Bessel-like SPDEs, Sorbonne Université (joint work with Henri Elad-Altman)
Squared Bessel processes Let δ, y, and (B t ) t a BM. By Yamada-Watanabe s Theorem, there exists a unique (strong) solution (Y t ) t of Y t = y + t and moreover Y so that Y = Y. 2 Y s db s + δ t, The transition semigroup is explicitly known and contains some Bessel functions and (Y t ) t is called a Squared Bessel Process, see Pitman-Yor. We define X t := Y t. What equation does X satisfy? The function y y is not smooth and the Itô formula can not be applied (too) naively.
Bessel processes For δ > we have where x := y. X t = x + δ 2 t More precisely, by the Itô-Tanaka formula X t = x + δ 2 t X s ds + B t X s ds + B t + 2 L t where (L a t ) t,a is defined by the occupation times formula t ϕ(x s ) ds = ϕ(a) L a t da for all ϕ C b (R). Here L and X is a semimartingale.
A very interesting SDE for δ > the drift x δ 2x is dissipative (i.e. decreasing) on R +, the SDE has pathwise uniqueness and the solution is Strong Feller X is known as the Bessel process. As δ the solution converges to the reflecting BM X t = X + L t + B t where L is continuous monotone non-decreasing, L =, X is continuous non-negative, and X t dl t =. For δ 2 a.s. X t > for all t >. For δ ], 2[ we have a.s. X t for some t >, but still Lt =. We can define diffusion local times (l a t ) t,a by t ϕ(x s ) ds = ϕ(a) l a t a δ da. Then (l t ) t is the inverse of a ( δ 2 )-stable subordinator.
δ < It turns out that in this situation the process X t := Y t solves this SDE X t = X + δ 2 l a t l t a a δ da + B t, where (l a t ) a,t is the family of diffusion local times. Formally this is equal to X t = X + δ t ds + l t + B t, 2 X s This SDE has a very exotic drift: an increasing singular non-linearity (the opposite of dissipative) and a reflection at but multiplied by an infinite constant. Indeed X is not a semimartingale and L t = +. The two infinite terms compensate each other in a renormalisation phenomenon. To my knowledge, there is no pathwise uniqueness result for such SDE. One can prove the Strong Feller property (see Henri s recent paper).
Bessel-like SPDEs with δ 3 u t = 2 u 2 x 2 + ξ + η where u : R + [, ] R +, ξ is a space-time white noise, η is a measure on R + [, ] s.t. u dη =. R + [,] This is the Nualart-Pardoux equation, whose invariant measure is the 3-Bessel bridge. The equation corresponding to the δ-bessel bridge for δ > 3 is u t = 2 u (δ )(δ 3) + 2 x2 8u 3 + ξ
Bessel-like SPDEs with δ 3 u t = 2 u 2 x 2 + ξ + η where u : R + [, ] R +, ξ is a space-time white noise, η is a measure on R + [, ] s.t. u dη =. R + [,] This is the Nualart-Pardoux equation, whose invariant measure is the 3-Bessel bridge. The equation corresponding to the δ-bessel bridge for δ > 3 is u t = 2 u (δ )(δ 3) + 2 x2 8u 3 + ξ What about δ < 3? This question has been open since 2.
Bessel-like SPDEs with δ < 3 We concentrate on the drift κ(δ) t u 3 (s, x) We introduce diffusion local times t and we can write κ(δ) t ϕ(u(s, x)) ds = (δ )(δ 3) ds, κ(δ) :=. 8 ϕ(a) l a t,x a δ da u 3 ds = κ(δ) (s, x) a 3 la t,x a δ da, which however diverges for δ 3 if l t,x >.
Bessel-like SPDEs with < δ < 3 Then, in analogy with Bessel processes, we write a renormalised version of the drift κ(δ) which may work as long as δ > 2. a 3 (la t,x l t,x) a δ da,
Bessel-like SPDEs with < δ < 3 Then, in analogy with Bessel processes, we write a renormalised version of the drift κ(δ) a 3 (la t,x l t,x) a δ da, which may work as long as δ > 2. For δ ], 2] one expects ( κ(δ) a 3 l a t,x l t,x a ) a la t,x a δ da, a=
Bessel-like SPDEs with < δ < 3 Then, in analogy with Bessel processes, we write a renormalised version of the drift κ(δ) a 3 (la t,x l t,x) a δ da, which may work as long as δ > 2. For δ ], 2] one expects ( κ(δ) a 3 l a t,x l t,x a ) a la t,x a δ da, a= however it turns out that a la t,x = a= so that the same expression is valid for δ ], 3[.
Bessel-like SPDEs with δ = The most important and interesting case is δ =, which, together with δ = 3, is a critical case. As δ, the previous expression can be seen to converge to 2 8 a 2 la t,x. a= We write therefore the SPDE for δ = : u t = 2 u 2 x 2 8 t 2 a 2 la t,x + ξ. a= Motivated by scaling limits of dynamical critical pinning models.
Bessel-like SPDEs with < δ < For δ ], [ one expects κ(δ) a 3 (l at,x l t,x a2 2 2 a 2 la t,x with a Taylor expansion of order 2 of a l a t,x. ) a δ da a=
A general formula For α > we define the measure on R + µ α (dx) := xα Γ(α) (x>) dx. For α we define the Schwartz distribution on R + if α N, and otherwise. µ α (ϕ) := Γ(α) µ α (ϕ) := ( ) α ϕ ( α) () x α ϕ(x) i α x i i! ϕ(i) () dx
A general formula Then we can write the above family of SPDEs in a unified way for all δ > u t = 2 u 2 x 2 + Γ(δ 3) κ(δ) µ δ 3(l ( ) t,x) + ξ
Results Most of the above is conjectural. We do have integration by parts formulae on the law of δ-bessel processes for δ < 3, see Henri s talk, which give the form of the equation. By Dirichlet forms methods, at least in the cases δ =, 2 we can construct (stationary) solutions to the SPDE. Major open questions pathwise uniqueness???? local times for SPDEs??? the Strong Feller property?? (Henri proved it for Bessel processes uniformly in δ) the associated Dirichlet forms? (for δ, 2)
Pathwise uniqueness Recall that for Bessel processes and δ <, we have pathwise uniqueness by setting Y t := X 2 t and applying the Itô formula in order to compute the SDE solved by Y, the Yamada-Watanabe theorem, since the diffusion coefficient of this SDE is only 2 -Hölder.
Pathwise uniqueness Recall that for Bessel processes and δ <, we have pathwise uniqueness by setting Y t := X 2 t and applying the Itô formula in order to compute the SDE solved by Y, the Yamada-Watanabe theorem, since the diffusion coefficient of this SDE is only 2 -Hölder. For space-time white noise driven SPDEs, Itô calculus is notoriously difficult. Carlo Bellingeri is investigating this with regularity structures.
Pathwise uniqueness Since the drift is proportional to u 3, it seems reasonable to set v := u 4 and study pathwise uniqueness for v. The equation is v t = 2 v 2 x 2 3 2 : ( ) 2 v (δ )(δ 3) : + + 4v 3 4 ξ x 2 This equation is impossible to solve today, but the exponent 3 4 has been shown by Mueller-Mytnik-Perkins to be critical for pathwise uniqueness, as for Yamada-Watanabe in one-dimensional diffusions.
Hitting Theorem (Dalang, Mueller, Z. (26)) Let δ 3 and k N such that k > 4 δ 2. Then P( t >, x,..., x k ], [: u(t, x i ) = ) =.
Hitting Theorem (Dalang, Mueller, Z. (26)) Let δ 3 and k N such that k > 4 δ 2. Then P( t >, x,..., x k ], [: u(t, x i ) = ) =. Now we can conjecture that the same formula holds for all δ 2!!!
Hitting Theorem (Dalang, Mueller, Z. (26)) Let δ 3 and k N such that k > 4 δ 2. Then P( t >, x,..., x k ], [: u(t, x i ) = ) =. Now we can conjecture that the same formula holds for all δ 2!!! Correctly, δ = 2 is the critical case for hitting infinitely many times in space.
Hitting Theorem (Dalang, Mueller, Z. (26)) Let δ 3 and k N such that k > 4 δ 2. Then P( t >, x,..., x k ], [: u(t, x i ) = ) =. Now we can conjecture that the same formula holds for all δ 2!!! Correctly, δ = 2 is the critical case for hitting infinitely many times in space. What about δ < 2, in particular δ =?