arXiv research

Unified kernel framework extends to stochastic systems, improving numerical stability.

In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we consider the case in which the components are independently but not identically distri…

A new method uses a product of experts with Dirichlet variables to approximate complex distributions.

The article constructs Feynman propagators for normally hyperbolic operators on curved spacetimes.

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

Holomorphic analogs of Feynman integrals are shown to be finite.

We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valu…

Three theorems about arbitrage bubbles in financial equations.

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…

The paper generalizes Feynman-Kac formula for volatility uncertainty.

New presentation of Goussarov-Habiro Lie algebra using primitive Feynman diagrams.

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…

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekin…

This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a finite dimensional case (which actually belongs more to multivariable calculus than …

New methods solve SPDEs for financial derivative pricing.

FKEE estimates expectations without samples, using diffusion bridges and PINNs.

New method recovers BSDE from financial data without ergodicity.

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 $L^2$ harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…

The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.

Functional-analytic method for stochastic parallel transport in bundles.

New method trains partial Bayesian neural networks efficiently.

Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in n-dimensional Euclidean space, for n greater than or equal to 4. At the end of their paper they conjecture that when n is odd, the terms on the antidiagonal on the second page precisely give the space …

New method steers protein design towards desired properties.

Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for computing multivariate Gaussian integrals. We apply our method to study training dyn…

In this paper, we aim to understand Residual Network (ResNet) in a scientifically sound way by providing a bridge between ResNet and Feynman path integral. In particular, we prove that the effect of residual block is equivalent to partial differential equation, and the ResNet transforming process can be equivalently co…

Study analyzes derivative-free loss method for solving PDEs and fluid problems.

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 …

In a previous paper [\AS], we used superspace techniques to prove that perturbation theory (around a classical solution with no zero modes) for Chern--Simons quantum field theory on a general $3$-manifold $M$ is finite. We conjectured (and proved for the case of $2$-loops) that, after adding counterterms of the expecte…

Improved diffusion models using energy distillation and sequential Monte Carlo.

Geometric approach to Dirac operator evolution on spacetimes.

Study on PDEs in Heston model with unique solution and convergence proof.

Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of Deligne-Mumford-Knudsen moduli spaces $\bar{M}_{g,n}$ of stable pointed algebraic curves; hence the…

We develop a new model for VIX derivatives with closed-form solutions.

Study an index theorem on manifolds with S^1 action using heat kernels and orbifolds.

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in "Grope cobordism of classical knots." We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the ``class'' is used to organize gropes. This implies that the grope cobordism equ…

We prove existence, regularity and a Feynman-Kač representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.

Study heat profiles and eigenfunctions using Brownian motion.

Study of mean curvature flows with conical singularities using mathematical techniques.

A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.

The paper develops methods to price and hedge options in path-dependent stock models.

Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

Stochastic delay differential equations (SDDE's) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fix…

At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process…

Formalizes quantum path integrals using groupoids and differential forms.

A new method integrates forms on Riemann surfaces, leading to modular forms.

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and struc…

Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.