Search ResultsShowing 1-20 of 53
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arXiv:2505.10175 (Published 2025-05-15)
From Combinatorics to Partial Differential Equations
Comments: Comments very welcome!Categories: math.PRThe optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a crucial dependance on dimension $d$, with $d=2$ being the critical dimension. This is revealed by adopting an analytical perspective, connecting e.\,g.~to Optimal Transportation. These short notes provide an introduction to the subject. The material presented here is based on a series of lectures held at the International Max Planck Research School during the summer semester 2022. Recordings of the lectures are available at https://www.mis.mpg.de/events/event/imprs-ringvorlesung-summer-semester-2022.
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arXiv:2410.18431 (Published 2024-10-24)
RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-Itô integrals
Categories: math.PRFractional Brownian motions(fBMs) are not semimartingales so the classical theory of It\^o integral can't apply to fBMs. Wick integration as one of the applications of Malliavin calculus to stochastic analysis is a fine definition for fBMs. We consider the fractional forward backward stochastic differential equations(fFBSDEs) driven by a fBM that have the Hurst parameter in (1/2,1) where $\int_{0}^{t} f_s \, dB_s^H$ is in the sense of a Wick integral, and relate our fFBSDEs to the system of partial differential equations by using an analogue of the It\^o formula for Wick integrals. And we develop a deep learning algorithm referred to as the RNN-BSDE method based on recurrent neural networks which is exactly designed for solving high-dimensional fractional BSDEs and their corresponding partial differential equations.
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arXiv:2410.12364 (Published 2024-10-16)
An informal introduction to the Parisi formula
Comments: This is an expanded version in English of an introductory article written in French for "La gazette de la soci\'et\'e math\'ematique de France". 19 pagesThis note is an informal presentation of spin glasses and of the Parisi formula. We also discuss some models for which the Parisi formula is not well-understood, and some partial progress that relies upon a connection with partial differential equations.
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arXiv:2409.07044 (Published 2024-09-11)
Tempered space-time fractional negative binomial process
Comments: 11 pagesCategories: math.PRIn this paper, we define a tempered space-time fractional negative binomial process (TSTFNBP) by subordinating the fractional Poisson process with an independent tempered Mittag-Leffler L\'{e}vy subordinator. We study its distributional properties and its connection to partial differential equations. We derive the asymptotic behavior of its fractional order moments and long-range dependence property. It is shown that the TSTFNBP exhibits overdispersion. We also obtain some results related to the first-passage time.
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arXiv:2407.07873 (Published 2024-07-10)
Dynamical Measure Transport and Neural PDE Solvers for Sampling
Jingtong Sun, Julius Berner, Lorenz Richter, Marius Zeinhofer, Johannes Müller, Kamyar Azizzadenesheli, Anima AnandkumarThe task of sampling from a probability density can be approached as transporting a tractable density function to the target, known as dynamical measure transport. In this work, we tackle it through a principled unified framework using deterministic or stochastic evolutions described by partial differential equations (PDEs). This framework incorporates prior trajectory-based sampling methods, such as diffusion models or Schr\"odinger bridges, without relying on the concept of time-reversals. Moreover, it allows us to propose novel numerical methods for solving the transport task and thus sampling from complicated targets without the need for the normalization constant or data samples. We employ physics-informed neural networks (PINNs) to approximate the respective PDE solutions, implying both conceptional and computational advantages. In particular, PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently, leading to significantly better mode coverage in the sampling task compared to alternative methods. Moreover, they can readily be fine-tuned with Gauss-Newton methods to achieve high accuracy in sampling.
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arXiv:2405.11510 (Published 2024-05-19)
On some open problems in reliability theory
We study a stochastic scheduling on an unreliable machine with general up-times and general set-up times which is described by a group of partial differential equations with Dirac-delta functions in the boundary and initial conditions. In special case that the random processing rate of job $i,$ the random up-time rate of job $i$ and the random repair rate of job $i$ are constants, we determine the explicit expression of its time-dependent solution and give the asymptotic behavior of its time-dependent solution. Our result implies that $C_0-$semigroup theory is not suitable for this model. In general case, we determine the Laplace transform of its time-dependent solution. Next, we convert the model into an abstract Cauchy problem whose underlying operator is an evolution family. Finally, we leave some open problems.
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arXiv:2404.07402 (Published 2024-04-11)
Schrödinger's bridges with stopping: Steering of stochastic flows towards spatio-temporal marginals
Comments: 6 pages, 2 figuresThe purpose of the present work is to expand substantially the type of control and estimation problems that can be addressed following the paradigm of Schr\"odinger bridges, by incorporating stopping (freezing) of a given stochastic flow. Specifically, in the context of estimation, we seek the most likely evolution realizing prescribed spatio-temporal marginals of stopped particles. In the context of control, we seek the control action directing the stochastic flow toward spatio-temporal probabilistic constraints. To this end, we derive a new Schr\"odinger system of coupled, in space and time, partial differential equations to construct the solution of the proposed problem. Further, we show that a Fortet-Sinkhorn type of algorithm is, once again, available to attain the associated bridge. A key feature of the framework is that the obtained bridge retains the Markovian structure in the prior process, and thereby, the corresponding control action takes the form of state feedback.
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arXiv:2312.14882 (Published 2023-12-22)
Sampling and estimation on manifolds using the Langevin diffusion
Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $d\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $\mu_\phi $ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $\phi$, first-order error bounds, in discretization step size, on the bias and variances of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $\mu_\phi$ and a stationary measure of the discretized Markov process. Generality of the proof techniques, which exploit links between two partial differential equations and the semigroup of operators corresponding to the Langevin diffusion, renders them amenable for the study of a more general class of sampling algorithms related to the Langevin diffusion. Conditions for extending analysis to the case of non-compact manifolds are discussed. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm.
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arXiv:2304.12101 (Published 2023-04-24)
Deviations of the intersection of Brownian Motions in dimension four with general kernel
Categories: math.PRIn this paper, we find a natural four dimensional analog of the moderate deviation results of Chen (2004) for the mutual intersection of two independent Brownian motions $B$ and $B'$. In this work, we focus on understanding the following quantity, for a specific family of kernels $H$, \begin{equation*} \int_0^1 \int_0^1 H (B_s - B'_t) \text{d}t \text{d}s . \end{equation*} Given $H(z) \propto \frac{1}{|z|^{\gamma}}$ with $0 < \gamma \le 2$, we find that the deviation statistics of the above quantity can be related to the following family of inequalities from analysis, \begin{equation} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{(1-\gamma/4)}_{L^2} \|\nabla f\|^{\gamma/4}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) H(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation} Furthermore, in the case that $H$ is the Green's function, the above will correspond to the generalized Gagliardo-Nirenberg inequality; this is used to analyze the Hartree equation in the field of partial differential equations. Thus, in this paper, we find a new and deep link between the statistics of the Brownian motion and a family of relevant inequalities in analysis.
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arXiv:2301.09996 (Published 2023-01-24)
Black-Scholes without stochastics or PDEs
We show how to derive the Black-Scholes model and its generalisation to the `exchange-option' (to exchange one asset for another) via the continuum limit of the Binomial tree. No knowledge of stochastic calculus or partial differential equations is assumed, as we do not use them.
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arXiv:2212.11884 (Published 2022-12-22)
The central limit theorem via doubling of variables
Comments: 15 pagesCategories: math.PRWe give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our technique also yields quantitative bounds for random variables with finite $2+\gamma$ moment for some $\gamma \in (0,1]$; when $\gamma=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$.
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arXiv:2212.05091 (Published 2022-12-09)
Analysis of some exactly solvable diminishing urn models
Comments: 11 pages, 2 figures; appeared in the Proceedings of the Formal Power Series and Algebraic Combinatorics (FPSAC 2007) Nankai University, Tianjin, China, 2007 (only printed proceedings). arXiv admin note: text overlap with arXiv:1110.2425We study several exactly solvable Polya-Eggenberger urn models with a \emph{diminishing} character, namely, balls of a specified color, say $x$ are completely drawn after a finite number of draws. The main quantity of interest here is the number of balls left when balls of color $x$ are completely removed. We consider several diminishing urns studied previously in the literature such as the pills problem, the cannibal urns and the OK Corral problem, and derive exact and limiting distributions. Our approach is based on solving recurrences via generating functions and partial differential equations.
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arXiv:2211.00719 (Published 2022-11-01)
A finite-dimensional approximation for partial differential equations on Wasserstein space
Categories: math.PRThis paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme to our context, as well as on a key precompactness result for semimartingale measures. We illustrate this result with the example of the Hamilton-Jacobi-Bellman and Bellman-Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.
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arXiv:2208.09732 (Published 2022-08-20)
Notes on tug-of-war games and the p-Laplace equation
The objective is the interplay between stochastic processes and partial differential equations. To be more precise, we focus on the connection between the nonlinear p-Laplace equation, and the stochastic game called tug-of-war with noise. The connection in this context was discovered roughly 15 years ago, and has provided novel insight and approaches ever since. These lecture notes provide a short introduction to the topic and to more research oriented literature. We also introduce the parabolic case side by side with the elliptic one, and cover some parts of the regularity theory.
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arXiv:2206.08334 (Published 2022-06-16)
On quasilinear parabolic systems and FBSDEs of quadratic growth
Subjects: 60H30Using probabilistic methods, we establish a-priori estimates for two classes of quasilinear parabolic systems of partial differential equations (PDEs). We treat in particular the case of a nonlinearity which has quadratic growth in the gradient of the unknown. As a result of our estimates, we obtain the existence of classical solutions of the PDE system. From this, we infer the existence of solutions to a corresponding class of forward-backward stochastic differential equations.
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arXiv:2109.05282 (Published 2021-09-11)
Well-posedness of path-dependent semilinear parabolic master equations
Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in games and control theory. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent master equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in stochastic control theory and games. We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupire's vertical derivative, and applying stochastic forward-backward system argument. Moreover, we consider a general non-smooth case with a functional mollifying method.
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arXiv:2108.05879 (Published 2021-08-12)
Feature Engineering with Regularity Structures
Comments: 21 pages, 5 figures, 2 tablesWe investigate the use of models from the theory of regularity structure as features in machine learning tasks. A model is a multi-linear function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs), even in low regularity regimes. Models can be seen as natural multi-dimensional generalisations of signatures of paths; our work therefore aims to extend the recent use of signatures in data science beyond the context of time-ordered data. We provide a flexible definition of a model feature vector associated to a space-time signal, along with two algorithms which illustrate ways in which these features can be combined with linear regression. We apply these algorithms in several numerical experiments designed to learn solutions to PDEs with a given forcing and boundary data. Our experiments include semi-linear parabolic and wave equations with forcing, and Burgers' equation with no forcing. We find an advantage in favour of our algorithms when compared to several alternative methods. Additionally, in the experiment with Burgers' equation, we noticed stability in the prediction power when noise is added to the observations.
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arXiv:2104.04372 (Published 2021-04-09)
Entropic regularisation of non-gradient systems
Comments: 31 pagesThe theory of (Wasserstein) gradient flows in the space of probability measures has made enormous progress over the last twenty years. Nonetheless, many partial differential equations (PDEs) of interest do not have gradient flow structure and, a priori, the theory is not applicable. In this paper, we develop a time-discrete entropic regularised variational scheme for a general class of such non-gradient PDEs. We prove the convergence of the scheme and illustrate the breadth of the proposed framework with concrete examples including the non-linear kinetic Fokker-Planck (Kramers) equation and a nonlinear degenerate diffusion of Kolmogorov type.
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arXiv:2102.09611 (Published 2021-02-18)
Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations
Comments: 26 pagesIn this work we recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov-Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.
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arXiv:2007.02174 (Published 2020-07-04)
1--Meixner random vectors
Comments: 45 pagesCategories: math.PRA definition of $d$--dimensional $n$--Meixner random vectors is given first. This definition involves the commutators of their semi--quantum operators. After that we will focus on the $1$-Meixner random vectors, and derive a system of $d$ partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete characterization of all non--degenerate three--dimensional $1$--Meixner random vectors. It must be mentioned that the three--dimensional case produces the first example in which the components of a $1$--Meixner random vector cannot be reduced, via an injective linear transformation, to three independent classic Meixner random variables.