Feynman Kac Semigroups Research Articles

Self-similar solution for Hardy operator

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.

Stability of estimates for fundamental solutions under Feynman–Kac perturbations for symmetric Markov processes

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations (see Definition 1.2), we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman–Kac semigroup of X. As a corollary, under the same assumptions, a weak type of global two-sided (upper) estimates holds for the fundamental solution of Feynman–Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman–Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

Large deviations principle via Malliavin calculus for the Navier–Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker–Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [17] in a discrete-time setting and extended in [26] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman–Kac semigroup, which we verify by using Malliavin calculus.

A Convergent Interacting Particle Method and Computation of KPP Front Speeds in Chaotic Flows

In this paper, we study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficients on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

Tamed spaces – Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with interior singularities and singular boundary behavior.

Limiting distributions for the maximal displacement of branching Brownian motions

We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schrödinger type operator. To establish our results, we show a sharp and locally uniform growth order of the Feynman–Kac semigroup.

Optimal Hardy inequality for the fractional Laplacian on Lp

For the fractional Laplacian we give Hardy inequality which is optimal in Lp for 1<p<∞. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on Lp.

Probabilistic approach to the quantum separation effect for the Feynman–Kac semigroup

Probabilistic approach to the quantum separation effect for the Feynman–Kac semigroup

A Feynman-Kac approach for logarithmic Sobolev inequalities

This note presents a method based on Feynman-Kac semigroups for logarithmic Sobolev inequalities. It follows the recent work of Bonnefont and Joulin on intertwining relations for diffusion operators, formerly used for spectral gap inequalities, and related to perturbation techniques. In particular, it goes beyond the Bakry-Émery criterion and allows to investigate high-dimensional effects on the optimal logarithmic Sobolev constant. The method is illustrated on particular examples (namely Subbotin distributions and double-well potentials), for which explicit dimension-free bounds on the latter constant are provided. We eventually discuss a brief comparison with the Holley-Stroock approach.

More on the long time stability of Feynman–Kac semigroups

Feynman–Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability analysis of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compactness, and a minorization condition ensuring some mixing. We show that these conditions are satisfied in a variety of situations, including stochastic differential equations. Illustrative examples are provided, where the stability of the non-linear semigroup arises either from the underlying dynamics or from the Feynman–Kac weight function. We also use our technique to provide uniform in the time step convergence estimates for discretizations of stochastic differential equations.

Derivative formula for the Feynman–Kac semigroup of SDEs driven by rotationally invariant [formula omitted]-stable process

By using the time change argument, we establish the derivative formula as well as gradient estimate for the Feynman–Kac semigroup of stochastic differential equations driven by rotationally invariant α-stable process with β-Hölder continuous coefficient, where α∈(0,2) and β>1−α∕2.

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

We consider the numerical analysis of the time discretization of Feynman–Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present error estimates à la Talay–Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman–Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

Functional Inequalities for Feynman–Kac Semigroups

Using the tools of stochastic analysis, we prove various gradient estimates and Harnack inequalities for Feynman–Kac semigroups with possibly unbounded potentials. One of the main results is a derivative formula which can be used to characterize a lower bound on Ricci curvature using a potential.

Intertwinings and Stein’s magic factors for birth–death processes

This article investigates second order intertwinings between semigroups of birth-death processes and discrete gradients on the space of natural integers N. It goes one step beyond a recent work of Chafai and Joulin which establishes and applies to the analysis of birth-death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth-death and Feynman-Kac semigroups and weighted gradients on N, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of discrete distributions. To illustrate the relevance of this approach, we also derive approximation results for the mixture of Poisson and geometric laws.

Formules de Feynmann-Kac pour le modèle de Nelson ultra-violet renormalisée

We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in the non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the $N$-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times $g^4N^3$, for all values of the coupling constant $g$. A variational argument also yields an upper bound of the same form for large $g^2N$. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.

$$L^p$$-independence of spectral radius for generalized Feynman–Kac semigroups

Under mild conditions on measures used in the perturbation, we establish the $$L^p$$ -independence of spectral radius for generalized Feynman–Kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric Markov processes. These results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for Markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized Feynman–Kac functionals developed by the second and third authors.

Gradient estimates for SDEs without monotonicity type conditions

We prove gradient estimates for transition Markov semigroups (Pt) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C1-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for DxPtφ of the formt|DxPtφ(x)|≤c(1+|x|k)‖φ‖∞,t∈(0,1], φ∈Cb(Rd), x∈Rd. We use two main tools. First, we consider a Feynman–Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut–Elworthy–Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for DxPtφ without weights do not hold.

Derivatives of Feynman–Kac Semigroups

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, although the assumptions are purely geometric.

A note on random walks with absorbing barriers and sequential Monte Carlo methods

ABSTRACTIn this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.

Uniform convergence of penalized time-inhomogeneous Markov processes

We provide a general criterion ensuring the exponential contraction of Feynman–Kac semi-groups of penalized processes. This criterion applies to time-inhomogeneous Markov processes with absorption and killing through penalization. We also give the asymptotic behavior of the expected penalization and provide results of convergence in total variation of the process penalized up to infinite time. For exponential convergence of penalized semi-groups with bounded penalization, a converse result is obtained, showing that our criterion is sharp in this case. Several cases are studied: we first show how our criterion can be simply checked for processes with bounded penalization, and we then study in detail more delicate examples, including one-dimensional diffusion processes conditioned not to hit 0 and penalized birth and death processes evolving in a quenched random environment.