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In this paper, we aim to develop a new weak formulation that ensures well-posedness for a broad range of stochastic partial differential equations with pseudo-differential operators whose symbols depend only on time and spatial frequencies. The main focus of this paper is to relax the conditions on the symbols of pseudo-differential operators and data while still ensuring that the stochastic partial differential equations remain well-posed in a weak sense. Specifically, we allow symbols to be random and remove all regularity and ellipticity conditions on them. As a result, our main operators include many interesting rough operators that cannot generate any regularity gain or integrability improvement from the equations. In addition, our data do not need to be regular or possess finite stochastic moments., Comment: 91 pages

This article investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic reaction-diffusion-advection equations (SRDAEs) with spatially homogeneous colored noises and variable-order nonlocal operators in mixed norm $L_q(L_p)$-spaces. We introduce a new condition (strongly reinforced Dalang's condition) on colored noise, which facilitates a deeper understanding of the complicated relation between nonlinearities and stochastic forces. Additionally, we establish the space-time H\"older type regularity of solutions., Comment: 47 pages

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-\Delta)$ and a second-order differential operator whose leading coefficients are not positive semi-definite., Comment: 49 pages

This paper investigates the existence, uniqueness, and regularity of solutions to evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces. We also explore trace embedding and continuity of solutions., Comment: 18 pages

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse H\"older inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights., Comment: 36 pages

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions., Comment: few glitches corrected, Appendix B for finite time case added

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$ \mathcal{L}^{\vec{a},\vec{0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb{R}}\left( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) \right) \frac{a_{i}(t,y^{i})}{|y^{i}|^{1+\alpha_{i}}} \mathrm{d}y^{i} . $$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calder\'on-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order., Comment: 47 pages

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(t,x)=\psi(t,-i\nabla)u(t,x)+f(t,x),\quad &(t,x)\in(0,T)\times\mathbb{R}^d,\\ u(0,x)=0,\quad & x\in\mathbb{R}^d \end{cases} \end{equation} in (Muckenhoupt) weighted $L_p$-spaces with time-measurable pseudo-differential operators \begin{equation} \label{ab op} \psi(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[\psi(t,\cdot)\mathcal{F}[u](t,\cdot)\right](x). \end{equation} More precisely, we find sufficient conditions of the symbol $\psi(t,\xi)$ (especially depending on the smoothness of the symbol with respect to $\xi$) to guarantee that equation is well-posed in (Muckenhoupt) weighted $L_p$-spaces. Here the symbol $\psi(t,\xi)$ is merely measurable with respect to $t$, and the sufficient smoothness of $\psi(t,\xi)$ with respect to $\xi$ is characterized by a property of each weight. In particular, we prove the existence of a positive constant $N$ such that for any solution $u$ to the equation, \begin{equation} \label{ab est} \int_0^T \int_{\mathbb{R}^d} |(-\Delta)^{\gamma/2} u(t,x) |^p (t^2 + |x|^2)^{\alpha/2} \mathrm{d}x\mathrm{d}t \leq N\int_0^T \int_{\mathbb{R}^d} |f(t,x)|^p (t^2 + |x|^2)^{\alpha/2} \mathrm{d}x\mathrm{d}t \end{equation} and \begin{equation} \label{ab est 2} \int_0^T \left(\int_{\mathbb{R}^d} |(-\Delta)^{\gamma/2} u(t,x) |^p |x|^{\alpha_2} \mathrm{d}x \right)^{q/p} t^{\alpha_1}\mathrm{d}t \leq N\int_0^T \left(\int_{\mathbb{R}^d} |f(t,x) |^p |x|^{\alpha_2} \mathrm{d}x \right)^{q/p} t^{\alpha_1}\mathrm{d}t, \end{equation} where $p,q\in(1,\infty)$, $-d-1<\alpha < (d+1)(p-1)$, $-1 < \alpha_1 < q-1$, $-d <\alpha_2< d(p-1)$, and $\gamma$ is the order of the operator $\psi(t,-i\nabla)$., Comment: 41 pages, We corrected some typos

We study the zero exterior problem for the elliptic equation $$ \Delta^{\alpha/2}u-\lambda u=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $\alpha\in (0,2)$, $\lambda \geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and H\"older estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and H\"older regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp., Comment: 42 pages; Title changed; The paper has been accepted for publication in "Discrete and Continuous Dynamical Systems (DCDS)"

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,\Lambda_t)_{t\geq0}$. Suppose that for any Schwartz function $\varphi$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[\varphi(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|\psi^{\mu}(r^{-1}D)\varphi\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_s$ is a positive constant related to $s$, $\mu$ is a symmetric L\'evy measure on $\mathbb{R}^d$, $\psi^{\mu}(r^{-1}D)\varphi(x)= \mathcal{F}^{-1} \left[ \psi^{\mu}(r^{-1}\xi) \mathcal{F}[\varphi]\right](x)$ and $$\psi^{\mu}(\xi):=\int_{\mathbb{R}^d}(e^{iy\cdot\xi}-1-iy\cdot\xi 1_{|y|\leq 1})\mu(dy).$$ In this paper, we establish the $L_p$-solvability to the initial value problem \begin{equation} \frac{\partial u}{\partial t}(t,x)=\mathcal{A}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in(0,T)\times\mathbb{R}^d, \end{equation} In other words, there exists a unique solution $u$ to equation satisfying $$ \|u\|_{L_q((0,T);H_p^{\mu;\gamma}(\mathbb{R}^d))}\leq N\|u_0\|_{B_{p,q}^{s;\gamma-\frac{2}{q}}(\mathbb{R}^d)}, $$ where $N$ is independent of $u$ and $u_0$, and the spaces $B_{p,q}^{s;\gamma-\frac{2}{q}}(\mathbb{R}^d)$ and $H_p^{\mu;\gamma}(\mathbb{R}^d)$ are scaled Besov spaces (see Definition 2.8) and generalized Bessel potential spaces (see Definition 2.3), respectively., Comment: 47 pages

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $\lambda \geq 0$ and the coefficients depend on $(\omega,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $\lambda$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$., Comment: 27 pages

We review the agent-based models (ABM) on social physics including econophysics. The ABM consists of agent, system space, and external environment. The agent is autonomous and decides his/her behavior by interacting with the neighbors or the external environment with the rules of behavior. Agents are irrational because they have only limited information when they make decisions. They adapt using learning from past memories. Agents have various attributes and are heterogeneous. ABM is a non-equilibrium complex system that exhibits various emergence phenomena. The social complexity ABM describes human behavioral characteristics. In ABMs of econophysics, we introduce the Sugarscape model and the artificial market models. We review minority games and majority games in ABMs of game theory. Social flow ABM introduces crowding, evacuation, traffic congestion, and pedestrian dynamics. We also review ABM for opinion dynamics and voter model. We discuss features and advantages and disadvantages of Netlogo, Repast, Swarm, and Mason, which are representative platforms for implementing ABM., Comment: 20pages, 2figures