Related papers: Quantitative heat kernel estimates for diffusions …
Consider the stochastic partial differential equation $$ \frac{\partial }{\partial t}u_t(\mathbf{x})= -(-\Delta)^{\frac{\alpha}{2}}u_t(\mathbf{x}) +b\left(u_t(\mathbf{x})\right)+\sigma\left(u_t(\mathbf{x})\right) \dot F(t, \mathbf{x}), \ \…
We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $\mathbb{R}^d$. An important special case is the time-fractional diffusion equation, which has seen much…
We consider a mixed stochastic differential equation $d{X_t}=a(t,X_t)d{t}+b(t,X_t) d{W_t}+c(t,X_t)d{B^H_t}$ driven by independent multidimensional Wiener process and fractional Brownian motion. Under Hormander type conditions we show that…
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C^{1,1} open sets D in R^d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided…
In this paper, we establish the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain $D$ in the setting of $L^2(D)$ space. The result is valid for all…
A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a…
We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with…
The first heat kernel coefficients are calculated for a dispersive ball whose permittivity at high frequency differs from unity by inverse powers of the frequency. The corresponding divergent part of the vacuum energy of the electromagnetic…
In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric…
We investigate the fractional Hardy-H\'enon equation with fractional Brownian noise $$ \partial_tu(t)+(-\Delta)^{\theta/2} u(t)=|x|^{-\gamma} |u(t)|^{p-1}u(t)+\mu \, \partial_t B^H(t), $$ where $\theta>0$, $p>1$, $\gamma\geq 0$, $\mu…
We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behaviour, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the…
The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $M_\gamma$, formally…
Starting from the Caldeira-Leggett (CL) model, we derive the equation describing the Quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms.…
Let $u(t,x)$ be the solution to a stochastic heat equation $$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$ with initial condition…
We present a detailed study of the quantum dissipative dynamics of a charged particle in a magnetic field. Our focus of attention is the effect of dissipation on the low- and high-temperature behavior of the specific heat at constant…
We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic upper bounds…
In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent…
Motivated by the spectral theory of relativistic atoms, we prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis is based on suitable supermedian functions,…
The stochastic dissipative Schrodinger equation is derived for an open quantum system consisting of a sub-system able to exchange energy with a thermal reservoir. The resultant evolution of the wave function also gives the evolution of the…
We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,…