Related papers: Quantitative heat kernel estimates for diffusions …
Timeseries generated from a dynamical source can often be modeled as sample paths of a stochastic differential equation (SDE). The timeseries thus reflects the motion of a particle which flows along the direction provided by a drift /…
We consider a system of $d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle $S^1$. We obtain sharp results on the H\"older continuity in time of the paths of the…
In the paper the principal result obtained is the estimate for the heat kernel associated to the Schr\"odinger type operator $(1+|x|^\alpha)\Delta-|x|^\beta$ \[ k(t,x,y)\leq Ct^{-\frac{\theta}{2}}\frac {\varphi(x)\varphi(y)}{1+|x|^\alpha},…
In these lecture notes, we explore the mathematical preliminaries and foundational concepts that connect stochastic processes with partial differential equations. We begin by investigating Brownian motion, which serves as a model for random…
We study the estimation of the invariant density of additive fractional stochastic differential equations with Hurst parameter $H \in (0,1)$. We first focus on continuous observations and develop a kernel-based estimator achieving faster…
For $d\geq 1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}; a \in [0, 1]\}$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+…
In this paper, we study a conditional distribution dependent stochastic differential equations driven by standard Brownian motion and fractional Brownian motion with Hurst exponent $H>\frac{1}{2}$ simultaneously. First, the existence and…
Avikainen provided a sharp upper bound of the difference $\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ by the moments of $|X-\widehat{X}|$ for any one-dimensional random variables $X$ with bounded density and $\widehat{X}$, and function of…
The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in…
We provide a stochastic fractional diffusion equation description of energy transport through a finite one-dimensional chain of harmonic oscillators with stochastic momentum exchange and connected to Langevian type heat baths at the…
For a semigroup $P_t$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_t(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on…
We prove some estimations of the correlation of two local observables in quantum spin systems (with Schr\"odinger equations) at large temperature. For that, we describe the heat kernel of the Hamiltonian for a finite subset of the lattice,…
The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions.…
In this article, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated…
Let $(M, g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 2$. Consider the conformal perturbation $\tilde{g}=h g$ where $h$ is a smooth bounded positive function on $M$. Denote by $\tilde{p}_t(x,y)$ the heat kernel of manifolds…
We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close…
We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.
In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in (1/2, 1)$ when the coefficient vector fields satisfy…
We consider a stochastic differential equation of the form \[dX_t=\theta a(t,X_t)\,dt+\sigma_1(t,X_t)\sigma_2(t,Y_t)\,dW_t\] with multiplicative stochastic volatility, where $Y$ is some adapted stochastic process. We prove…
We prove upper and lower bounds of the heat kernel for the operator $\Delta-\nabla (\frac{1}{|x|^{\alpha}})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $\alpha >0$. We obtain these bounds from an isoperimetric inequality for a…