In this short article, we shall study one-dimensional local Dirichlet spaces. One result, which has its independent interest, is to prove that irreducibility implies the uniqueness of symmetrizing measure for right Markov processes. The other result is to give a representation for any 1-dim local, irreducible and regular Dirichlet space and a necessary and sufficient condition for a Dirichlet space to be regular subspace of another Dirichlet space.
A new result for the strong uniqueness for catalytic branching diffusions is established, which improves the work of Dawson, D.A.; Fleischmann, K.; Xiong, J.[Strong uniqueness for cyclically symbiotic branching diffusions. Statist. Probab. Lett. 73, no. 3, 251--257 (2005)].
A new proof of pathwise uniqueness for SDEs with Sobolev diffusion and integrable drift term is introduced by extending a method from E. Fedrizzi and F. Flandoli (Pathwise uniqueness and continuous dependence of SDEs with non-regular drift, Stochastics 83 (2011), pp. 241--257) to the case of nonconstant diffusion.
We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Young graph with the Jack edge multiplicities. Each of the limit Markov processes is ergodic and its stationary distribution is a symmetrizing measure. The infinitesimal generators of the processes are exp…
In this paper we prove a new strong uniqueness result and a weak existence result for possibly {\it degenerate} multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples with Hölder diffusion coefficients are provided to show our results.
A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of "reflection structure" which ensures the existence of such couplings. In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases. We also prove that a Kendall-Cranston coupling is maximal under the reflection structure.
In this paper we are concerned with the stochastic partial differential equations of super-fast diffusion processes describing behavior of plasma dX(t)-Δln(X(t)+1)dt=\surd(Q)dW(t), in (0,T)\timesO, where O is a bounded open subset of R. We define a strong solution adequate to the properties of the natural logarithm and we prove the corresponding existence and uniqueness result.
We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given smooth surface and having the properties of skewing and delaying. The theorem on the existence of no more than one solution to the problem is proved.
Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.
This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes $(X(t),Λ(t))$, where $Λ(t)$ is a component representing discrete events taking values in a countably infinite set. Considering the corresponding stochastic differential equations, our main focus is on treating those with non-Lipschitz coefficients. We first show that there exists a unique strong solution to the corresponding stochastic differential equation. Then Feller and strong Feller properties are investigated.
This paper is concerned with the nonlinear filtering problem for a general Markovian partially observed system (X,Y), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t, the sigma-algebra generated by the observation process Y provides all the available information about the signal X. The central goal of stochastic filtering is to characterize the filter which is the conditional distribution of X, given the observed data. It has been proved in Ceci-Colaneri (2012) that the filter is the unique probability measure-valued process satisfying a nonlinea…
In this paper, we develop a general methodology to prove weak uniqueness for stochastic differential equations with coefficients depending on some path-functionals of the process. As an extension of the technique developed by Bass \& Perkins [BP09] in the standard diffusion case, the proposed methodology allows one to deal with processes whose probability laws are singular with respect to the Lebesgue measure. To illustrate our methodology, we prove weak existence and uniqueness in two examples : a diffusion process with coefficients depending on its running symmetric local time and a diffusio…
One proves the uniqueness of distributional solutions to nonlinear Fokker--Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean--Vlasov stochastic differential equation (SDE).
We construct the Dirichlet form associated with the dynamical $Φ^4_3$ model obtained in [Hai14, CC13] and [MW16]. This Dirichlet form on cylinder functions is identified as a classical gradient bilinear form. As a consequence, this classical gradient bilinear form is closable and then by a well-known result its closure is also a quasi-regular Dirichlet form, which means that there exists another (Markov) diffusion process, which also admits the $Φ^4_3$ field measure as an invariant (even symmetrizing) measure.
We study the martingale problem associated with the operator $L u = \partial_s u + 1/2 \sum_{i,j=1}^{d_0} a^{ij} \partial_{ij} u + \sum_{i,j=1}^d B^{ij} x^j \partial_i u$, where $d_0 \leq d$. We show that the martingale problem is well-posed when the function $a$ is continuous and strictly positive-definite on $\bb R^{d_0}$ and the matrix $B$ takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when $B$ is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.
We consider stochastic differential equations with (oblique) reflection in a $2$-dimensional domain that has a cusp at the origin, i..e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leqδ_0,ψ_1(x_1)<x_2<ψ_ 2(x_1)\}$, with $ψ_1(0)=ψ_2(0)=0$, $ψ_1'(0)=ψ_2'(0)=0$. Given a vector field $γ$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $γ^i(0):=\lim_{x_1\rightarrow 0^{+}}γ(x_1,ψ_i(x_1))$, $ i=1,2,$ and assuming there exists a vector $e^{*}$ such that $\langle e^{*},γ^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>…
In this paper we consider stochastic differential equations with discontinuous diffusion coefficient of varying sign, for which weak existence and uniqueness holds but strong uniqueness fails. We introduce the notion of $φ$-strong solution, and we show that under certain conditions on the diffusion coefficient a $φ$-strong solution exists and it is unique. We also give a construction of a $φ$-strong solution and a weak solution in terms of the notion of \emph{i.i.d. sign choice} introduced in the paper, and we give an explicit representation of the set of all weak solutions, which in particula…
We consider a transport-diffusion equation forced by random noise of three types: additive, linear multiplicative in It$\hat{\mathrm{o}}$'s interpretation, and transport in Stratonovich's interpretation. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport-diffusion equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.
In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions to SDE's. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate Markov property. To prove uniqueness we solve a general martingale problem for \cadlag processes. This result is of independent interest. In the last section we present application of our results considering generalized exponential Levy model.
We study the inverse boundary crossing problem for diffusions. Given a diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[τ>t]$, where $τ$ is the first hitting time of $X_t$ to the boundary $b(t)$. The approach taken is analytic, based on solving a parabolic variational inequality to find $b$. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary $b$ does indeed have $p$ as its boundary cros…
We investigate the existence and uniqueness of strong solutions up to an explosion time for regime-switching diffusion processes in an infinite state space. Instead of concrete conditions on coefficients, our existence and uniqueness result is established under the general assumption that the diffusion in every fixed environment has a unique non-explosive strong solution. Moreover, non-explosion conditions for regime-switching diffusion processes are given. The strong Feller property is proved by further assuming that the diffusion in every fixed environment generates a strong Feller semigroup…
We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to $\mathbb{R}^d$, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in energy markets. We prove existence and uniqueness of strong solutions. In addition we study the strong convergence order of the Euler-Maruyama scheme and recover the optimal rate $1/2$.
This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples include the unit ball, the product of the unit cube and nonnegative orthant, and the unit simplex.
For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability measures. Consequently, the generator of the regime-switching diffusions process has an extension in the $L^1$-space w.r.t the invariant probability measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness of the extension. Regularities and the uniqueness of the invariant probability density w.r.t the nice reference probability measure ar…
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $λu-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as so…
For continuous γ, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}γ(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise uniqueness holds when γand g are Lipschitz and \frac{g(1)}{γ^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with γ=\sqrt2 and g\equiv c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C^3 boundary and suppose h:\barD\to R Lipschitz on \barD…
Using elliptic and parabolic regularity results in $L^p$-spaces and generalized Dirichlet form theory, we construct for every starting point weak solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt process wit…
Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set $[0,\infty)^n\times\mathbb{R}^m$, and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero proportional to the distance to the boundary of the domain, and …
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.
The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtain…
In this article we show that the ordinary stochastic differential equations of K.Itô maybe considered as part of a larger class of second order stochastic PDE's that are quasi linear and have the property of translation invariance. We show using the `monotonicity inequality' and the Lipshitz continuity of the coefficients $σ_{ij}$ and $b_i$, existence and uniqueness of strong solutions for these stochastic PDE's. Using pathwise uniqueness, we prove the strong Markov property.
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particular to CR Sasakian manifolds with Tanaka-Webster-Ricci curvature bounded from below and Carnot groups of step two.
We prove strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised kernels. We apply these results to the rough Heston model, with square-root diffusion coefficient, recently proposed in Mathematical Finance to model the volatility of asset prices.
We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hörmander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Va…
Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes
Al.I. Cuza University（阿尔·伊·库扎大学）；Octav Mayer Institute of Mathematics of Romanian Academy（罗马尼亚科学院奥克塔夫·迈尔数学研究所）；Faculty of Mathematics, Bielefeld University（比勒费尔德大学数学学院）；Academy for Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳）数据科学学院）
In this article, we study stochastic partial differential equations with two reflecting walls, driven by space-time white noise with non-constant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. The strong Feller property is also obtained.
The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and be almost in VMO (vanishing mean oscillation) with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic Itô equations in the class of "good" solutions (which is nonempty). This also implies uniqueness of a Green's function in the class of "good" ones (which is always nonempty)…
We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure $L^2$ condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function $a(t,x,u)=Cu^γ$, $1/2\leγ\le1$ and $C\ne0$. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.
