Related papers: It\^{o}'s formula for linear fractional PDEs

We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the…

We use the nonstandard Fourier transform method, along with an established nonstandard approach to ODE's, to find a solution to the heat equation, on $(0,\infty)\times\mathcal{R}$, with a given boundary condition $g$ at $t=0$. We use this…

In the present work, an attempted was made to develop a numerical algorithm by the use of new orthogonal hybrid functions formed from hybrid of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal…

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(\beta^k)_k$ of i.i.d.…

We use the path integral approach to a two-dimensional noncommutative harmonic oscillator to derive the partition function of the system at finite temperature. It is shown that the result based on the Lagrangian formulation of the problem,…

Fourier expansion of the integrand in the path integral formula for the partition function of quantum systems leads to a deterministic expression which, though still quite complex, is easier to process than the original functional integral.…

We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators…

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional…

For degenerate stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H>1/2$, the derivative formulas are established by using Malliavin calculus and coupling method, respectively. Furthermore, we find…

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$ d X= b(X) dt + d Z, \qquad X(0)=x, $$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued L\'evy processes. Using the…

We present analytical formula along with its existence theorem for solution of inverse heat conduction problem of semi-infinite bar, equivalent to a Volterra integral equation of first kind, as an infinite series of fractional derivatives.…

This paper is devoted to a construction of the stochastic It\^o integral with respect to infinite dimensional cylindrical Wiener process. The construction given is an alternative one to that introduced by DaPrato and Zabczyk [3]. The…

Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose…

In this paper we provide a physical interpretation of It\^o-process resulting in thermal equilibrium distribution of a Brownian particle experiencing coordinate dependent diffusion. Since the local quantities like diffusivity would go…

In this article, we first establish derivative formulae for fractional Gruschin type process, which generalize the result of Wang (J Theor Probab 27:80--95, Theorem 1.1, 2012). Since we work on a non-Markovian context, some technical…

This paper establishes by employing analytic and probabilistic techniques estimates concerning the {\it heat content} for the fractional Schr\"odinger operator $\F+\ind$ with $0<\alpha\leq 2$ in $\Rd$, $d\geq 2$ and $\dom$ a Lebesgue…

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the…

We consider non-negative solutions to the semilinear space-fractional diffusion problem $(\partial_t+(-\Delta)^{\alpha/2})u=\rho(x)u^p$ on whole space $R^n$ with nonnegative initial data and with $(-\Delta)^{\alpha/2}$ being the…

We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath and external non-linear force field. For the force free case the velocity damping follows the Mittag-Leffler relaxation and the diffusion is…

Stochastic integration \textit{wrt} Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic modeling, biomedicine and finance. The aim of this work is to define and…