A unified fixed point approach to study the existence and uniqueness of solutions to the generalized stochastic functional equation emerging in the psychological theory of learning

<abstract><p>The model of decision practice reflects the evolution of moral judgment in mathematical psychology, which is concerned with determining the significance of different options and choosing one of them to utilize. Most studies on animals behavior, especially in a two-choice situation, divide such circumstances into two events. Their approach to dividing these behaviors into two events is mainly based on the movement of the animals towards a specific choice. However, such situations can generally be divided into four events depending on the chosen side and placement of the food. This article aims to fill such gaps by proposing a generic stochastic functional equation that can be used to describe several psychological and learning theory experiments. The existence, uniqueness, and stability analysis of the suggested stochastic equation are examined by utilizing the notable fixed point theory tools. Finally, we offer two examples to substantiate our key findings.</p></abstract>

Effect of quantum fluctuations on soliton regimes in microlasers

We present a theoretical investigation of effect of quantum fluctuations on laser solitons. Derivation of the stochastic equation, linearized with respect to quantum perturbations is carried out and the solutions are found. Explicit expressions are obtained for the time dependence of the soliton coordinates and momentum dispersion (variance) for perturbations averaged over the reservoir. It is shown that the dispersion of the soliton momentum becomes constant. It is shown that the dispersion of quantum perturbations tends to infinity near the Andronov-Hopf bifurcation threshold. The magnitude of quantum perturbations near the threshold of the appearance of hysteresis is estimated. It is shown that quantum perturbations do not significantly noise the soliton profile even with a very low intensity tending to zero. The number of photons in such solitons without supporting radiation, can reach unity.

MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

Quantum State Evolution in an Environment of Cosmological Perturbations

We study the pure and thermal states of quantized scalar and tensor perturbations in various epochs of Universe evolution. We calculate the density matrix of non-relativistic particles in an environment of these perturbations. We show that particle’s motion can be described by a stochastic equation with a noise coming from the cosmological environment. We investigate the squeezing of Gaussian wave packets in different epochs and its impact on the noise of quantized cosmological perturbations.

Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays

<p style='text-indent:20px;'>We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.</p>

Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

Mild Solution for a Stochastic Partial Differential Equation with Noise

This paper focuses on the study of the existence of a mild solution to time and space-fractional stochastic equation perturbed by multiplicative white noise. The required results are obtained by means of Sadovskii’s fixed point theorem.

On the distribution of fluxes of gamma-ray blazars: hints for a stochastic process?

ABSTRACT We examine a model for the observed temporal variability of powerful blazars in the γ-ray band in which the dynamics is described in terms of a stochastic differential equation, including the contribution of a deterministic drift and a stochastic term. The form of the equation is motivated by the current astrophysical framework, accepting that jets are powered through the extraction of the rotational energy of the central supermassive black hole mediated by magnetic fields supported by a so-called magnetically arrested accretion disc. We apply the model to the γ-ray light curves of several bright blazars and we infer the parameters suitable to describe them. In particular, we examine the differential distribution of fluxes (dN/dFγ) and we show that the predicted probability density function for the assumed stochastic equation naturally reproduces the observed power-law shape at large fluxes $\mathrm{ d}N/\mathrm{ d}F_{\gamma } \propto F_{\gamma }^{-\alpha }$ with α &gt; 2.