Search ResultsShowing 1-20 of 47
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arXiv:2410.19173 (Published 2024-10-24)
Probabilistic Representation of Commutative Quantum Circuit Models
In commuting parametric quantum circuits, the Fourier series of the pairwise fidelity can be expressed as the characteristic function of random variables. Furthermore, expressiveness can be cast as the recurrence probability of a random walk on a lattice. This construction has been successfully applied to the group composed only of Pauli-Z rotations, and we generalize this probabilistic strategy to any commuting set of Pauli operators. We utilize an efficient algorithm by van den Berg and Temme (2020) using the tableau representation of Pauli strings to yield a unitary from the Clifford group that, under conjugation, simultaneously diagonalizes our commuting set of Pauli rotations. Furthermore, we fully characterize the underlying distribution of the random walk using stabilizer states and their basis state representations. This would allow us to tractably compute the lattice volume and variance matrix used to express the frame potential. Together, this demonstrates a scalable strategy to calculate the expressiveness of parametric quantum models.
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arXiv:2409.04207 (Published 2024-09-06)
Probabilistic Representation for Viscosity Solutions to Double-Obstacle Quasi-Variational Inequalities
Comments: arXiv admin note: text overlap with arXiv:2402.17541, arXiv:2210.02417We prove the existence and uniqueness of viscosity solutions to quasi-variational inequalities (QVIs) with both upper and lower obstacles. In contrast to most previous works, we allow all involved coefficients to depend on the state variable and do not assume any type of monotonicity. It is well known that double obstacle QVIs are related to zero-sum games of impulse control, and our existence result is derived by considering a sequence of such games. Full generality is obtained by allowing one player in the game to randomize their control. A by-product of our result is that the corresponding zero-sum game has a value, which is a direct consequence of viscosity comparison. Utilizing recent results for backward stochastic differential equations (BSDEs), we find that the unique viscosity solution to our QVI is related to optimal stopping of BSDEs with constrained jumps and, in particular, to the corresponding non-linear Snell envelope. This gives a new probabilistic representation for double obstacle QVIs. It should be noted that we consider the min-max version (or equivalently the max-min version); however, the conditions under which solutions to the min-max and max-min versions coincide remain unknown and is a topic left for future work.
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arXiv:2407.09034 (Published 2024-07-12)
Numerical approximation of ergodic BSDEs using non linear Feynman-Kac formulas
In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantee the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions.
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arXiv:2404.05969 (Published 2024-04-09)
On the random generation of Butcher trees
We provide a probabilistic representation of the solutions of ordinary differential equations (ODEs) by random generation of Butcher trees. This approach complements and simplifies a recent probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees allows one to improve numerical accuracy by Monte Carlo iterations whereas the finite order truncation of Butcher series can have higher time complexity, as shown in examples.
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arXiv:2312.07096 (Published 2023-12-12)
Probabilistic representation of the gradient of a killed diffusion semigroup: The half-space case
We introduce a probabilistic representation of the derivative of the semigroup associated to a multidimensional killed diffusion process defined on the half-space. The semigroup derivative is expressed as a functional of a process that is normally reflected when it hits the hyperplane. The representation of the derivative also involves a matrix-valued process which replaces the Jacobian of the underlying process that appears in the traditional pathwise derivative of a classical diffusion. The components of this matrix-valued process become zero except for those on the first row every time the reflected process touches the boundary. The results in this paper extend those in recent work of the authors, where the one-dimensional case was studied.
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arXiv:2312.07084 (Published 2023-12-12)
A probabilistic representation of the derivative of a one dimensional killed diffusion semigroup and associated Bismut-Elworthy-Li formula
Categories: math.PRWe provide a probabilistic representation for the derivative of the semigroup corresponding to a diffusion process killed at the boundary of a half interval. In particular, we show that the derivative of the semi-group can be expressed as the expected value of a functional of a reflected diffusion process. Furthermore, as an application, we obtain a Bismut-Elworthy-Li formula which is also valid at the boundary.
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arXiv:2311.08564 (Published 2023-11-14)
Probabilistic Representations of Ordered Exponentials: Vector-Valued Schrödinger Semigroups and the Combinatorics of Anderson Localization
Comments: 54 pages, 10 figures; comments welcomeWe provide two applications of an elementary (yet seemingly unknown) probabilistic representation of matrix ordered exponentials, which generalizes the Feynman-Kac formula in finite dimensions and the change of measure formula between two continuous-time Markov processes on a finite state space. Our first and main application consists of a new Feynman-Kac formula for a class of vector-valued Schr\"odinger operators on the line, which is driven by two sources of randomness: The usual Brownian motion, and a continuous-time Markov process on a finite state space. An important feature of these formulas -- which is at the core of our motivation -- is that they enable the calculation of the joint moments of the semigroup kernels when the matrix potential function contains a continuous Gaussian noise. In particular, our moment formulas shed new light on what the joint moments of the Feynman-Kac kernels of the multivariate stochastic Airy operators of Bloemendal and Vir\'ag (Ann. Probab., 44(4):2726--2769, 2016.) should be; we state a precise conjecture to that effect, which we pursue in a forthcoming paper. Our second application consists of Feynman-Kac formulas for the expected square modulus $\mathbf E\big[|\Psi(t,x)|^2\big]$ of the solutions of the Schr\"odinger equation $\partial_t\Psi=-\mathsf i\mathcal H(t)\Psi$ with a time-dependent Hamiltonian $\mathcal H(t)$. Using this, we show that when we take $\mathcal H(t)=-\Delta+q(t,x)$ restricted to a finite box within $\mathbb Z^d$, where $q(t,x)$ is a possibly time-dependent Gaussian process, $\mathbf E\big[|\Psi(t,x)|^2\big]$ can be written as a relatively simple expectation that involves self- and mutual-intersections of random walks. In particular, this formula hints at a unified combinatorial mechanism that explains the occurrence of localization for both time-dependent and time-independent noises.
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arXiv:2306.10913 (Published 2023-06-19)
Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation
Comments: arXiv admin note: text overlap with arXiv:2110.09941, arXiv:2106.12127We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation based on $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.
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arXiv:2302.05104 (Published 2023-02-10)
Monte Carlo Neural Operator for Learning PDEs via Probabilistic Representation
Rui Zhang et al.Neural operators, which use deep neural networks to approximate the solution mappings of partial differential equation (PDE) systems, are emerging as a new paradigm for PDE simulation. The neural operators could be trained in supervised or unsupervised ways, i.e., by using the generated data or the PDE information. The unsupervised training approach is essential when data generation is costly or the data is less qualified (e.g., insufficient and noisy). However, its performance and efficiency have plenty of room for improvement. To this end, we design a new loss function based on the Feynman-Kac formula and call the developed neural operator Monte-Carlo Neural Operator (MCNO), which can allow larger temporal steps and efficiently handle fractional diffusion operators. Our analyses show that MCNO has advantages in handling complex spatial conditions and larger temporal steps compared with other unsupervised methods. Furthermore, MCNO is more robust with the perturbation raised by the numerical scheme and operator approximation. Numerical experiments on the diffusion equation and Navier-Stokes equation show significant accuracy improvement compared with other unsupervised baselines, especially for the vibrated initial condition and long-time simulation settings.
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arXiv:2210.02417 (Published 2022-10-05)
Probabilistic Representation of Viscosity Solutions to Quasi-Variational Inequalities with Non-Local Drivers
Comments: arXiv admin note: text overlap with arXiv:2112.09350We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.
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arXiv:2203.09787 (Published 2022-03-18)
A New Probabilistic Representation of the Alternating Zeta Function and a New Selberg-like Integral Evaluation
In this paper, we present two new representations of the alternating Zeta function. We show that for any s $\in$ C this function can be computed as a limit of a series of determinant. We then express these determinants as the expectation of a functional of a random vector with Dixon-Anderson density. The generalization of this representation to more general alternating series allows us to evaluate a Selberg-type integral with a generalized Vandermonde determinant.
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Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations
Comments: This version is 39 pages long and has been accepted to Stochastics who will publish the final versionCategories: math.PRKeywords: backward doubly stochastic differential equation, generalized backward doubly stochastic differential, variational generalized backward doubly stochastic, parabolic stochastic variational inequality, probabilistic representationTags: journal articleWe derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a probabilistic representation (in stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet-Neumann conditions.
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arXiv:2101.06024 (Published 2021-01-15)
A probabilistic representation for heat flow of harmonic map on manifolds with time-dependent Riemannian metric
Categories: math.PRIn this paper we will give a probabilistic representation for the heat flow of harmonic map with time-dependent Riemannian metric via a forward-backward stochastic differential equation on manifolds. Moreover, we can provide an alternative stochastic method for the proof of existence of a unique local solution for heat flow of harmonic map with time-dependent Riemannian metric.
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arXiv:2011.10453 (Published 2020-11-20)
Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift
Categories: math.PRIn this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.
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arXiv:2010.09027 (Published 2020-10-18)
Probabilistic representation of helicity in viscous fluids
It is shown that the helicity of $3D$ viscous incompressible flow can be identified with the overall linking of the fluid's initial vorticity to the expectation of a stochastic mean field limit. The relevant mean field limit is obtained by following the Lagrangian paths in the stochastic Hamiltonian interacting particle system of [8].
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arXiv:1908.04550 (Published 2019-08-13)
Integration by parts formula for killed processes: A point of view from approximation theory
Comments: 38 pages. Accepted for publication in the Electronic Journal of ProbabilityCategories: math.PRIn this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li's type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.
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arXiv:1902.01110 (Published 2019-02-04)
Regarding the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs
Categories: math.PROn a finite weighted graph $G$, the effective resistance between two vertices $x$ and $y$ admits the probabilistic representation \[ R(x,y) = \frac{1}{c_x \cdot \mathbb{P}_x[\tau_y < \tau_x^+]} \] using the the random walk $\mathbb{P}_x$ on $G$ starting in $x$. We show that the same representation does not hold in general for the free effective resistance of an infinite graph. More precisely, we show that a transient graph admits such a representation if and only if it is a subgraph of an infinite line.
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arXiv:1808.04964 (Published 2018-08-15)
A Probabilistic Proof of the Perron-Frobenius Theorem
The Perron-Frobenius theorem plays an important role in many areas of management science and operations research. This paper provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic representation of the Perron-Frobenius eigenvalue and eigenvectors in terms of the dynamics of a Markov chain. The proof provides conditions in both the finite-dimensional and infinite-dimensional settings under which the Perron-Frobenius eigenvalue and eigenvectors exist. Furthermore, the probabilistic representations that arise can be used to produce a Monte Carlo algorithm for computing the Perron-Frobenius eigenvalue and eigenvectors that will be explored elsewhere.
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arXiv:1801.10510 (Published 2018-01-31)
Probabilistic representation for solutions to nonlinear Fokker-Planck equations
Comments: 21 pagesCategories: math.PROne obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker-Planck equation in $\mathbb R^d$ with multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear stochastic differential equation. The case of a nonlinear Fokker-Planck equation with linear space dependent drift is also studied.
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arXiv:1712.00332 (Published 2017-12-01)
On a skewed and multifractal uni-dimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence
We construct, for the first time to our knowledge, a one-dimensional stochastic field $\{u(x)\}_{x\in \mathbb{R}}$ which satisfies the following axioms which are at the core of the phenomenology of turbulence mainly due to Kolmogorov: (i) Homogeneity and isotropy: $u(x) \overset{\mathrm{law}}= -u(x) \overset{\mathrm{law}}=u(0)$ (ii) Negative skewness (i.e. the $4/5^{\mbox{\tiny th}}$-law): \\ $\mathbb{E}{(u(x+\ell)-u(x))^3} \sim_{\ell \to 0+} - C \, \ell\,,$ \, for some constant $C>0$ (iii) Intermittency: $\mathbb{E}{|u(x+\ell)-u(x) |^q} \asymp_{\ell \to 0} |\ell|^{\xi_q}\,,$ for some non-linear spectrum $q\mapsto \xi_q$ Since then, it has been a challenging problem to combine axiom (ii) with axiom (iii) (especially for Hurst indexes of interest in turbulence, namely $H<1/2$). In order to achieve simultaneously both axioms, we disturb with two ingredients a underlying fractional Gaussian field of parameter $H\approx \frac 1 3 $. The first ingredient is an independent Gaussian multiplicative chaos (GMC) of parameter $\gamma$ that mimics the intermittent, i.e. multifractal, nature of the fluctuations. The second one is a field that correlates in an intricate way the fractional component and the GMC without additional parameters, a necessary inter-dependence in order to reproduce the asymmetrical, i.e. skewed, nature of the probability laws at small scales.