[Paper Review] Towards a characterization of Markov processes enjoying the time-inversion property

Stephan Lawi|ArXiv.org|Jun 1, 2005

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper provides a necessary and sufficient condition for homogeneous Markov processes in $\mathbb{R}^n$ to enjoy the time-inversion property of degree $\alpha$, characterized by a specific form of semigroup densities involving functions $\Phi$, $\theta$, and $\rho$ with homogeneity and scaling properties. The result generalizes known processes like Bessel and Dunkl processes and introduces a new matrix-valued jump process (skew-Wishart) that satisfies the time-inversion property.

ABSTRACT

We give a necessary and sufficient condition for a homogeneous Markov process taking values in $\R^n$ to enjoy the time-inversion property of degree $α$. The condition sets the shape for the semigroup densities of the process and allows to further extend the class of known processes satisfying the time-inversion property. As an application we recover the result of Watanabe in \cite{Wa1975} for continuous and conservative Markov processes on $\R_+$. As new examples we generalize Dunkl processes and construct a matrix-valued process with jumps related to the Wishart process by a skew-product representation.

Motivation & Objective

• To characterize all homogeneous Markov processes in $\mathbb{R}^n$ that enjoy the time-inversion property of degree $\alpha$.
• To provide a necessary and sufficient condition on semigroup densities for time-invariance under time inversion.
• To extend the class of known processes with this property, including those with jumps and matrix-valued paths.
• To recover Watanabe's result on diffusions on $\mathbb{R}_+$ using the new characterization.
• To construct new examples, such as a skew-Wishart process, via skew-product representations.

Proposed method

• Derive the transitional density of the time-inverted process $t^\alpha X_{1/t}$ using the original semigroup densities $p_t(x,y)$.
• Use Doob's $h$-transform to identify transformations preserving the time-inverted process, leading to equivalence classes of processes.
• Establish a functional form for $p_t(x,y)$ such that the time-inverted process remains homogeneous, involving $\Phi$, $\theta$, and $\rho$ with specific homogeneity degrees.
• Impose scaling conditions: $\Phi(\lambda x,y) = \Phi(x,\lambda y)$, $\rho(\lambda x) = \lambda^{2/\alpha}\rho(x)$, $\theta(\lambda y) = \lambda^\beta \theta(y)$.
• Verify that the semigroup densities of the time-inverted process match those of a homogeneous Markov process under this form.
• Construct a matrix-valued process with jumps (skew-Wishart) by combining a Wishart process with a Poisson process, showing it satisfies the time-inversion condition.

Experimental results

Research questions

• RQ1 What necessary and sufficient condition on semigroup densities ensures that a homogeneous Markov process on $\mathbb{R}^n$ enjoys the time-inversion property of degree $\alpha$?
• RQ2 How can the class of processes with the time-inversion property be extended beyond diffusions to include jump processes and matrix-valued processes?
• RQ3 Can the characterization recover Watanabe's result on conservative diffusions on $\mathbb{R}_+$?
• RQ4 What is the role of Doob's $h$-transform in classifying processes with equivalent time-inverted dynamics?
• RQ5 How can a skew-product representation generate new matrix-valued processes with the time-inversion property?

Key findings

• The paper establishes that a Markov process enjoys the time-inversion property of degree $\alpha$ if and only if its semigroup densities are of the form $p_t(x,y) = t^{-n\alpha/2} \Phi(x/t^{\alpha/2}, y/t^{\alpha/2}) \theta(y/t^{\alpha/2}) \exp(\rho(x/t^{\alpha/2}) + \rho(y/t^{\alpha/2}))$, with $\Phi$, $\theta$, and $\rho$ satisfying specific homogeneity conditions.
• The symmetry condition $\Phi(x,y) = \Phi(y,x)$ implies a relation between the time-inverted semigroup and the original one: $q_t^{(x)}(a,b) = \frac{\Phi(x,b)}{\Phi(x,a)} \exp(t\rho(x)) p_t(a,b)$.
• The skew-Wishart process, constructed via a skew-product of a Wishart process and a Poisson process, is shown to satisfy the time-inversion property of degree $\alpha=2$.
• The generalized Dunkl process fits the characterization, extending known results to a broader class of jump processes.
• The characterization recovers Watanabe's result on diffusions on $\mathbb{R}_+$, providing an alternative proof via the functional form of semigroup densities.
• The semigroup densities of the skew-Wishart process are explicitly derived using hypergeometric functions and zonal polynomials, confirming their consistency with the time-inversion condition.