## 1. Introduction

[2] In the theory of wave propagation solutions of the Helmholtz equation

are customarily sought in the product form Φ() = *u*() e, where the phase *S*() satisfies the eikonal equation [*S*()]^{2} = κ^{2}(), and the amplitude *u*() satisfies the complete transport equation

[3] There are at least three reasons justifying the use of such a representation of the wave field: (1) The eikonal equation admits a constructive solution by use of the canonical Hamilton-Jacobi method of analytical mechanics; (2) the structure of the eikonal is closely connected with the intuitively clear idea of propagation along rays; and (3) in many cases the amplitude *u*() can be approximated sufficiently well by the ray theory solution *u*_{0} of the equation obtained from (2) by dropping its first term.

[4] If the first term in (2) is dropped then the resulting first-order equation has the solution

where *J*() is a characteristic of the vector field (*S*) widely known in the literature as its “geometrical divergence,” and _{t} = − *S*(_{t})*t* is the solution of the ordinary differential equation d_{t} = − *S*(_{t})d*t*, _{0} = , so that the trajectory of _{t} is the ray along which the wave arrives at .

[5] It is clear that the approximation *u* ≈ *u*_{0} is accurate only when *k* ≫ 1 and when the geometrical divergence *J*() has no singularities in the vicinity of the ray passing through the observation point, which is not the case in many important situations arising, for example, in problems of propagation of low-frequency waves, problems of diffraction, and problems of wave propagation in nonhomogeneous media. Such limitations naturally have generated numerous attempts to improve the elementary approximation ϕ ≈ *u*_{0} either by constructing a series ϕ ≈ *u*_{0} + *u*_{1} + …, or by a more complicated choice of the initial approximation *u*_{0}, or by a combination of both of these ideas.

[6] Although much progress has been achieved in finding asymptotic or approximate solutions of the complete transport equation (2), it is nevertheless instructive and useful to observe that the exact solution *u*() can be represented by explicit probabilistic formulas which are exact in the same rigorous sense as *f*(*x*) = sin(*x*) is an exact solution of *f*″ + *f* = 0. Correspondingly, these solutions do not fail anywhere including at caustics, and they do not loose any information which may be used for the analysis of the physical phenomena.

[7] The basic ideas of the probabilistic approach to wave propagation are traced back to the 1920s–1930s, when the link between partial differential equations and Brownian motion was first observed [*Philips and Wiener*, 1923; *Courant et al.*, 1928; *Petrovsky*, 1934], but the rapid development in this area was made only after the publications of the landmark papers of Feynman [*Feynman*, 1942, 1948] and *Kac* [1949] which presented similar but at the same time very different results: *Feynman* [1942, 1948] represented solutions of the Schrödinger equation by heuristically introduced path integrals which did not admit probabilistic interpretation, and *Kac* [1949] adapted Feynman's formula to the heat conduction equation which was solved by means of the rigourously justified Wiener integration in a functional space which had a clear probabilistic sense.

[8] Since the Schrödinger equation is closely related to the Helmholtz equation, it is not surprising that there have been attempts to employ Feynman's path integral for the analysis of wave propagation. In the first papers exploring this direction [*Buslaev*, 1967; *Keller and KcLaughlin*, 1975] the ray approximation of the wave field was derived from the path integral solution of the Helmholtz equation. Later, the path integrals were used for numerical simulations of acoustical [*Schlottmann*, 1999] and electromagnetic [*Nevels et al.*, 2000] waves, but as mentioned in the survey [*Galdi et al.*, 2000], the perspectives of broader application of the path integrals to wave propagation were limited, presumably because of the notorious difficulty of computation of the Feynman path integrals.

[9] It is well known [*Feynman*, 1998] that the probabilistic formulas employed in Kac's solution [*Kac*, 1949] of the heat conduction equation rest on a rigorous mathematical foundation and admit efficient numerical simulation, but this equation is not directly connected to the Helmholtz equation describing wave propagation. Nevertheless, it has been recently found that there is a natural way of solve the Helmholtz equation by a probabilistic “random walk” method which is based on Kac's formula and provides a direct improvement of the simple ray approximation of the theoretically exact solution of the Helmholtz equation.

[10] In the next section we briefly discuss the principles of random walk and of its relationship with differential equations. Then we derive solutions of the Helmholtz equation which directly improve the approximation provided by ray theory. Finally, to illustrate an application of the random walk method to problems of diffraction we derive a probabilistic solution of the two dimensional problem of diffraction by a wedge with impedance boundary conditions.