Stochastic Analysis on Manifolds

By Y. Gliklikh. 2011

473 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/74f31c3e0310489d904064d4856b2ad4635fe349

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In Computational Theory and Mathematics

Citations in the corpus

Option pricing with linear market impact and nonlinear Black–Scholes equations (2018, unknown): “Since u is only locally smooth up to time T , we first need to introduce a notion of local solution to (34), (35), (36), defined up to a stopping time; see [19], Chapter 1.”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “In fact Ricci curvature also arises in Bismut type formula [30] and its applications to gradient estimates [10, 9], and to the construction of couplings between Brownian motions [32, 35].”
A class of globally solvable Markovian quadratic BSDE systems and applications (2016, methodology): “We refer the reader to the books [28] and [35] for more details.”
A survey of Ricci curvature for metric spaces and Markov chains (2010, methodology): “A probabilistic proof of the Lichnerowicz theorem using couplings of Brownian motions can be found, for example, in [Hsu02] (Theorem 6.”
One Point Isometric Matching with the Heat Kernel (2010, unknown): “The heat kernel has an alternative interpretation as the transition density function of Brownian motion on M (see e.g. [Hsu02] for details).”
SE(3) diffusion model with application to protein backbone generation (2023, methodology): “The forward process (R)t≥0 is simply the Brownian motion on SO(3), and pt|0 is defined by the heat kernel, see Hsu (2002). We obtain pt|0 analytically as a series as a special case of the decomposition of the heat kernel for compact Lie groups.”
Causal transport plans and their Monge–Kantorovich problems (2013, background): “A well known example of such isomorphisms is the Itô map of stochastic differential geometry (see [7], [17] or [3]).”
Riemannian Score-Based Generative Modeling (2022, methodology): “We deﬁne semimartingales and SDEs on manifold through the lens of their Since the previous SDE is deﬁned w.r.t the Stratanovitch integral we have that if , where Φ ? is the pushforward operation (see Hsu, 2002, Proposition 1.2.4).”
A spectral notion of Gromov–Wasserstein distance and related methods (2011, background): “For detailed exposition of the material presented here, we refer the reader to the excellent surveys [63–66].”
Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, background): “9 and the martingale problem on differentiable manifolds [24].”
The Signature of a Rough Path: Uniqueness (2014, methodology): “The function F (x) was used in [36] to construct SDEs on M driven by classical Brownian motion.”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “The estimates (i) and (iii) were already known in Riemannian geometry (see [33] and [43] respectively) while (ii) appears to be new even in the Riemannian context.”
Stochastic and geometric analysis of two infinite-dimensional groups (2010, unknown): “In general, to construct a Brownian motion on a Lie group, one might solve a Stratonovich stochastic differential equation (SDE) on such a group [20, 16].”
Probability Distribution in the SABR Model of Stochastic Volatility (2015, methodology): “This formula was originally derived by McKean [16] (see also [14] and references therein).”
Latent Space Non-Linear Statistics (2018, background): “milies of distributions in the geometric spaces, and the ability to evaluate likelihoods. One example of such a distribution is the transition distribution of the Riemannian Brownian motion, see e.g. [9]. In this section, we will show how likelihoods of data in the latent space Zunder this distribut”
Some constructions for the fractional Laplacian on noncompact manifolds (2012, background): “Rotationally symmetric manifolds are important in the sense that they serve as models for more general problems (see, for instance, [34] for the study of manifolds having a pole, [47] for Brownian motion and probability aspects).”