We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp $\LL\_p$-mean error bounds, and laws of large numbers for $U$-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schrödinger semigroups are also discussed.
In this paper we show that two-sided heat kernel estimates for a class of (not necessarily symmetric) diffusions with jumps are stable under non-local Feynman-Kac perturbations.
We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic partial differential equations (SPDEs) with space-time noise. As an application of the Feynman-Kac formulae, random periodic solutions and stationary solutions to certain SPDEs are obtained.
We present a Feynman-Kac formula for the $1$-dimensional stochastic heat equation (SHE) driven by a time-homogeneous Gaussian white noise potential, where the noise is interpreted in the Wick-Itô-Skorokhod sense. Our approach consists in constructing a Wong-Zakai-type approximation for the SHE from which we are able to obtain an "approximating Feynman-Kac" representation via the reduction of the approximated SHE to a deterministic partial differential equation (PDE). Then we will show that those "approximating Feynman-Kac" converge to a well defined object we will call "formal Feynman-Kac" rep…
The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution.
We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a non-conservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness. The solution of that type of equation can be approached via a weighted particle system.
We consider the continuous parabolic Anderson model with the Gaussian fields under the measure-valued initial conditions, the covariances of which are nonhomogeneous in time and fractional rough in space. We mainly study the spatial behaviors for the Feynman-Kac formulas in Stratonovich's sense. Benefited from the application of Feynman-Kac formula based on Brownian bridge, the precise spatial asymptotics can be obtained in the broader conditions than before.
We study the parabolic integral kernel associated with the weighted Laplacian and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and estimates for them and for their derivatives, given in terms of a Gaussian term and the semi-classical bridge. Assumptions are on the Riemannian data.
We provide a generic way of deducing non-asymptotic error bounds for Sequential MCMC methods from suitable stability properties of Feynman-Kac propagators. We show how to derive this type of stability from mixing conditions for the MCMC dynamics, namely, spectral gaps and hyperboundedness, and from upper bounds on the relative densities in the sequence of distributions.
We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solut…
Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein [2] in the context of discrete generation models to continuous time F…
Let $X$ be a symmetric strong Markov process on a Luzin space. In this paper, we present criteria of the $L^p$-independence of spectral bounds for generalized non-local Feynman-Kac semigroups of $X$ that involve continuous additive functionals of $X$ having zero quadratic variations and discontinuous additive functionals of $X$.
In this paper we obtain a Feynman-Kac formula for the solution of a fractional stochastic heat equation driven by fractional noise. One of the main difficulties is to show the exponential integrability of some singular nonlinear functionals of symmetric stable Lévy motion. This difficulty will be overcome by a technique developed in the framework of large deviation. This Feynman-Kac formula is applied to obtain the Hölder continuity and moment formula of the solution.
In this article, given $y :[0,η)\rightarrow H$ a continuous map into a Hilbert space $H$ we study the equation \[\hat y(t) = e^{\int_0^tc(s,\hat y)}y(t)\] where $c(s,\cdot)$ is a given `potential' on $C([0,η),H)$. Applying the transformation $y \rightarrow \hat y$ to the solutions of the SPDE and PDE underlying a diffusion, we study the Feynman-Kac formula.
The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.
In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on $\mathbb{R}$. The bounds for $p$-th moments proved in [3] are simplified. We validate the Feynman-Kac formula for the $p$-point correlation function of the solutions to this equation with measure-valued initial data.
In this paper we show the existence and form uniqueness of a solution for multidimensional backward stochastic differential equations driven by a multidimensional Lévy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for multidimensional Lévy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equations provide an analogue of the famous Black-Scholes partial differential equation and thus can be used f…
This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [4], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases than were not covered previously in the literature.
In this paper we show that Dirichlet heat kernel estimates for a class of (not necessarily symmetric) Markov processes are stable under non-local Feynman-Kac perturbations. This class of processes includes, among others, (reflected) symmetric stable-like processes on closed $d$-sets in $\bR^d$, killed symmetric stable processes, censored stable processes in $C^{1, 1}$ open sets as well as stable processes with drifts in bounded $C^{1, 1}$ open sets.
In this paper we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman-Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE.Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.
The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas a…
We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities
We consider the (unique) mild solution $u(t,x)$ of a 1-dimensional stochastic heat equation on $[0,T]\times\mathbb R$ driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by $\partial_x u(t,x)$, and its representation as a Feynman-Kac type closed form. The chaos expansion of $\partial_x u(t,x)$ makes it possible to find its (optimal) Hölder regularity especially in space.
In the framework of stochastic functional differential equations (SFDE's) and the corresponding calculus developed in the recent years by F. Yan and S. Mohammed, we provide a series of representation formulae for a variety of highly degenerate functional differential equations of the type of the Feynman-Kac formulae. More precisely, we study the stochastic process satisfying regular SFDE's with killing and absorbing boundary, we give the differential equation to be solved in order to compute the distribution of the first exit time from a regular domain and apply our results to a model describi…
This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear natu…
A complex notion of backward stochastic differential equation (BSDE) is proposed in this paper to give a probabilistic interpretation for linear first order complex partial differential equation (PDE). By the uniqueness and existence of regular solutions to complex BSDE, we deduce that there exists a unique classical solution $\{\mathbb{U}(t,x)$ to complex PDE and $\{\mathbb{U}(t,x)$ is analytic in $x$ for each $t$. Thus we extend the well known real Feynman-Kac formula to a complex version. It is stressed that our complex BSDE corresponds to a linear PDE without the second order term.
In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local paraboli…
This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables rela…
It is well-known since the work of Pardoux and Peng [12] that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs, see also Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a…
In the first part of this paper, we give a useful criterion for uniform integrability of exponential martingales in the context of Markov processes. The condition of this criterion is easy to verify and is, in general, much weaker than the commonly used Novikov's condition. In the second part of this paper, we present a new approach to the study of spectral bounds of Feynman-Kac semigroups for a large class of symmetric Markov processes. We first establish criteria for the $L^p$-independence of spectral bounds for Feynman-Kac semigroups generated by continuous additive functionals, using gauge…
We find Feynman-Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.
In this paper we investigate the speed of convergence of the fluctuations of a general class of Feynman-Kac particle approximation models. We design an original approach based on new Berry-Esseen type estimates for abstract martingale sequences combined with original exponential concentration estimates of interacting processes. These results extend the corresponding statements in the classical theory and apply to a class of branching and genealogical path-particle models arising in nonlinear filtering literature as well as in statistical physics and biology.
We design exact polynomial expansions of a class of Feynman--Kac particle distributions. These expansions are finite and are parametrized by coalescent trees and other related combinatorial quantities. The accuracy of the expansions at any order is related naturally to the number of coalescences of the trees. Our results include an extension of the Wick product formula to interacting particle systems. They also provide refined nonasymptotic propagation of chaos-type properties, as well as sharp $\mathbb{L}_p$-mean error bounds, and laws of large numbers for $U$-statistics.
In this paper, we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form, which is associated with a right Markov process $X$ satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional $M$ of $X$, the subprocess $X^M$ of $X$ killed by $M$ also satisfies the sector condition and we then characterize th…
An efficient discrete time and space Markov chain approximation employing a Brownian bridge correction for computing curvilinear boundary crossing probabilities for general diffusion processes was recently proposed in Liang and Borovkov (2021). One of the advantages of that method over alternative approaches is that it can be readily extended to computing expectations of path-dependent functionals over the event of the process trajectory staying between two curvilinear boundaries. In the present paper, we extend the scheme to compute expectations of the Feynman--Kac type that frequently appear…
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.
For a (non-symmetric) strong Markov process $X$, consider the Feynman--Kac semigroup \[T_t^Af(x):=\mathbb {E}^x\bigl[e^{A_t}f(X_t)\bigr],\quad x\in {\mathbb {R}^n}, t>0,\] where $A$ is a continuous additive functional of $X$ associated with some signed measure. Under the assumption that $X$ admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to $T_t^A$ possesses the density $p_t^A(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for $p_t^A(x,y)$. Some examples are provided.
We give the solution of certain parabolic evolution problems (time-depending perturbations of the heat equation for the harmonic oscillator) as explicit integrals on the Wiener space.
We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman-Kac formula, where the underlying stochastic processes satisfy variational inequalities modelling elasto-plastic and obstacle…
In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations, 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 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…
In this paper, we provide an integral equation characterization of the solution to a Cauchy problem associated to the Feynman-Kac formula for a regime-switching diffusion. We give a sufficient condition to guarantee the uniqueness of solutions to the integral equation and provide an example in the context of option pricing under the Ornstein-Uhlenbeck regime-switching model.
We introduce a stochastic process and functional that should describe the semigroup generated by the stochastic Bessel operator. Recently Gorin and Shkolnikov showed that the largest eigenvalues for certain random matrix ensembles with soft edge behavior can be understood by analyzing large powers of tridiagonal matrices, which converge to operators in the stochastic Airy semigroup. In this article we make some progress towards realizing Gorin and Shkolnikov's program at the random matrix hard edge. We analyze large powers of a suitable tridiagonal matrix model (a slight modification of the $β…
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 with $L^\infty$-potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.
In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.
Suppose that $α\in (0,2)$ and that $X$ is an $α$-stable-like process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the class $\bf{J_{d,α}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup $\{S_{t}^{F}:t \ge 0\}$ defined by $$ S_{t}^{F}f(x)=\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-\frac{d}α}(1 \w…
In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter $H<1/2$. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman--Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution…
We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of forward backward stochastic differential equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (…
A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.
This paper describes an algorithm of interest. This is a preliminary version and we intend on writing a better descripition of it and getting bounds for its complexity.
We introduce a class of time-inhomogeneous transition operators of Feynman-Kac type that can be considered as a generalization of symmetric Markov semigroups to the case of a time-dependent reference measure. Applying weighted Poincaré and logarithmic Sobolev inequalities, we derive L^p-L^p and L^p-L^q estimates for the transition operators. Since the operators are not Markovian, the estimates depend crucially on the value of p. Our studies are motivated by applications to sequential Markov Chain Monte Carlo methods.
