In this paper, we study a particular model of distorted Brownian motion (dBM) on state spaces with varying dimension. Roughly speaking, the state space of such a process consists of two components: a $3$-dimensional component and a $1$-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the $3$- or $1$-dimensional component receives a strong "push" towards the origin. On each component, the "magnitude" of the "push" can be parametrized by a constant $γ>0$. In this article, using probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any $0<t<\infty$.
The Liouville Brownian motion was introduced in \cite{GRV} as a time changed process $B_{A_t^{-1}}$ of a planar Brownian motion $(B_t)_{t \ge 0}$, where $(A_t)_{t \ge 0}$ is the positive continuous additive functional of $(B_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville measure. We first consider a distorted Brownian motion $(X_t)_{t\ge0}$ starting from all points in $\R^2$ associated to a Dirichlet form $(\E, D(\E))$ (see \cite{ShTr14}). We show that the positive continuous additive functional $(F_t)_{t \ge 0}$ of $(X_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville distorted m…
Using Girsanov transformations we construct from sticky reflected Brownian motion on $[0,\infty)$ a conservative diffusion on $E:=[0,\infty)^n$, $n \in \mathbb{N}$, and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process, we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our s…
Roughly speaking, a space with varying dimension consists of at least two components with different dimensions. In this paper we will concentrate on the one, which can be treated as $\mathbb{R}^3$ tying a half line not contained by $\mathbb{R}^3$ at the origin. The aim is twofold. On one hand, we will introduce so-called distorted Brownian motions on this space with varying dimension (dBMVDs in abbreviation) and study their basic properties by means of Dirichlet forms. On the other hand, we will prove the joint continuity of the transition density functions of these dBMVDs and derive the short…
For any starting point in $\mathbb{R}^d$, we identify the stochastic differential equation that is satisfied by distorted Brownian motion with respect to a certain discontinuous Muckenhoupt $A_2$-weight $ψ$. The discontinuities of $ψ$ typically take place on a sequence of level sets of the Euclidean norm $D_k:=\{x\in \mathbb{R}^d\, | \;\|x\|=d_k\}$, $k\in\mathbb{Z}$, where $(d_k)_{k\in\mathbb{Z}}\subset (0,\infty)$ may have accumulation points and each level set $D_k$ plays the role of a permeable membrane.
Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of \cite…
We systematically develop general tools to apply Fukushima's absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density w.r.t. the reference measure and methods to estimate drift potentials comfortably. We then apply our results to distorted Brownian motions and construct weak solutions to singular stochastic differential equations, i.e. equations with possibly unbounded and discontinuous drift and reflection terms which may be the sum of countably many local times. The solutions…
We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E:=[0,\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary (sticky boundary behavior). In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly for quasi every starting point with respect to the associated Dirichlet form. That the boundary behavior of the constructed process indeed is …
We obtain the viscosity solution of G-heat equation with the initial condition $ϕ(x)=x^{n}$ for each integer $n\geq1$ using the method of G-Brownian motion.
In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and po…
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 Dirichlet heat kernel estimates hold for all t>0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C^{1,1} open sets, of such subordinate Brownian motions with Gaussian components.
We investigate the 3rd term of spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a, b) in a real line when the underlying subordinators are stable subordinators with index αis in (1, 2) or α= 1. We prove that in the 3rd term of spectral heat content, one can observe the length b-a of the interval D.
The spectral heat content is investigated for time-changed killed Brownian motions on C1,1 open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at \infty with index β\in (0, 1). In the case of inverse subordinators, the asymptotic limit of the spectral heat content is shown to involve a probabilistic term depending only on β\in (0, 1). In contrast, in the case of subordinators, this universality holds only when β\in ( 1/2 , 1).
We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $α/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $α\in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $α\in [d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of the open set. The m…
In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and Laplace exponent of the corresponding subordinator only.
In this paper we study the small time asymptotic behavior of the spectral heat content $\widetilde{Q}_D^{(α)}(t)$ of an arbitrary bounded $C^{1,1}$ domain $D$ with respect to the \textit{subordinate killed Brownian motion} in $D$ via an $(α/2)$-stable subordinator. For all $α\in (0,2)$, we establish a two-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$ in all dimensions. When $α\in (1,2)$ and $d\geq 2$, we establish a three-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$.
For a class of Laplace exponents we derive the heat trace asymptotics of the generator of the corresponding subordinate Brownian motion on Euclidean space. The terms in the asymptotic expansion are found to depend both on the geometry of Euclidean space and probabilistic properties of the subordinator. The key assumption is the existence of a suitable density for the Levy measure of the subordinator. An intermediate step is the computation of the zeta function of the generator. We employ methods from the theory of classical pseudodifferential operators on Euclidean space. The analysis is highl…
We show the existence of the scaling exponent $χ\in (0,4[(1+γ^2/4)- \sqrt{1+γ^4/16}]/γ^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $γ<2$ on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$ \lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^γ(u,v)| }{|\log t|}=\fracχ{2-χ}, \ \mbox{\rm a.s.} $$
In this article we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, e.g. \[ ϕ(λ)=\fracλ{\log(1+λ)}-1 \quad \text{ or }\quad ϕ(λ)=\fracλ{\log(1+λ^{β/2})},\ β\in (0,2)\, \] that correspond to subordinate Brownian motions with scaling order that is not necessarily strictly between 0 and 2. These est…
In this paper we consider $β[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus β[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $β[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus β[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus β[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$.
We prove tightness of a family of path measures $ν_{\varepsilon}$ on tubes $L(\varepsilon)$ of small diameters around a closed and connected submanifold $L$ of another Riemannian manifold $M$. Together with a convergence result for Dirichlet semigroups on tubular neighbourhoods, that implies weak convergence of the measures as the tube radius $\varepsilon$ tends to zero to a measure supported by the path space of the submanifold. As a consequence, we obtain weak convergence of the measures obtained by conditioning Brownian motion to stay within the tubes $L(\varepsilon)$ up to a finite time $T…
Consider the stochastic partial differential equation $\partial_t u = Lu+σ(u)ξ$, where $ξ$ denotes space-time white noise and $L:=-(-Δ)^{α/2}$ denotes the fractional Laplace operator of index $α/2\in(\nicefrac12\,,1]$. We study the detailed behavior of the approximate spatial gradient $u_t(x)-u_t(x-\varepsilon)$ at fixed times $t>0$, as $\varepsilon\downarrow0$. We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.
In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of $r^α$ and $t^{\frac{-α}{α+ 1}}$ respectively. These volume fluctuations are the same order as for t…
We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion.
We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, [G. Cannizzaro, L. Haunschmid-Sibitz, F. Toninelli, preprint arXiv:2106.06264] proved sharp $\sqrt{log}$-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-d Gaussian Free Field (GFF) $\underlineω$. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of $\underlineω$. Adapting their method, we show that if $s\ge1$, with $s=1$ corresponding to the standard stochastic heat equa…
In this paper we address again the problem of the connection between multitime Brownian sheet and heat type PDEs. The main results include: the volumetric character of the solutions of the forward (backward) diffusion-like PDEs; the forward mean value of a Brownian process as the solution of the forward heat PDE; the backward mean value of a Brownian process as the solution of the backward heat PDE; the multitime stochastic processes with volumetric dependence; the stochastic partial derivative of a stochastic process with respect to a multitime Wiener process; Hermite polynomials stochastic p…
We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger-Yau type lower bound for the heat kernels of such manifolds and also sharp Cheng's type estimates for the Dirichlet eigenvalues of metric balls.
In this paper we introduce and study Brownian motion on state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density functions (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Holder re…
Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of M with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in M. In particular, we prove a Gaussian type central limit theorem showing that when rescaled appropriately, the fluctuations of the abelianized winding are normally distributed with an explicit covariance matrix.
We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.
In [4], it is proved that we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard $d$-dimensional Brownian motion in moving boundaries in $\mathbb{R}^{d}$, $d\geq 2$, under a $C^{3}$-diffeomorphism. Similarly as in [4], by using a property of local time of standard $d$-dimensional Brownian motion and the heat equation with Diri…
Let us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=ρ\,{\rm sgn}(x)|x|^α/t^β$. This process can be viewed as a distorted Brownian motion in a potential, possibly singular, depending on time. After obtaining results on existence and uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters $ρ,α$ and $β$, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and e…
In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory.
We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.
The fundamental solution of the heat equation on $R^n$ is known as the heat kernel which is also the transition density of a Brownian motion. Similar statements hold when $\R^n$ is replaced by a Lie group. We briefly demonstrate how the results on $R^n$ concerning the heat kernel and Brownian motion may be easily transferred to compact Lie groups using the wrapping map of Dooley and Wildberger.
The main objective of the work is to provide sharp two-sided estimates of $λ$-Green function of hyperbolic Brownian motion of a half-space. We strongly rely on recent results obtained by K. Bogus and J. Malecki [3], regarding precise estimates of the Bessel heat kernel of half-lines.
We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2…
We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the restrictions in [11] (or [3]) that the underlying state space has to be bounded, and that the initial distribution of the limiting continuous process has to be its invariant distribution. The approach in this paper is that we first obtain heat kernel upper bounds for the approximating random walks that are uniform in their mesh size, by establishing a Nash-type inequal…
The aim of this paper is two-fold. On one hand, we will study the distorted Brownian motion on $\mathbb{R}$, i.e. the diffusion process $X$ associated with a regular and strongly local Dirichlet form obtained by the closure of $\mathscr{E}(f,g)=\frac{1}{2}\int_\mathbb{R} f'(x)g'(x)ρ(x)dx$ for $f,g\in C_c^\infty(\mathbb{R})$ on $L^2(\mathbb{R}, \mathfrak{m})$, where $\mathfrak{m}(dx)=ρ(x)dx$ and $ρ$ is a certain positive function. After figuring out the irreducible decomposition of $X$, we will present a characterization of that $X$ becomes a semi-martingale by virtue of so-called Fukushima's d…
In this paper, we study a two-point boundary value problem consisting of the heat equation on the open interval $(0,1)$ with boundary conditions which relate first and second spatial derivatives at the boundary points. Moreover, the unique solution to this problem can be represented probabilistically in terms of a sticky Brownian motion. This probabilistic representation is attained from the stochastic differential equation for a sticky Brownian motion on the bounded interval $[0,1]$.
We take the spaces introduced by Laakso in 2000 and building on the work of Barlow, Bass, Kumagai, and Teplyaev prove the existence and uniqueness of a local symmetry invariant diffusion via heat kernel estimates. This work also builds upon works of Cheeger, Barlow and Bass, as well as the author.
In this paper, we consider the strong convergence order of the exponential integrator for the stochastic heat equation driven by an additive fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$. By showing the strong order one of accuracy of the exponential integrator under appropriote assumptions, we present the first super-convergence result in temporal direction on full discretizations for stochastic partial differential equations driven by infinite dimensional fractional Brownian motions with Hurst parameter $H\in(\frac12,1)$. The proof is a combination of Malliavin calculus, …
A generical formalism for the discussion of Brownian processes with non-constant particle number is developed, based on the observation that the phase space of heat possesses a product structure that can be encoded in a commutative unit ring. A single Brownian particle is discussed in a Hilbert module theory, with the underlying ring structure seen to be intimately linked to the non-differentiability of Brownian paths. Multi-particle systems with interactions are explicitly constructed using a Fock space approach. The resulting ring-valued quantum field theory is applied to binary branching Br…
Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Saïd--Kobayashi--Ørsted $(k,a)$-generalized Laguerre semigroup theory
We study measures associated to Brownian motions on infinite-dimensional Heisenberg-like groups. In particular, we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness.
We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.
We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space-time white noise.
Let $X$ be a symmetric jump process on $\R^d$ such that the corresponding jumping kernel $J(x,y)$ satisfies $$J(x,y)\le \frac{c}{|x-y|^{d+2}\log^{1+\varepsilon}(e+|x-y|)}$$ for all $x,y\in\R^d$ with $|x-y|\ge1$ and some constants $c,\varepsilon>0$. Under additional mild assumptions on $J(x,y)$ for $|x-y|<1$, we show that $C\sqrt{r\log \log r}$ with some constant $C>0$ is an upper rate function of the process $X$, which enjoys the same form as that for Brownian motions. The approach is based on heat kernel estimates of large time for the process $X$. As a by-product, we also obtain two-sided he…
We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equation under…
Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion
Technische Universität Braunschweig（布伦瑞克工业大学）；Kobe University（神户大学）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理科学研究所在）；Massachusetts Institute of Technology（麻省理工学院）；University of Cambridge（剑桥大学）
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 time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos development for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.
We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.
Under a complete Ricci flow, we construct a coupling of two Brownian motion such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49--84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.
