Feynman-Kac Functionals Research Articles

Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin–Wentzell small noise asymptotics. Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. The rate functions can be interpreted as the value functions of certain stochastic control problems in which there are two types of controls; one of the controls is random and nonanticipative and arises from the aggregated contributions of the individual Brownian noises, whereas the second control is nonrandom and corresponds to the small common Brownian noise that impacts all particles. We also study large deviation behavior of interacting particle systems approximating various types of Feynman–Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes

Feynman–Kac penalizations of rotationally symmetric [formula omitted]-stable processes

In Yano et al. (2009), limit theorems (so called, Feynman–Kac penalizations) for the one-dimensional recurrent symmetric α-stable process weighted and normalized by negative (killing) Feynman–Kac functionals with small potential are studied. In this paper, we deal with the same problem for negative potentials diverging at infinity.

$$L^p$$-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.

Lifschitz singularity for subordinate Brownian motions in presence of the Poissonian potential on the Sierpiński gasket

We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman–Kac functionals.

General analytic characterization of gaugeability for Feynman-Kac functionals

We give a general analytic characterization of the gaugeability for generalized Feynman-Kac functionals involving continous additive functionals locally of zero energy generalizing the earlier work on the analytic characterization of the gaugeability of Feynman-Kac functional for absorbing Brownian motion on a bounded domain given by Aizenman-Simon (Commun Pure Appl Math 35(2), 209–273, 1982). The result also extends the previous known results on the analytic characterization of the gaugeability for generalized Feynman-Kac functionals in the framework of symmetric Markov processes satisfying irreducibility condition and absolute continuity condition.

Convergence of measures penalized by generalized Feynman-Kac transforms

We prove the existence of limiting laws for symmetric stable-like processes penalized by generalized Feynman-Kac functionals and characterize them by the gauge functions and the ground states of Schrodinger type operators.

Analytic characterizations of gaugeability for generalized Feynman-Kac functionals

We give analytic characterizations of gaugeability for generalized Feynman-Kac functionals including continuous additive functional of zero quadratic variation in the framework of symmetric Markov processes. Our result improves the previous work on the analytic characterization due to Z.-Q. Chen (2003) even if we restrict ourselves to deal with non-local perturbations. We also prove that such a characterization is also equivalent to semi-conditional gaugeability and to the subcriticality of the Schrödinger operator associated to our generalized Feynman-Kac semigroup under the conditional (semi-)Green-tightness of related measures.

Large deviation principles for generalized Feynman-Kac functionals and its applications

Large deviation principles of occupation distribution for generalized Feyn-man-Kac functionals are presented in the framework of symmetric Markov processes having doubly Feller or strong Feller property. As a consequence, we obtain the $L^p$-independence of spectral radius of our generalized Feynman-Kac functionals. We also prove Fukushima's decomposition in the strict sense for functions locally in the domain of Dirichlet form having energy measure of Dynkin class without assuming no inside killing.

Large time asymptotics of Feynman–Kac functionals for symmetric stable processes

Let μ be a positive Kato measure on associated with the Green kernel of the transient symmetric α‐stable process, the Markov process with generator (). Let be the positive continuous additive functional in the Revuz correspondence with μ. If, in addition, μ is of compact support, we give exact large time asymptotics of the expectation of the Feynman–Kac functional, .

Feynman-Kac penalization problem for critical measures of symmetric $\alpha $-stable processes

We consider the Feynman-Kac penalization problem for critical measures of symmetric $\alpha $-stable processes via large time asymptotics of Feynman-Kac functionals.

A refinement of analytic characterizations of gaugeability for generalized Feynman–Kac functionals

A refinement of analytic characterizations of gaugeability for generalized Feynman–Kac functionals

Semi-Dirichlet forms, Feynman–Kac functionals and the Cauchy problem for semilinear parabolic equations

In the first part of the paper we prove various results on regularity of Feynman–Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.

Large deviations for symmetric stable processes with Feynman-Kac functionals and its application to pinned polymers

Let $\nu$ and $\mu$ be positive Radon measures on ${\boldsymbol R} ^d$ in Green-tight Kato class associated with a symmetric $\alpha$-stable process $(X_t , P_x)$ on ${\boldsymbol R}^d$, and $A_t ^\nu$ and $A_t ^\mu$ the positive continuous additive functionals under the Revuz correspondence to $\nu$ and $\mu$. For a non-negative $\beta$, let $P_{x,t} ^{\beta \mu}$ be the law $X_t$ weighted by the Feynman-Kac functional $\exp(\beta A_t ^\mu)$, i.e., $P_{x,t} ^\mu =(Z_{x,t} ^\mu)^{-1}\exp(\beta A_t ^\mu)P_x$, where $Z_{x,t} ^\mu$ is a normalizing constant. We show that $A_t ^\nu /t$ obeys the large deviation principle under $P_{x,t}^{\beta \mu}$. We apply it to a polymer model to identify the critical value $\beta _{\rm cr}$ such that the polymer is pinned under the law $P^{\beta \mu} _{x,t} $ if and only if $\beta$ is greater than $\beta_{\rm cr}$. The value $\beta _{\rm cr} $ is characterized by the rate function.

Feynman-Kac Penalisations of Symmetric Stable Processes

In K. Yano, Y. Yano and M. Yor (2009), limit theorems for the one-dimensional symmetric α-stable process normalized by negative (killing) Feynman-Kac functionals were studied. We consider the same problem and extend their results to positive Feynman-Kac functionals of multi-dimensional symmetric α-stable processes.

Stochastic Feynman–Kac Equations Associated to Lévy–Itô Diffusions

We consider linear parabolic stochastic integro-PDEs of Feynman–Kac type associated to Lévy–Itô diffusions. The solution of such equations can be represented as certain Feynman–Kac functionals of the associated diffusion such that taking expectation yields the deterministic Feynamn–Kac formula. We interpret the problem in the framework of white noise analysis and consider differentiation in the sense of stochastic distributions. This concept allows for relaxed assumptions on the equation coefficients, identically to those required in problems of similar deterministic integro-PDEs.

Gaugeability for Feynman-Kac functionals with applications to symmetric 𝛼-stable processes

For symmetric α \alpha -stable processes, an analytic criterion for a measure being gaugeable was obtained by Z.-Q. Chen (2002), M. Takeda (2002) and M. Takeda and T. Uemura (2004). Applying it, we consider the ultracontractivity of Feynman-Kac semigroups and expectations of the number of branches hitting closed sets in branching symmetric α \alpha -stable processes.

Partition function of a spinor gas

For a spinor gas, i.e., a mixture of identical particles with several internal degrees of freedom, we derive the partition function in terms of the Feynman-Kac functionals of polarized components. As an example we study a spin-1 Bose gas with the spins subjected to an external magnetic field and confined by a parabolic potential. From the analysis of the free energy for a finite number of particles, we find that the specific heat of this ideal spinor gas as a function of temperature has two maxima: one is related to a Schottky anomaly, due to the lifting of the spin degeneracy by the external field, the other maximum is the signature of Bose-Einstein condensation.

Integration with respect to the product of a measure and a closable set function

The notion of a set function being closable with respect to a family of measures was introduced in [1] and applied there to the integration of Feynman-Kac functionals. In [2], it is shown how the interchange of integrals for the product of an operator valued measure and a scalar measure can be used to solve operator equations associated with the perturbation of semigroups. The purpose of this article is to develop the analogous machinery for closable set functions with the view of applying it to Schrödinger’s equation.

Feynman-Kac functionals and positive solutions of 1/2?u+qu=0

Let D be a bounded C2 domain in ℝ d and let q be a bounded Borel function in D. For x∃D and z∃∂D suppose (Xt) under the law Px;z is Brownian motion in D starting at x and conditioned to converge to z. Let Τ be the lifetime of (Xt). We show that if the quantity \(E^{x;z} \left\{ {\exp \left[ {\int\limits_0^t {q(X_s )ds} } \right]} \right\}\) is finite for one x∃D and one z∃∂D, then this quantity remains bounded as x varies over D and z varies over ∂D. This may be considered one quantitative expression of the qualitative statement that no matter where Brownian motion in D eventually hits ∂D, it goes all over D before it gets there. We apply this result to show that if the equation 1/2δu+qu=0 admits a non-negative solution in D, which is strictly positive on a subset of ∂D of positive harmonic measure, then for any non-negative bounded Borel function f on ∂D it admits a unique bounded solution u satisfying u=f on ∂D, and this solution u is non-negative.