Abstract. In this paper the heat equation with a random right side is examined. In particular, we give conditions for the existence of solutions in the case when the right side is a random field, sample continuous with probability one from the space L p ( Ω ) $L_{p}(\Omega )$ . Estimations for the distribution of the supremum of solutions of such equations are found.

This paper mainly focuses on the estimates for distribution of supremum for the normalized φ-sub-Gaussian random fields defined on the unbounded domain. In particular, we obtain the estimates for distribution of supremum for the normalized solution of the hyperbolic equation of mathematical physics, which will be useful to construct modeless. By using this result, we can approximate the solutions of such equation with given accuracy and reliability in the uniform metric.

The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random factors is a classical problem of the parabolic type of mathematical physics. In this paper, the heat equation with random right side is examined. In particular, we give conditions of existence with probability, one classical solutions in the case when the right side is a random field, sample continuous with probability one from the space () Sub ϕ Ω. Estimation for the distribution of the supremum of solutions of such equations is founded.

The heat equation with random conditions is a classical problem of mathematical physics. Recently, a number of works appeared, which in many ways investigated this equation according to the type of random initial conditions. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on the plane with a random right part. We consider the right part as a random function of the Orlicz space. The conditions of existence with probability one classical solution of the problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random conditions is a classical problem of mathematical physics. In this paper we consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space Subφ(Ω). The conditions of existence with probability one generalized solution of the problem are investigated.