Related papers: Probabilistic representations of solutions to the …

We consider the heat equation defined by a generalized measure theoretic Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure…

In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coeffcient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this…

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by…

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable…

In this paper, we consider the heat equation with the natural polynomial non-linear term; and with two different cases in the diffusion term. The first case (anomalous diffusion) concerns the fractional Laplacian operator with parameter…

We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to…

We consider the heat equation on the $N$-dimensional cube $(0,1)^N$ and impose different classes of integral conditions, instead of usual boundary ones. Well-posedness results for the heat equation under the condition that the moments of…

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the…

We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time…

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We…

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit…

We construct a class of exponential type solutions for the linear, delayed heat equation. These representations may be used to provide a priori ansatzes for certain boundary and/or initial-value problems arising in heat transfer. Several of…

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\phi$-heat equation and prove Gaussian upper and lower bounds for the $\phi$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci…

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_n$ related to…

In the present study examines the statistical structure of the generated randomized density of the normal distribution and the Cauchy distribution. The study put the allegation that a randomized probability density of the normal…

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it…

In Stochastic Thermodynamics, heat is a random variable with a probability distribution associated. Studies of the distribution of heat are mostly in the overdamped regime and in one dimension. Here we solve the heat distribution in the…

In this work we find a solution to problem of the heat equation which is annihiliated at a cubic boundary $f$. The solution turns out to be the convolution between the fundamental solution of the heat equation and a function $\phi$ which…

We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the…

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the…