The heat equation with random initial conditions is a well-studied topic in both the mathematical and physical literature. De Fériet [Reference De Fériet and Neyman7] and Rosenblatt [Reference Rosenblatt9] introduced rigorous probabilistic methods to study heat equations with stationary initial conditions and gave the spectral representations of their solutions as stochastic integrals. Random fields with singular spectra appear in various problems, including rescaling linear diffusion equations with singular initial conditions (see, for example, Anh and Leonenko [Reference Anh and Leonenko5, Reference Anh and Leonenko6] and Leonenko and Woyczyński [Reference Leonenko and Woyczyński8]).

This thesis studies the asymptotic behaviour of random fields that are solutions of partial differential equations with random initial conditions exhibiting long-range and cyclic long-range dependence. The principal focus is on investigating multiscaling limits and their associated spectral and covariance structures.

The analysis includes the cases of solutions $u(t,x)$ to fractional, high-order, heat and Riesz–Bessel type equations with random initial conditions exhibiting both classical and cyclic long-range dependence. The results are obtained for the cases of arbitrary multiple spectral singularities, which have not been considered before.

The initial conditions considered in the thesis are given by

$$ \begin{align*} u(0,x)=\xi(x), \end{align*} $$

where $\xi (x)$ is a Gaussian random field with the covariance function

$$ \begin{align*} r(x)=\sum_{j=0}^n A_j \frac{\cos(w_j x)}{(1+x^2)^{\kappa_j/2}}, \end{align*} $$

which corresponds to the spectral density with singularities at zero and nonzero frequencies (see [Reference Alghamdi, Leonenko and Olenko1–Reference Alghamdi and Olenko3]).

The first class of equations studied in the thesis comprises high-order heat equations with random initial conditions whose spectra have singularities at zero and at nonzero frequencies. Two cases are considered.

For the case of even order,

$$ \begin{align*} \frac{\partial u(t,x)}{\partial t} = (-1)^{m/2+1} \frac{\partial^m u(t,x)}{\partial x^m}, \quad t>0,\ x\in \mathbb R,\ m = 2k\ (k\in \mathbb{N}), \end{align*} $$

the corresponding solutions have the spectral form

$$ \begin{align*} u(t,x)=M(x)+\int_{\mathbb{R}} e^{i\lambda x-\lambda^m t} Z(d\lambda). \end{align*} $$

Their multiscaling limits are obtained and investigated. For even-order equations, the rescaled solutions converge to Gaussian random fields that are stationary in space but nonstationary in time.

For odd-order heat equations, such as

$$ \begin{align*} \frac{\partial u(t, x)}{\partial t} = \mu \frac{\partial^m u(t, x)}{\partial x^m}, \quad t> 0,\ x \in \mathbb{R},\ \mu=\pm1,\ m = 2k + 1 \ (k\in \mathbb{N}), \end{align*} $$

standard scaling limits do not exist. To overcome this difficulty, kernel-averaged solutions are introduced, leading to well-defined nondegenerate limits.

The second class considered comprises fractional differential equations of the form

$$ \begin{align*} \frac{\partial^{\,\beta} u(t,x)}{\partial t^{\,\beta}} = -\mu (I-\Delta)^{\gamma/2}(-\Delta)^{\alpha/2} u(t,x), \quad t>0, \end{align*} $$

where $\Delta $ is the Laplace operator, and $-(I-\Delta )^{\gamma /2}$ and $(-\Delta )^{\alpha /2}$ are interpreted as the inverse operators of Bessel and Riesz potentials. Their solutions admit the spectral representation

$$ \begin{align*} u(t,x)=\int_{\mathbb{R}} e^{i\lambda x} E_\beta(-\mu |\lambda|^\alpha (1+|\lambda|^2)^{\gamma/2} t^{\,\beta}) Z(d\lambda). \end{align*} $$

The asymptotic behaviour of the rescaled solutions

$$ \begin{align*} U_\varepsilon(t,x) =\varepsilon^{-\beta/2\alpha} u\bigg(\frac{t}{\varepsilon},\frac{x}{\varepsilon^{\,\beta/\alpha}}\bigg) \end{align*} $$

is investigated. It is shown that, as $\varepsilon \to 0$ , the finite-dimensional distributions of $U_\varepsilon (t,x)$ converge to spatiotemporal Gaussian random fields that are stationary in space and nonstationary in time,

$$ \begin{align*} U_0(t,x) =\int_{\mathbb{R}} e^{i\lambda x} E_\beta(-\mu t^\beta |\lambda|^\alpha)\,W(d\lambda). \end{align*} $$

Many results of the modern asymptotic theory use regular variation to generalise classical limit theorems (see [Reference Alodat, Leonenko and Olenko4]). This thesis establishes multiscaling limit theorems for the case of regularly varying asymptotics described by slowly varying functions $L_{j}(\cdot )$ . It is interesting that the limit behaviour is different and depends on the locations of spectral singularities. Consider, for example, the case of a singularity at the origin. If $\varepsilon \to 0$ , the finite-dimensional distributions of the random fields with $A_0\neq 0$ ,

$$ \begin{align*} U^{\,(L)}_\varepsilon(t,x) = \frac{\varepsilon^{-{\kappa_0\beta}/{2\alpha}}}{L_{0}({\varepsilon^{-\beta/\alpha}})}u\bigg( \frac{t}{\varepsilon}, \frac{x}{\varepsilon^{\,\beta/\alpha}} \bigg), \quad t>0, \ x\in \mathbb{R}, \end{align*} $$

converge weakly to the finite-dimensional distributions of the zero-mean Gaussian field.

The results demonstrate that cyclic spectral singularities substantially affect the asymptotic behaviour of solutions and lead to new types of multiscaling limit theorems. The limit random fields obtained are stationary in space but not in time, and their covariance and spectral structures are explicitly derived. Various numerical examples are presented in the thesis to illustrate the theoretical results.