Existence and uniqueness of reflecting diffusions in cusps

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 $\{(x_1,x_2):0<x_1\leq\delta_0,\psi_1(x_1)<x_2<\psi_ 2(x_1)\}$, with $\psi_1(0)=\psi_2(0)=0$, $\psi_1'(0)=\psi_2'(0)=0$. Given a vector field $\gamma$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $\gamma^i(0):=\lim_{x_1\rightarrow 0^{+}}\gamma (x_1,\psi_i(x_1))$, $ i=1,2,$ and assuming there exists a vector $e^{*}$ such that $\langle e^{*},\gamma^i(0)\rangle>0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin. Our proof uses a new scaling result and a coupling argument.

See Section 2 for the complete formulation of our assumptions.
Here we characterize the reflecting diffusion as the solution of a stochastic differential equation with reflection (SDER) which will always be a semimartingale. In particular, we recover the results by DeBlassie and Toby (1993a) and DeBlassie and Toby (1993b) for the cases when the process is a semimartingale, except for the case when γ 1 (0) and γ 2 (0) point at each other and β 2 < 2.
First, we show that our conditions imply that, starting away from the origin, the origin is never reached. Therefore we easily obtain strong existence and uniqueness of the reflecting diffusion from known results on existence and uniqueness in smooth domains (Section 3).
Moreover, the fact that, starting away from the origin, the reflecting diffusion is well defined for all times allows us to obtain a weak solution of the SDER starting at the origin as the limit of solutions starting away from the origin (Section 4.1). To this end, we employ a random time change of the SDER (the same that is used in Kurtz (1990) to obtain a solution of a patchwork martingale problem from a solution of the corresponding constrained martingale problem) that makes it particularly simple to prove relative compactness of the processes.
The main result of this paper, however, is weak uniqueness of the solution to the SDER starting at the origin (Section 4.3). Our assumptions on the direction of reflection guarantee that any solution starting at the origin immediately leaves it. Since the distribution of a solution starting away from the origin is uniquely determined, the distribution of a solution starting at the origin is determined by its exit distribution from an arbitrarily small neighborhood of the origin. The crucial ingredient that allows us to understand the behavior of the process near the origin is a scaling result (Section 4.2). Combined with a coupling argument based on Lindvall and Rogers (1986), this scaling result shows that indeed all solutions starting at the origin must have the same exit distribution from every neighborhood of the origin. For a more detailed discussion of our approach, see the beginning of Section 4. Some technical lemmas that are needed in our argument are proved in Section 5. The Feller property is proved in Section 4.4.
The most general uniqueness result for SDER in piecewise C 1 domains can be found in Dupuis and Ishii (1993). Reflecting diffusions in piecewise smooth domains are characterized as solutions of constrained martingale problems in Kurtz (1990), and Costantini and Kurtz (2015) reduces the problem of proving uniqueness for the solution of a constrained martingale problem (as well as of a martingale problem in a general Polish space) to that of proving a comparison principle for viscosity semisolutions of the corresponding resolvent equation. None of these results applies to the situation we are considering here. In particular, Dupuis and Ishii (1993) makes the assumption that the convex cone generated by the normal vectors at each point does not contain any straight line, which is violated at the tip of the cusp.
Finally, we wish to mention that our work was partly motivated by diffusion approximations for some queueing models where domains with cusplike singularities appear (in particular ). These models are in higher dimensions, but this paper is intended as a first contribution in the direction of understanding reflecting diffusions in such domains.
The notation used in the paper is collected in Section 6.

Formulation of the problem and assumptions
We are interested in studying diffusion processes with oblique reflection in the closure of a simply connected 2-dimensional domain D ⊂ [0, ∞) × R with a boundary ∂D that is C 1 except at a single point (which we will take to be the origin 0), where the domain has a cusp. More precisely D satisfies the following.
c) There exists a δ 0 > 0 and continuously differentiable functions ψ 1 and ψ 2 with ψ 1 ≤ ψ 2 and ψ 1 (0) = ψ 2 (0) = 0, ψ ′ The direction of reflection is assigned at all points of the boundary except the origin and is given by a unit vector field γ verifying the following condition. The mappings are Lipschitz continuous and hence the limits b) Let Γ(0) be the convex cone generated by There exists e * ∈ R 2 such that e * , γ > 0, ∀γ ∈ Γ(0).
Of course, without loss of generality, we can suppose that |e * | = 1.
Remark 2.3 Condition 2.2(b) can be reformulated as follows. In a neighborhood of the origin, we can view D as being the intersection of three C 1 domains, with unit inward normal vector at the origin, respectively, Then, letting the normal cone at the origin, N(0), be the closed, convex cone generated by {ν 1 (0), ν 2 (0), ν 3 (0)}, Condition 2.2(b) is equivalent to requiring that there exists e * ∈ N(0) such that e * , γ > 0, ∀γ ∈ Γ(0), where we can think of Γ(0) as the closed, convex cone generated by the directions of reflection at the origin for each of the three domains. In other terms, Condition 2.2(b) is the analog of the condition usually assumed in the literature for polyhedral domains (see e.g. Varadhan and Williams (1985), Taylor and Williams (1993) or Dai and Williams (1996)). Note that, in contrast, the condition that there exists e * * ∈ Γ(0) such that e * * , ν > 0, ∀ν ∈ N(0), can never be satisfied at a cusp, because ν 2 (0) = −ν 1 (0).
We seek to characterize the diffusion process with directions of reflection γ as the solution of a stochastic differential equation driven by a standard Brownian motion W : where Λ is nondecreasing, Γ 1 (0) is the convex hull of γ 1 (0) and γ 2 (0) and for x ∈ ∂D − {0} Γ 1 (x) := {γ(x)} and γ is almost surely measurable. We make the following assumptions on the coefficients. b) (σσ T )(0) is nonsingular.
} is the filtration generated by W and Z.
Definition 2.7 Given a standard Brownian motion W and X(0) ∈ D independent of W , (X, Λ) is a strong solution of (2.1) if (X, Λ) is adapted to the filtration generated by X(0) and W and the equation is satisfied.
(X, Λ, W ), defined on some probability space, is a weak solution of (2.1) if W is a standard Brownian motion, (X, Λ) is compatible with W , and the equation is satisfied.
Given an initial distribution µ ∈ P(D), weak uniqueness or uniqueness in distribution holds if for all weak solutions with P {X(0) ∈ ·} = µ, X has the same distribution on Strong uniqueness holds if for any standard Brownian motion W and weak solutions (X, Λ, W ), ( X, Λ, W ) such that X(0) = X(0) a.s. and (X, Λ, X, Λ) is compatible with W , X = X a.s.
Remark 2.8 Of course, any strong solution is a weak solution. Existence of a weak solution and strong uniqueness imply that the weak solution is a strong solution (c.f., Yamada and Watanabe (1971) and Kurtz (2014)).
For processes starting away from the tip, existence and uniqueness follows from results of Dupuis and Ishii (1993) and the fact that under our conditions, the solution never hits the tip. For processes starting at the tip, we only prove weak existence and uniqueness. The proof is based on rescaling of the process near the tip and a coupling argument.
3 Strong existence and uniqueness starting at x 0 = 0 Our first result is that, for every x 0 ∈ D − {0}, (2.1) has a unique strong solution with X(0) = x 0 , well-defined for all times. In fact, by Dupuis and Ishii (1993), for each n > 0, the solution, X, is well-defined up to We will do this by means of a modification of the Lyapunov function used in Section 2.2 of Varadhan and Williams (1985).
Theorem 3.1 Let W be a standard Brownian motion. Then, for every Proof. As anticipated above, by Dupuis and Ishii (1993) there is one and only one stochastic process X that satisfies (2.1) for t < lim n→+∞ τ n , where τ n is defined by (3.1). Therefore, we only have to prove (3.2). Define where ϑ(z) ∈ (−π, π] is the angular polar coordinate of z and . Then one can check that, if p is taken sufficiently close to 1, Therefore there exists δ > 0 such that Without loss of generality, we can assume that x 0 1 > δ. By Itô's formula, for n −1 < δ/2, Consequently, if X 1 hits δ/2, then with probability one, it hits δ before it hits 0. In particular, with probability one, X 1 never hits 0. Remark 3.2 Theorem 3.1 implies existence and uniqueness of a strong solution to (2.1) for every initial condition such that P(X(0) ∈ D − {0}) = 1 which in turn implies existence and uniqueness in distribution of a weak solution to (2.1) for every initial distribution µ such that µ(D − {0}) = 1.
4 Weak existence and uniqueness starting at x 0 = 0 In this section we prove weak existence and uniqueness for the solution of (2.1) starting at the origin. In order to prove existence (Theorem 4.1), we start with a sequence of solutions to (2.1) starting at x n ∈ D − {0}, where {x n } converges to the origin. For every n, we consider a random time change of the solution, the same time change that is used in Kurtz (1990) to construct a solution to a patchwork martingale problem from a solution to the corresponding constrained martingale problem. The time changed processes and the time changes are relatively compact, and any limit point satisfies the time changed version of (2.1) with X(0) = 0. The key point of the proof is to show that the limit time change is invertible. The process obtained is a weak solution to (2.1) defined for all times.
Weak uniqueness of the solution of (2.1) starting at the origin (Theorem 4.6 below) is the main result of this paper. Our proof takes inspiration from the one used in Taylor and Williams (1993) for reflecting Brownian motion in the nonnegative orthant. The argument of that paper, in the case when the origin is not reached, can essentially be reformulated as follows: First, it is shown that, for any solution of the SDER starting at the origin, the exit time from B δ (0), δ > 0, is finite and tends to zero as δ → 0, almost surely, and that any two solutions of the SDER, starting at the origin, that have the same exit distributions from B δ (0), for all δ > 0 sufficiently small, have the same distribution; next it is proved that, for any ξ ∈ ∂B 1 (0), ξ in the nonnegative orthant, letting X δξ be the solution of the SDER starting at δξ and τ 2δ be its exit time from B 2δ (0), P X δξ (τ 2δ )/(2δ) ∈ · is independent of δ and hence defines the transition kernel of a Markov chain on ∂B 1 (0). This Markov chain is shown to be ergodic and that in turn ensures that, for any initial distribution µ n on D ∩ ∂B δ/2 n (0) , the exit distribution of X µ n from B δ (0) converges, as n goes to infinity, to a uniquely determined distribution. Consequently, any two solutions of the SDER starting at the origin have the same exit distributions from B δ (0).
The first part of our argument is the same as in Taylor and Williams (1993), except that we find it more convenient to use the exit distribution from {x : x 1 < δ} rather than from B δ (0). We prove that, for any solution of (2.1) starting at the origin, the exit time from {x : x 1 < δ}, δ > 0, is finite and tends to zero as δ → 0, almost surely (Lemma 4.3), and that any two solutions of (2.1) starting at the origin that have the same exit distributions from {x : x 1 < δ}, for all δ > 0 sufficiently small, have the same distribution (Lemma 4.5).
The second part of our argument consists in showing that, for {δ n }, a sequence of positive numbers decreasing to zero, any two solutions X, X of (2.1) starting at the origin satisfy where τ X δn and τ X δn are the corresponding exit times from [0, δ n ). This fact cannot be proved by the arguments used in Taylor and Williams (1993). Instead, it is achieved by the rescaling result of Section 4.2, together with a coupling argument based on Lindvall and Rogers (1986).

Existence
Theorem 4.1 There exists a weak solution to (2.1) starting at x 0 = 0.
Then H n 0 , H n 1 are nonnegative and nondecreasing, where M n is a continuous, square integrable martingale with With reference to Theorem 5.4 in Kurtz and Protter (1991), let Since U n , H n 0 , and H n 1 are all Lipschitz with Lipschitz constants bounded by 1, {(Y n , U n , M n , H n 0 , H n 1 )} is relatively compact in distribution in the appropriate space of continuous function. Taking a convergent subsequence with limit (Y, U, M, H 0 , H 1 ), Y satisfies where M(t) = W (H 0 (t)) for a standard Brownian motion W . Since |U n (r) − U n (t)| ≤ |H n 1 (r) − H n 1 (r)|, the same inequality holds for U and H 1 and hence It remains only to characterize η.
Note that the second term on the right of (4.9) converges to zero. By applying the same time-change argument as in Theorem 4.1, we see that X n is relatively compact and (q nX n 1 (s) + δ n+1 , q nX n 2 (s)) → 0.
(4.13) Lemma 4.3 For δ sufficiently small, for any solution, X, of (2.1) starting at the origin Proof. Here X is fixed, so we will omit the superscript X. Let e * be the vector in Condition 2.2(b) and Then Observe that and hence, for δ sufficiently small, Therefore, for δ sufficiently small, which yields the assertion by taking the limit as t goes to infinity.
Remark 4.4 By looking at the proof of Lemma 4.3, we see that we have proved, more generally, that, for every x 0 ∈ D with x 0 1 < δ, for every solution X of (2.1) starting at x 0 , Lemma 4.5 Suppose any two weak solutions, X, X, of (2.1) starting at the origin satisfy L(X(τ X δ )) = L( X(τ X δ )), (4.14) for all δ sufficiently small (recall that, by Lemma 4.3, τ X δ and τ X δ are almost surely finite). Then the solution of (2.1) starting at the origin is unique in distribution.
Theorem 4.6 The solution of (2.1) starting at x 0 = 0 is unique in distribution.

The Feller property
We conclude with the observation that the family of distributions P x x∈D , where P x is the distribution of the unique weak solution of (2.1) starting at x, enjoys the Feller property.
Proposition 4.7 Let X x be the unique weak solution of (2.1) starting at x. Then the mapping x ∈ D → X x is continuous in distribution.
Proof. The proof is exactly the same as that of Theorem 4.1. In fact, once it is known that the weak solution of (2.1) starting at the origin is unique, the proof of Lemma 4.1 amounts to showing that X x is continuous in distribution at the origin.
Since the first exit time from (−∞, 1) × R is a continuous functional on a set of paths that has probability one under the distribution ofX, by the continuous mapping theorem we may assume thatX n 2 (τ n ) converges in distribution toX 2 (τ 1 ). Then and the assertion follows by (5.1) and by the arbitrariness of {x n 2 }. The following lemma, which uses the coupling of Lindvall and Rogers (1986), may be of independent interest. Lemma 5.3 Let β : R d → R d and ς : R d → R d × R d be Lipschitz continuous and bounded and let ςς T be uniformly positive definite. Suppose that
(ii) Let E 1 and E 2 be complete separable metric spaces and P be a transition function from E 1 to E 2 , and let P µ(dy) = E 1 P (x, dy)µ(dx), µ ∈ P(E 1 ).

Notation
·, · denotes the scalar product of two vectors. For any matrix M (or vector v), M T (v T ) denotes its transpose. trM denotes the trace of a matrix. For vectors v 1 , v 2 , ..., v k ∈ R h , C(v i , i = 1, ..., k) denotes the closed convex cone generated by v 1 , v 2 , ..., v k .
I E is the indicator function of a set E. For E ⊆ R h d(x, E) is the distance of a point x from E. B r (x) ⊆ R h is a ball of radius r centered at x.
For f : R h → R m with first order partial derivatives, Df denotes the Jacobian matrix of f . For f : R h → R with second order partial derivatives D 2 f denotes the Hessian matrix.
| · | denotes indifferently the absolute value of a number, the norm of a vector or of a matrix, while · denotes the supremum norm of a bounded, reaal valued function.
For an open set E ⊆ R h , C i (E) denotes the set of real valued functions defined on E with continuous partial derivatives up to the order i. L(ξ) denotes the law (distribution) of a random variable ξ. · T V denotes the total variation norm of a finite, signed measure.
Throughout the paper c and C denote positive constants depending only on the data of the problem. When necessary, they are indexed c 0 , c 1 , ..., C 0 , C 1 , ... and the dependence on the data or other parameters is explicitely pointed out.