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We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \[ \begin{cases} \partial_{t}u(t,\phi)+\mathcal{L}u(t,\phi)+f(t,\phi,u(t,\phi),\partial_{x}u(t,\phi) \sigma(t,\phi),(u(\cdot,\phi))_{t})=0,\;t\in[0,T),\;\phi\in\mathbb{\Lambda}\, ,u(T,\phi)=h(\phi),\;\phi\in\mathbb{\Lambda}, \end{cases} \] where $\mathbb{\Lambda}=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,\phi))_{t}:=(u(t+\theta,\phi))_{\theta\in[-\delta,0]}$ and \[ \mathcal{L}u(t,\phi):=\langle b(t,\phi),\partial_{x}u(t,\phi)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[\sigma(t,\phi)\sigma^{\ast}(t,\phi)\partial_{xx} ^{2}u(t,\phi)\big]. \] The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via $g$-expectations are also provided., Comment: 45 pages

We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing for fairly general coupling functions becomes relevant when the matter-radiation system is confined to cavities with inward pointing boundary singularities., Comment: 48 pages. Second version: references updated, typos removed plus a small number of other trivial improvements

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington-Kirkpatrick spin-glass model without external magnetic field to the quantum case with a "transverse field" of strength $b$. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $v>0$ is smaller than the temperature $1/\beta$, then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $b/v\geq0$. The macroscopic annealed free energy (times $\beta$) turns out to be non-trivial and given, for any $\beta v>0$, by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $\beta v, Comment: v1: 36 pages, 1 figure; v2: 38 pages, 1 figure, high-field limit added, 3 references added, some misprints corrected, minor stylistic improvements, back references included; v3: 41 pages, 3 figures, information added, references added or updated, misprints corrected, stylistic improvements; v4: final version after minor editing

In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

For a stochastic differential equation of the heat equation type, we obtain a Feynman–Kac formula to which the method of analytic continuation with respect to a parameter can be applied under certain assumptions.

The Navier–Stokes equations written in the vector potential can be recast as non-linear Schrodinger equations at imaginary times, i.e. heat equations with a potential term, using the Cole–Hopf tran...

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\Delta+\xi$ in large sets recently proved by the first two authors., Comment: 69 pages, version to appear in Prob. Theory Rel. Fields

In this article given y : [0, η) → H, a continuous map into a Hilbert space H, we study the equation $$\displaystyle \hat y(t)= e^{ \int \limits _0^t c(s,\hat y)ds}y(t), $$ where c(s, ⋅) is a given “potential” on C([0, η), H). Applying the transformation \(y \rightarrow \hat y\) to the solutions of the SPDE and SDE underlying a diffusion, we study the Feynman–Kac formula.

For the heat equation driven by a smooth, Gaussian random potential: \begin{equation*}\partial_{t}u_{\varepsilon}=\frac{1}{2}\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}),\quad t>0,x\in\mathbb{R},\end{equation*} where $\xi_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon}$ is a diverging constant chosen properly, we prove that $u_{\varepsilon}$ converges in $L^{n}$ to the solution of the stochastic heat equation for any $n\geq1$. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux (J. Math. Soc. Japan 67 (2015) 1551–1604), for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.

The beginning of the diabetic nephropathy in humans is related out to the abundance of proteins like albumin fluxing out through the urine as consequence of the permanent filtering of these negatively charged giant proteins that can escape from the micro vessels inside of the renal glomerulus mainly due to the presence of large densities of glucose in the blood. An indicator that is used by the nephrologist is known as the Albumin Excretion Rate (AER). In this paper we perform numerical estimates of this indicator by using the Feynman-Kac formula by which the flux of albumin is seen from the angle of propagation instead of diffusion. Since this anomalous flux out of albumin is seen as a stochastic dynamics so that the theory of random paths and the Feynman propagator would fit to model the very beginning of the diabetic nephropathy. We present numerical estimates that are compared with clinic data in order to validate the present approach.

A tilted Liouville-master equation in Hilbert space is presented for Markovian open quantum systems. We demonstrate that it is the unraveling of the tilted quantum master equation. The latter is widely used in the analysis and calculations of stochastic thermodynamic quantities in quantum stochastic thermodynamics., 1 figure

We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

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., Comment: 23 pages

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

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 -Holder continuous coefficient, where α ∈ ( 0 , 2 ) and β > 1 − α ∕ 2 .

We consider the parabolic Anderson model driven by fractional noise: $$ \frac{\partial}{\partial t}u(t,x)= \kappa \boldsymbol{\Delta} u(t,x)+ u(t,x)\frac{\partial}{\partial t}W(t,x) \qquad x\in\mathbb{Z}^d\;,\; t\geq 0\,, $$ where $\kappa>0$ is a diffusion constant, $\boldsymbol{\Delta}$ is the discrete Laplacian defined by $\boldsymbol{\Delta} f(x)= \frac{1}{2d}\sum_{|y-x|=1}\bigl(f(y)-f(x)\bigr)$, and $\{W(t,x)\;;\;t\geq0\}_{x \in \mathbb{Z}^d}$ is a family of independent fractional Brownian motions with Hurst parameter $H\in(0,1)$, indexed by $\mathbb{Z}^d$. We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation \begin{equation} u(t,x)=\mathbb{E}^x\Bigl[u_o(X(t))\exp \int_0^t W\bigl(\mathrm{d}s, X(t-s)\bigr)\Bigr]\,, \end{equation} is a mild solution to this problem. Here $u_o(y)$ is the initial value at site $y\in\mathbb{Z}^d$, $\{X(t)\;;\;t\geq0\}$ is a simple random walk with jump rate $\kappa$, started at $x \in \mathbb{Z}^d$ and independent of the family $\{W(t,x)\;;\;t\geq0\}_{x\in\mathbb{Z}^d}$ and $\mathbb{E}^x$ is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter $H\in (0,1)$.

We discuss the probabilistic representation of the solutions of the heat equation perturbed by a repulsive point interaction in terms of a perturbation of Brownian motion, via a Feynman–Kac formula involving a local time functional. An application to option pricing is given, interpolating between the extreme cases of classical Black–Scholes options and knockouts having the barrier situated exactly at the exercise price.

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.

International audience; With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.

In this paper, we consider the Fourier-type functionals onWiener space. We then establish the analytic Feynman integrals involv-ing the ⋄-convolutions. Further, we give an approach to solution of theSchro¨dinger equation via Fourier-type functionals. Finally, we use this ap-proach to obtain solutions of the Schro¨dinger equations for harmonic oscil-lator and double-well potential. The Schro¨dinger equations for harmonicoscillator and double-well potential are meaningful subjects in quantummechanics. 1. IntroductionLet C 0 [0,T] denote the one-parameter Wiener space, that is, the space ofcontinuous real-valued functions xon [0,T] with x(0) = 0. In 1948, Feynmanassumed the existence of an integral over a space of paths and used this integralin a formal way in his approach to quantum mechanics [9]. A number ofmathematicians have attempted to give rigorously meaningful definitions ofthe Feynman integral with appropriate existence theorems and have expressedsolutions of the Schr¨odinger equation in terms of their integrals. One of theseapproaches is based on the similarity between the Wiener and the Feynmanintegrals, where procedures are developed to obtain the Feynman integralsfrom the Wiener integrals by an analytic extension from the real axis to theimaginary axis.Consider a differential equation(1.1)∂∂tψ(u,t) =12λ∆ψ(u,t) −V(u)ψ(u,t)with the initial condition ψ(u,0) = ϕ(u), where ∆ is the Laplacian and V is anappropriate potential function. For λ>0, this is the diffusion equation with

In this work is introduced the harmonic oscillator process that in probability is known as Ornstein-Uhlenbeck process or velocities process of Ornstein-Uhlenbeck and is established that the mathematical expectation of certain functionals of this process coincides with the expected value of certain observables in the fundamental state or basis state to the oscillator and with it is found a relation between the spaces of classical and quantum probability. Furthermore is established a certain algebra of operators that gives rise to the dynamics of quantum harmonic oscillator and to consider certain class of perturbations of the Hamiltonian corresponding to their harmonic oscillator is obtained a version of the Feynman-Kac formula with a dilatation generated by the unitary group of translations for a co-cycle. Finally are given some generalizations of the Feynman-Kac formula.

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $\alpha-$stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P\{M\geq x \}\sim (2/\alpha)^{1/2}x^{-\alpha /2}$ as $x \rightarrow \infty$., Comment: Second version fixes several minor errors in the original version. (Thanks to Renming Song for pointing these out.)

Three magnetic relativistic Schrödinger operators are considered corresponding to the classical relativistic Hamiltonian symbol with magnetic vector and electric scalar potentials. We discuss their difference in general and their coincidence in the case of constant magnetic fields, as well as whether they are covariant under gauge transformation. Then results are surveyed on path integral representations for their respective imaginary-time relativistic Schrödinger equations, i. e. heat equations, by means of the probability path space measure coming from the Lévy process concerned. © 2012 Springer.

We consider the Nelson model on some static space–times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass m ( x ) tends to 0 at spatial infinity. Using path space techniques, we show that if m ( x ) ⩽ C | x | − μ at infinity for some C > 0 and μ > 1 then the Nelson Hamiltonian has no ground state.

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class., 26 pages; will appear in a slightly different form in the Journal of Mathematical Analysis and Applications

Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula for Schrodinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Ito formula on manifolds for Schrodinger operators with magnetic fields.

We consider birth-and-death stochastic particle systems in continuum which are under a self-regulation mechanism controlling configurations of particles via a pairwise interaction between them. The latter is reflected in a potential perturbation of the free generator. We show that the ground state renormalization scheme in the considered model leads to an invariant measure, a renormalized generator and resulting equilibrium birth-and-death stochastic dynamics for the system. The proof is based on the Gibbs-type representation for related path space measure. This measure has OS-positivity property and is constructed via the cluster expansion method.

Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski’s equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo’s work (Proc. Natl. Acad. Sci. USA 98(7):3658–3661, [2001]) and Hatano-Sasa equality of steady state thermodynamics (Phys. Rev. Lett. 86:3463–3466, [2001]) to the general multidimensional case.

We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable. Keywords: Airy point process; Brownian bridge; Brownian excursion; Dumitriu–Edelman model; Feynman–Kac formula; Gaussian beta ensemble; intersection local time; moment method; path transformation; quantile transform; random matrix soft edge; random walk bridge; stochastic Airy operator; strong invariance principle; trace formula; Vervaat transform, National Science Foundation (U.S.) (Grant DMS-1407562), National Science Foundation (U.S.) (Grant DMS-1664619)

Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel.

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. G\"uneysu, J.S. M{\o}ller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued L^p-spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter., Comment: 76 pages

We consider the asymptotic almost sure behavior of the solution of the equation u ( t , x ) = u 0 ( x ) + κ 2 ∫ 0 t Δ u ( s , x ) d s + ∫ 0 t u ( s , x ) ∂ W x ( s ) , where { W x : x ∈ R d } is a field of Brownian motions. In fact, we establish existence of the Lyapunov exponent, λ ( κ ) = lim t → ∞ 1 t log u ( t , x ) . We also show that c 1 κ 1 / 3 ⩽ λ ( κ ) ⩽ c 2 κ 1 / 5 as κ ↘ 0 under the assumption that the correlation function of the background field { W x : x ∈ R d } is C β for 1 β ⩽ 2 .

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of the generating Schroedinger operator also act as Carleman operators with continuous integral kernels. Applications to Schroedinger operators with rather general random scalar potentials include a rigorous justification of an integral-kernel representation of their integrated density of states - a relation frequently used in the physics literature on disordered solids., Comment: 41 pages. Final version. Dedicated to Volker Enss on the occasion of his 60th birthday

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q − q′|) ≡ ‹V(q)V(q′)›, where q ∈ ℝd and d is the dimension of the Euclidean space ℝd. For the value ‹G(q,t;q0)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q0)› as |q − q0| → ∞ and t → ∞.

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

The Belavkin equation, describing the continuous measurement of the position of a quantum particle, is studied. A rigorous representation of its solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the complex Cameron-Martin space is given.

Dans cet article nous etudions l’equation de la chaleur lineaire stochastique multidimensionnelle, connue aussi comme model d’Anderson parabolique, perturbee par un bruit gaussien qui est blanc en temps et qui a une covariance correlee en espace. Le noyau de Riesz en dimension quelconque et la covariance du mouvement Brownien fractionnaire avec parametre de Hurst $H\in(\frac{1}{4},\frac{1}{2}]$ en une dimension, sont des examples d’une telle covariance. D’abord, on etablit l’existence d’une solution d’evolution unique et on obtient une formule de Feynman–Kac pour les moments de la solution, en utilisant une famille de ponts browniens independants et en supposant une condition generale d’integrabilite sur la condition initiale. Dans la deuxieme partie du travail nous calculons les exposants de Lyapunov et les exposants superieur et inferieur de croissance exponentielle en fonction d’une quantite variationnelle. La derniere partie du travail est consacre a l’etude de la transition de phase pour le model d’Anderson.

It is known that the rescaled position of a tagged particle in symmetric simple exclusion processes converges to a diffusion. If now the tracer particle is driven by a small force, then it picks up a velocity. The Einstein relation states that in the limit, this velocity is proportional to the small force, and the constant of proportionality can be computed from the diffusion matrix of the tracer particle with no driving force. Such a relation is believed to be generally valid. In this article we establish its validity for all symmetric simple exclusion processes in dimension \(\) and we prove a density property for certain invariant states of the driven system.

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space ${\mathcal F}_{(e^{\theta^{\ast}})^{\ast}}^{\ast}({\mathcal N}')$.

Motivated by the work of Baras-Goldstein (1984), we discuss when expectations of the Feynman-Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $\alpha$-stable process. In connection with the work of Ishige-Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion., Comment: This is the version accepted for publication in Journal of Mathematical Society of Japan

The Green's function of the relativistic Aharonov-Bohm-Coulomb system is given by the Feynman-Kac formula. The earlier treatment is based on the multiple-valued transformation of Levi-Civit´. The method used in this contribution involves only the explicit form of a simple Green's function and an explicit path integral is avoided.

We provide a proof of the Feynman–Kac formula for Brownian motion, using excursion theory up to an independent exponential time θ. Call g(θ) the last zero before θ. The independence of the pre-g(θ) process and the post-g(θ) process and the representation of their laws in terms of the integrals of Wiener measure up to inverse local time, or first hitting times allow to recover a formulation of the Feynman–Kac formula via excursion theory.