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, althou…
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48 results for “Feynman--Kac semigroups”
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.
problem Analyzing perturbations of order ≤ 1 in noncommutative geometry.
method Develops a Feynman-Kac formula for differential operators of order ≤ 1 on complex metric vector bundles over Riemannian manifolds.
result Explicit Feynman-Kac type formula for holomorphic semigroups generated by .
We extend the Feynman-Kac formula for Schrödinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then u…
Generates semigroups for differential expressions on Riemannian manifolds.
problem Analyzing differential expressions on Riemannian manifolds.
method Study of generalized Ornstein-Uhlenbeck differential expressions and their maximal realizations.
result Generates analytic quasi-contractive semigroups in weighted -spaces.
We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman…
We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitabl…
This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators…
Flat semigroups can represent normal weighted homogeneous surface singularities.
problem Representability of flat semigroups in normal weighted homogeneous surface singularities.
method Study of numerical semigroups associated with surface singularities and prove representability conditions.
result A numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
The paper generalizes Feynman-Kac formula for volatility uncertainty.
problem Calculating sublinear expectation under volatility uncertainty.
method Generalization of Feynman-Kac formula under different hypotheses.
result G-conditional expectation is a viscosity solution of a nonlinear PDE.
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
Concave Distortion Semigroupsq-fin.RM
The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a conca…
New methods solve SPDEs for financial derivative pricing.
problem Deriving the price of financial derivatives using SPDEs.
method Developed a conditional Feynman-Kac formula to solve SPDEs.
result Established new numerical methods for mixed Monte-Carlo PDEs.
This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…
The paper studies a semigroup generated by finite intervals and characterizes its properties.
problem Characterizing the semigroup generated by finite intervals.
method Analyzing the semigroup , showing Green relations coincide, isomorphic to partial convex order isomorphisms, and studying shift-continuous topologies.
result The semigroup is isomorphic to the semigroup of partial convex order isomorphisms and admits only Rees congruences.
FKEE estimates expectations without samples, using diffusion bridges and PINNs.
problem Estimating expectations without large sample sizes.
method Diffusion bridge models and Feynman-Kac operator approximation using PINNs.
result Significantly reduces variance and improves efficiency.
New method recovers BSDE from financial data without ergodicity.
problem Discovering probabilistic laws from financial data.
method Stochastic SINDy method under risk-neutral measure.
result Recovery of BSDE from limited financial data.
We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Functional-analytic method for stochastic parallel transport in bundles.
problem Stochastic parallel transport in Hermitian bundles over Riemannian manifolds.
method Purely functional-analytic construction.
result Obtained a general Feynman-Kac formula in vector bundles.
Intertwining curvature bounds for graphs and quantum Markov semigroups verified.
problem Intertwining curvature bounds for graphs and quantum Markov semigroups.
method Introducing and verifying curvature bounds in various examples.
result Improved entropic curvature bounds for depolarizing semigroups and qubits.
New method trains partial Bayesian neural networks efficiently.
problem Challenges in approximating multi-modal latent variable distributions in pBNNs.
method Formulates pBNN training as a Feynman--Kac model and uses sequential Monte Carlo samplers.
result Proposed training scheme outperforms state of the art in predictive performance.
The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.
problem Understanding the algebraic structure of numerical semigroups through topological representations.
method Associaing iterated torus knots to free numerical semigroups and analyzing their knot complements and Alexander polynomials.
result Alexander polynomials of knots associated with free numerical semigroups coincide with the semigroup's Poincaré series.
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely gene…
Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia setsmath.DS
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if and are two connected compo…
New method steers protein design towards desired properties.
problem Challenges in designing proteins with specific structures and properties.
method Feynman-Kac framework applied to RFdiffusion models with guiding potentials.
result Significant improvement in predicted interface energetics and binder designability.
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick…
Unified kernel framework extends to stochastic systems, improving numerical stability.
problem Extending kernel methods to stochastic dynamical systems with diffusion.
method Unified kernel framework, Feynman-Kac path-integral representations, collocation-based computational framework.
result Kernel equivalence under uniform ellipticity assumptions and improved numerical stability with moderate diffusion.
Graphs approximate semigroups for diffusion on Riemannian manifolds.
problem Approximating semigroups for diffusion on Riemannian manifolds.
method Discretized approximation using random walks on proximity graphs.
result Quantitative error estimates for convergence of discrete semigroups to continuous semigroups.
Study analyzes derivative-free loss method for solving PDEs and fluid problems.
problem Solving elliptic PDEs and fluid problems using neural networks.
method Derivative-free loss method with Feynman-Kac formulation and stochastic walkers.
result Training loss bias scales with time interval and spatial gradient, inversely with walker size.
Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the…
DM uses semigroup property to tune diffusion time for better data analysis.
problem Difficulty in tuning diffusion time for optimal data analysis.
method Proposes a semigroup criterion to select diffusion time.
result Effective and robust method for picking diffusion time.
Constructs free semigroups with critical exponents close to but less than ambient groups.
problem Creating free semigroups with critical exponents close to but less than ambient groups.
method Constructing finitely generated free subsemigroups with specific properties.
result Free semigroups with critical exponents arbitrarily close to but strictly less than ambient groups.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet …
Improved diffusion models using energy distillation and sequential Monte Carlo.
problem Training instability and inferior performance in energy parameterized diffusion models.
method Introduced a novel training regime for energy functions through distillation of pre-trained diffusion models, and cast the sampling procedure as a Feynman Kac model.
result Demonstrated improved performance and new sampling techniques.
Paper proposes deep learning for operators in semigroups, improving dynamical system modeling.
problem Modeling unknown autonomous dynamical systems using time series data at varying time lags.
method Novel deep learning approach embedding semigroup property into data-driven learning process.
result Framework reduces data dependency, improves accuracy, robustness, and stability for long-time prediction.
The aim of this paper is to show that the dynamics of heat semigroups () on a symmetric space of non-compact type is very different from the dynamics of the heat semigroups if . To see this, it is shown that certain shifts of the heat semigroups have a chaotic behavior if and that …
We look at the semigroup generated by a system of heat equations. Applications to testing normality and option pricing are addressed.
Paper analyzes a new Hopf-Lax semigroup in metric spaces.
problem Analyzing a new Hopf-Lax semigroup in metric spaces.
method Using continuous sections of quotient maps and variational problems.
result The 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation.
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
Paper improves robustness and sparsity in adversarially trained DNNs.
problem Developing efficient compression algorithms for robustly trained DNNs.
method Pruning weights using relaxed augmented Lagrangian algorithms for both structured and unstructured levels, leveraging Feynman-Kac formalism.
result At least doubles channel sparsity of adversarially trained ResNet20 for CIFAR10 classification.
Study on PDEs in Heston model with unique solution and convergence proof.
problem Analyzing PDEs in the Heston model for financial applications.
method Regularity results, verification theorem, unique viscosity solution, convergence proof.
result Unique viscosity solution for wide initial and source data.
Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…
Study estimates gaps in semigroup products, proving embedding properties.
problem Estimating singular value gaps in semigroup products.
method Lower estimates for singular value gaps of free products of semigroups in ping-pong position.
result Groups generated by semigroups in ping-pong position are quasi-isometrically embedded.
The space of -generator postcritically bounded polynomial semigroups and random complex dynamicsmath.DS
We investigate the dynamics of -generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space of such semigroups. We show that for a parameter in the intersection of , the hyperbolicity locus ${\c…