Skew SHE with Local-Time Drift

• Skew SHE with local-time drift is a singular stochastic PDE that generalizes the classical heat equation by incorporating a spatially varying local time drift.
• It employs rigorous Dirichlet form theory and finite-dimensional Galerkin projection to construct a Markov process capturing skew interactions.
• The analysis provides practical insights into approximation schemes for non-log-concave Gibbs measures and links to skew Brownian motion.

The skew stochastic heat equation (skew SHE) with local-time drift is a singular stochastic partial differential equation (SPDE) characterized by the introduction of a non-standard drift term involving the local time of the solution process. This equation generalizes the classical stochastic heat equation by incorporating a spatially-varying, potentially discontinuous drift that acts via the spatial local time—a quantitative measure of how much time the process spends at particular spatial levels. The model is closely related to multi-dimensional analogues of skew Brownian motion and reflects fine probabilistic and analytic properties. The primary rigorous study and construction for the skew SHE with local-time drift is detailed in the work by Bounebache and Zambotti (Bounebache et al., 2011).

1. Formal Definition and Equation Structure

Consider the spatial domain $x\in[0,1]$ with homogeneous Dirichlet boundary conditions and initial data $u_0\in L^2(0,1)$. The skew SHE with local-time drift is given by

$$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$

with boundary and initial conditions

$$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$

Here, $\dot W(t,x)$ denotes space-time white noise, and $f:\R\to\R$ is a function of bounded variation, admitting a Lebesgue–Stieltjes decomposition

$$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$

where $f_0$ is a bounded continuous density and the sum represents atomic components. For each $t\ge0$, $x\in[0,1]$, the map $u_0\in L^2(0,1)$0 admits a jointly continuous local time

$u_0\in L^2(0,1)$1

and $u_0\in L^2(0,1)$2 denotes its spatial derivative, in the sense of distributions when $u_0\in L^2(0,1)$3 has atoms.

2. Construction and Interpretation of the Local-Time Drift

The local-time drift captures singular interactions of the solution $u_0\in L^2(0,1)$4 with prescribed spatial levels, as encoded by the function $u_0\in L^2(0,1)$5. The occupation-time formula,

$u_0\in L^2(0,1)$6

renders the distribution of $u_0\in L^2(0,1)$7 in terms of local times, for $u_0\in L^2(0,1)$8. The drift term

$u_0\in L^2(0,1)$9

serves as a spatially singular "skew forcing", where the sign and amplitude of $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$0 determine the direction and intensity of the drift at given levels. The analogy with skew Brownian motion is precise: the measure $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$1 assigns "weights" to the local time at each level $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$2, thus biasing the SPDE's excursions above and below these levels.

The definition of the drift relies on the regularity and tightness properties of the local times. Well-posedness is established using an integration-by-parts formula for functions $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$3 (with $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$4), involving the invariant measure (see below), and a priori estimates for occupation measures and local times.

3. Markovian Structure and Dirichlet Formulation

The probabilistic construction hinges on Dirichlet form theory. The reference Gaussian structure is given by the law $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$5 of the standard Brownian bridge on $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$6, which is a centered Gaussian measure on $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$7 with covariance $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$8, where $$\partial_t u(t,x) = \tfrac12\,\partial^2_{x}u(t,x) + \dot W(t,x) + \frac12\int_\R f(da)\,\partial_x L^a_t(u(\cdot,x)),$$9 with Dirichlet boundary data. The unperturbed Dirichlet form is

$$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$0

with core given by cylindrical exponentials. The addition of the skew local-time drift introduces a Gibbs-type product structure, leading to a new invariant measure.

The full Dirichlet form for the equation is

$$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$1

where $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$2 is the non-log-concave Gibbs measure. By a perturbative analysis utilizing the boundedness of $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$3, the Dirichlet form is shown to be closable and quasi-regular, ensuring the existence of an associated Markov process $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$4 on $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$5.

A main conclusion is as follows: There exists a quasi-regular symmetric Dirichlet form $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$6 in $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$7 whose associated Hunt process $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$8 solves the skew SHE in the weak (martingale) sense, with the singular drift specified by the local times. Uniqueness in law remains unresolved in full generality due to the possible lack of log-concavity of $$u(t,0) = u(t,1) = 0,\qquad u(0,x) = u_0(x),\quad x\in[0,1].$$9, although the constructed process is canonical as the limit of natural approximation schemes.

4. Invariant Measure and Stationarity

The explicit invariant measure for the skew SHE is a Gibbs measure of the form

$\dot W(t,x)$0

with $\dot W(t,x)$1 (the Brownian-bridge reference) and normalizing constant $\dot W(t,x)$2. The corresponding potential,

$\dot W(t,x)$3

is generally non-convex, potentially featuring both upward and downward jumps via the atomic components of $\dot W(t,x)$4. Stationarity of the process under $\dot W(t,x)$5 is verified via the integration-by-parts identity and the symmetry of the Dirichlet form. The drift induced by the local time acts as the generalized gradient of the potential $\dot W(t,x)$6 in a distributional (local-time) sense.

5. Approximation Methods and Convergence Theory

Two principal families of approximation are employed to construct the skew SHE with local-time drift:

5.1 Regularization of the Singular Drift

A sequence of smooth, bounded, bounded-variation functions $\dot W(t,x)$7 is chosen with $\dot W(t,x)$8 pointwise outside the jump set of $\dot W(t,x)$9. Defining

$f:\R\to\R$0

the corresponding Dirichlet forms $f:\R\to\R$1 in $f:\R\to\R$2 are built analogously. Using Mosco (or T-) convergence of Dirichlet forms, it is shown that $f:\R\to\R$3, implying convergence in law of the stationary solutions under $f:\R\to\R$4 (in $f:\R\to\R$5) to those under $f:\R\to\R$6.

5.2 Finite-Dimensional Galerkin Projection

Letting $f:\R\to\R$7 be the subspace of piecewise-constant functions on dyadic partitions, with $f:\R\to\R$8 the orthogonal projector, consider $f:\R\to\R$9 (the finite-dimensional projection of the Brownian bridge law). Defining

$$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$0

on $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$1, and

$$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$2

it is shown that $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$3 (Mosco convergence), with the finite-dimensional Markov processes being systems of interacting skew Brownian motions with discrete local-time interactions at the jump levels of $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$4. These processes converge in law (in $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$5) to the infinite-dimensional skew SHE.

6. Analytical Estimates and Technical Framework

Key assumptions are that $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$6 is a bounded, bounded-variation function, the initial datum $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$7 (or under $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$8 for stationarity), and that the jump magnitudes $$f(da) = f_0(a)\,da + \sum_{j=1}^m a_j\,\delta_{y_j}(da),$$9 satisfy $f_0$0, ensuring the well-posedness of the finite-dimensional skew Brownian motions.

Technical control is achieved through a priori estimates of the modulus of continuity for the stationary infinite-dimensional Markov process $f_0$1 in $f_0$2. For Sobolev–Slobodeckii exponents $f_0$3 and $f_0$4 with $f_0$5 and $f_0$6, one obtains

$f_0$7

for some $f_0$8. Stationarity and Kolmogorov’s criterion yield tightness in the required function spaces. Core analytical tools include the Lyons–Zheng decomposition and Burkholder–Davis–Gundy inequalities, providing control in $f_0$9-type norms and, via Sobolev embeddings, uniform-space control.

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For a detailed and rigorous exposition of the skew stochastic heat equation with local-time drift, including proofs and full technical development, see Bounebache and Zambotti, "A skew stochastic heat equation" (Bounebache et al., 2011).