The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but thi...

This paper is intended to provide a stochastic approach involving the combined use of the Feynman-Kac formula and Monte Carlo method as a solution algorithm for estimating the timedependent effective diffusivity in a one-dimensional parabolic inverse problem. The inverse problem is purposed to design a mathematical model for the gas-diffusion process within the alveolar region of the human's lu...

These classical inequalities allow one to estimate the number of negative eigenvalues and the sums Sγ = ∑ |λi| for a wide class of Schrödinger operators. We provide a detailed proof of these inequalities for operators on functions in metric spaces using the classical Lieb approach based on the Kac-Feynman formula. The main goal of the paper is a new set of examples which include perturbations o...

In this paper we study a class of backward stochastic differential equations with reflections (BSDER, for short). Three types of discretization procedures are introduced in the spirit of the so-called Bermuda Options in finance, so as to first establish a Feynman–Kac type formula for the martingale integrand of the BSDER, and then to derive the continuity of the paths of the martingale integran...

It is well known that Zk,m(t) is related to the virial coefficients of a quantum system of obstacles K at inverse temperature t. For example, G. E. Uhlenbeck [15] calculated the asymptotic behaviour of Zk,m(t) as t → ∞ in the special case where k = 1, m = 3, and K is a ball. I. McGillivray [14] obtained the full asymptotic series as t → ∞ for k = 1, m ≥ 3, and K an arbitrary, non-polar compact ...

A Feynman-Kac-type formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the Pauli-Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation invariant and ...

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc. 28, 317–321 (2000)], we study one-parameter semigroups generated by (the negative of) rather general Schrödinger 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 m...

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman-Kac formula is taken here by subordinated Brownian motion. As specific examples, fractional and r...

In this paper we investigate a class of decoupled forward–backward SDEs, where the volatility of the FSDE is degenerate and the terminal value of the BSDE is a discontinuous function of the FSDE. Such an FBSDE is associated with a degenerate parabolic PDE with discontinuous terminal condition. We first establish a Feynman–Kac type representation formula for the spatial derivative of the solutio...

We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. In this paper we consider self-adjoint Berezin-Toeplitz operators associated with semibounded quadratic forms. Following a concept of Da...