Kac Formula Research Articles

Teaching renormalization, scaling, and universality with an example from quantum mechanics

We discuss the quantum mechanics of a particle restricted to the half-line with potential energy for . It is known that two scale-invariant theories may be defined. By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)-space of Hamiltonians. A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points. Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points. Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory. By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value. It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator. Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions. In the former example, V appears as part of an expectation value in the Feynman–Kac formula. In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.

A Feynman–Kac formula for differential forms on manifolds with boundary and geometric applications

We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct $L^2$ harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity property expressed in terms of a certain stochastic average of the Weitzenb\"ock operator $R_p$ acting on $p$-forms and the second fundamental form of the boundary. This extends previous work by Elworthy-Li-Rosenberg on closed manifolds to this setting. As an application we find a geometric obstruction to the existence of metrics with 2-convex boundary and positive $R_2$ in this stochastic sense. We also discuss a version of the Feynman-Kac formula for spinors under suitable boundary conditions.

First order Feynman–Kac formula

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge.

Cole–Hopf-Feynman–Kac formula and quasi-invariance for Navier–Stokes equations

We make a refined comparison between the Navier–Stokes equations and their dynamically-scaled Leray equations solely on the basis of their scaling property. Previously it was observed using the vector potentials that they differ only by one drift term (Ohkitani 2017 J. Phys. A: Math. Theor. 50 045501). The Duhamel principle recasts the equations in path integral forms, which differ by two Maruyama–Girsanov densities. In this brief paper we simplify the concept of quasi-invariance (or, near-invariance) by combining the result with a Cole–Hopf transform and the Feynman–Kac formula. That way, as a multiplicative characterisation we can place those equations just one Maruyama–Girsanov density apart. Furthermore, as an additive characterisation we express the difference in terms of the Malliavin H-derivative.

The Feynman–Kac formula and Harnack inequality for degenerate diffusions

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.

Nonlinear stochastic receding horizon control: stability, robustness and Monte Carlo methods for control approximation

ABSTRACTThis work considers the stability of nonlinear stochastic receding horizon control when the optimal controller is only computed approximately. A number of general classes of controller approximation error are analysed including deterministic and probabilistic errors and even controller sample and hold errors. In each case, it is shown that the controller approximation errors do not accumulate (even over an infinite time frame) and the process converges exponentially fast to a small neighbourhood of the origin. In addition to this analysis, an approximation method for receding horizon optimal control is proposed based on Monte Carlo simulation. This method is derived via the Feynman–Kac formula which gives a stochastic interpretation for the solution of a Hamilton–Jacobi–Bellman equation associated with the true optimal controller. It is shown, and it is a prime motivation for this study, that this particular controller approximation method practically stabilises the underlying nonlinear process.

Brownian motion and finite approximations of quantum systems over local fields

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

The stochastic solution to a Cauchy problem for degenerate parabolic equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ∈(0,1], the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement δ≥1/2. Uniqueness results, including a Feynman–Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

Estimates on Functional Integrals of Quantum Mechanics and Non-relativistic Quantum Field Theory

We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form $${E[\rm {exp}(\mathcal{A}_T)]}$$ , the (effective) action $${\mathcal{A}_T}$$ being a function of particle trajectories up to time T. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman–Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schrodinger operator with $${1/|x|^2}$$ potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.

An Itô calculus for a class of limit processes arising from random walks on the complex plane

Within the framework of the previous paper (Bonaccorsi and Mazzucchi, 2015), we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman–Kac formula and a representation formula for higher derivatives of analytic functions are also given.

Continuity of the Feynman–Kac formula for a generalized parabolic equation

It is well–known 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. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang in 1998. 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 more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied Pardoux and Zhang in their paper from 1998, as well as those considered by Maticiuc and Răşcanu in their paper from 2010, are Corollaries of our main results.

SDE Based Regression for Linear Random PDEs

A simulation based method for the numerical solution of PDEs with random coefficients is presented. By the Feynman--Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE) driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behavior.

Field Determination Near Plasmonic Structures by the Feynman–Kac Stochastic Representation

Feynman–Kac stochastic representation for elliptic equations is used to construct the solution of Helmholtz equation in domains containing plasmonic structures. The method involves initiation of Brownian motion at an interior point in the plasmonic structure and relating the field there to that on the structure boundary (equivalent surface) through the Feynman–Kac formula. The field exterior to the plasmonic structure can be related to the field on the equivalent surface through a variety of numerical methods depending on the complexity of the exterior domain. Finally, the field on the equivalent surface is determined by the imposition of boundary conditions. Comparison is shown here with the exact solution for the total field on the surface of a circular nanocylinder and a nanosphere driven by localized sources and where the permittivity is described by the Drude model.

From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman–Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman–Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.

Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes

We consider a classical finite horizon optimal control problem for continuous-time pure jump Markov processes described by means of a rate transition measure depending on a control parameter and controlled by a feedback law. For this class of problems the value function can often be described as the unique solution to the corresponding Hamilton–Jacobi-Bellman equation. We prove a probabilistic representation for the value function, known as nonlinear Feynman–Kac formula. It relates the value function with a backward stochastic differential equation (BSDE) driven by a random measure and with a sign constraint on its martingale part. We also prove existence and uniqueness results for this class of constrained BSDEs. The connection of the control problem with the constrained BSDE uses a control randomization method recently developed by several authors. This approach also allows to prove that the value function of the original non-dominated control problem coincides with the value function of an auxiliary dominated control problem, expressed in terms of equivalent changes of probability measures.

Continuous-time Markov decision processes with risk-sensitive finite-horizon cost criterion

This paper studies continuous-time Markov decision processes with a denumerable state space, a Borel action space, bounded cost rates and possibly unbounded transition rates under the risk-sensitive finite-horizon cost criterion. We give the suitable optimality conditions and establish the Feynman–Kac formula, via which the existence and uniqueness of the solution to the optimality equation and the existence of an optimal deterministic Markov policy are obtained. Moreover, employing a technique of the finite approximation and the optimality equation, we present an iteration method to compute approximately the optimal value and an optimal policy, and also give the corresponding error estimations. Finally, a controlled birth and death system is used to illustrate the main results.

Logarithmic minimal models with Robin boundary conditions

We consider general logarithmic minimal models , with coprime, on a strip of N columns with the (r, s) Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. On the lattice, these models are Yang–Baxter integrable loop models that are described algebraically by the one-boundary Temperley–Lieb algebra. The (r, s) Robin boundary conditions are a class of integrable boundary conditions satisfying the boundary Yang–Baxter equations which allow loop segments to either reflect or terminate on the boundary. The associated conformal boundary conditions are organized into infinitely extended Kac tables labelled by the Kac labels and . The Robin vacuum boundary condition, labelled by , is given as a linear combination of Neumann and Dirichlet boundary conditions. The general (r, s) Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an (r, s)-type seam consisting of an r-type seam of width w columns and an s-type seam of width d = s − 1 columns. The r-type seam admits an arbitrary boundary field which we fix to the special value where is the crossing parameter. The s-type boundary introduces d defects into the bulk. We consider the commuting double-row transfer matrices and their associated quantum Hamiltonians and calculate analytically the boundary free energies of the (r, s) Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes , the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of N for and restricting to the ground state sequences , with the inverse , we find that the conformal weights take the values where is given by the usual Kac formula. The (r, s) Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label . Level degeneracies support the conjecture that the characters of the associated (reducible or irreducible) representations are given by Virasoro Verma characters.

Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps

In this work, we are concerned with multistep schemes for solving forward backward stochastic differential equations with jumps. The proposed multistep schemes admit many advantages. First of all, motivated by the local property of jump diffusion processes, the Euler method is used to solve the associated forward stochastic differential equation with jump, which reduce dramatically the entire computational complexity, however, the quantities of interests in the backward stochastic differential equations (with jump) are still of high order rate of convergence. Secondly, in each time step, only one jump is involved in the computational procedure, which again reduces dramatically the computational complexity. Finally, the method applies easily to partial-integral differential equations (and some nonlocal PDE models), by using the generalized Feynman---Kac formula. Several numerical experiments are presented to demonstrate the effectiveness of the proposed multistep schemes.

Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman---Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorokhod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres method, approximation of the boundary local time is obtained and the Feynman---Kac formula is then calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman–Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman–Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson’s model.