We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution.

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

The purpose of this paper consists in proposing a generalized solution for a porous media type equation on a half-line with Neumann boundary condition and prove a probabilistic representation of this solution in terms of an associated microscopic diffusion. The main idea is to construct a stochastic differential equation with reflection which has a solution in law and whose marginal law densiti...

Consider the heat equation driven by a smooth, Gaussian random potential: ∂tuε = 1 2 ∆uε + uε(ξε − cε), t > 0, x ∈ R, where ξε converges to a spacetime white noise, and cε is a diverging constant chosen properly. For any n > 1, we prove that uε converges in Ln to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of...

the notion of a probabilistic metric  space  corresponds to thesituations when we do not know exactly the  distance.  probabilistic metric space was  introduced by karl menger. alsina, schweizer and sklar gave a general definition of  probabilistic normed space based on the definition of menger [1]. in this note we study the pn spaces which are  topological vector spaces and the open mapping an...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

In this paper, travelling wave solutions to the nonlinearly dispersive Schrödinger equation are computed in the case of onedimensional non-relativistic electron confined to a cylindrical quantum well. Investigations gave evidence to the possibility of simplified continuous solutions which are in good agreement with the probabilistic interpretation of this equation.