings. [less ▲] Detailed reference viewed: 299 (61 UL)Derivatives of Feynman-Kac Semigroups Thompson, James in Journal of Theoretical Probability (2018) We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of ... [more ▼] We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, although the assumptions are purely geometric. [less ▲] Detailed reference viewed: 197 (62 UL)First Order Feynman-Kac Formula ; Thompson, James in Stochastic Processes & Their Applications (2018) We study the parabolic integral kernel associated with the weighted Laplacian and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and estimates for them and for their derivatives ... [more ▼] We study the parabolic integral kernel associated with the weighted Laplacian and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and estimates for them and for their derivatives, given in terms of a Gaussian term and the semi-classical bridge. Assumptions are on the Riemannian data. [less ▲] Detailed reference viewed: 169 (33 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: 177 (41 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: 180 (27 UL) |
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