The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure and Feynman's Operational Calculus

We investigate what happens if in the Feynman-Kac functional, we perform the time integration with respect to a Borel measure η rather than ordinary Lebesgue measure l. Let u(t) be the operator associated with this functional through path integration. We show that u(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue-Stieltjes measure η.” One recovers the classical Feynman-Kac formula by letting η = l. We deduce from the integral equation that u(t) satisfies a differential equation associated with the continuous part μ of η when η = μ = l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum-mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete part v of η: the function u(t) undergoes a discontinuity at every point in the support of v, assumed here to be finite. Further, one obtains an explicit expression for u(t) in terms of operators alternatively associated with μ and v. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operator u(t). We provide natural physical interpretations of our results in both the diffusion and quantum-mechanical cases. We also suggest further connections with Feynman?s operational calculus for noncommuting operators.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula

In this article, we consider a complex-valued and a measure-valued measure on , the space of all real-valued continuous functions on . Using these concepts, we establish the measure-valued Feynman-Kac formula and we prove that this formula satisfies a Volterra integral equation. The work here is patterned to some extent on earlier works by Kluvanek in 1983 and by Lapidus in 1987, but the present setting requires a number of new concepts and results.

     

An integral equation with a difference kernel

In this note a new method of solving a class of integral equations with difference kernels is given. It is based on establishing a connection between the solution of the given equation and that of the corresponding equation on the half-axis. This method allows us to reduce the given equation to a new integral equation with the kernel of a simple structure.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 927–932, June, 1976.     

Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula

We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.

     

Heat equation with a general stochastic measure on nested fractals

A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3.     

The general solution of a non-linear integral equation of convolution type

All continuous, non-negative solutions of a non-linear convolution equation are explicitly determined.

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Zusammenfassung Alle stetigen nichtnegativen Lösungen einer nichtlinearen Konvolutionsgleichung werden explizit bestimmt.

     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

An integral formula on the Heisenberg group

Let $\mathbb{H }^n$ denote the $(2n+1)$ -dimensional (sub-Riemannian) Heisenberg group. In this note, we shall prove an integral identity (see Theorem 1.2) that generalizes a formula obtained in the Seventies by Reilly (Indiana Univ Math J 26(3):459–472, 1977). Some first applications will be given in Sect. 4.     

Linear relations generated by an integral equation with Nevanlinna operator measure

We define families of maximal and minimal linear relations generated by an integral equation with Nevanlinna operator measure and prove their holomorphic property. We also prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We apply the obtained results for proving the constancy of deficiency indices of some integral and differential equations.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Boundary integral equation method in thermoelasticity part I: general analysis

The boundary integral equation formulation of thermoelasticity problems is presented. Fundamental solutions of governing differential equations are given for various problems of thermoelasticity according to the classification made here. As well as the representations of the temperature and displacement fields those of the traction vector and heat flux are given in a regularized form. A new type of boundary equation has been found due to such regularization. The boundary element method is applied to solve all the presented boundary equations.     

An integral equation in conformal geometry

Motivated by Carleman's proof of the isoperimetric inequality in the plane, we study the problem of finding a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class on a compact manifold with boundary. We derive a criterion for the existence and make a related conjecture.     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.